BigIdeasinMathematicsforFutureElementaryTeachers

BigIdeasinDataAnalysisandProbability

JohnBeam,JasonBelnap,EricKuennen,AmyParrott,CarolE.Seaman,andJenniferSzydlik

(UpdatedSummer2016)

1

ThisworkislicensedundertheCreativeCommonsAttribution-NonCommercial-NoDerivatives4.0InternationalLicense.Toviewacopyofthislicense,visithttp://creativecommons.org/licenses/by-nc-nd/4.0/orsendalettertoCreativeCommons,POBox1866,MountainView,CA94042,USA.

2

DearFutureTeacher,Wewrotethisbooktohelpyoutoseethestructurethatunderlieselementarymathematics,togiveyouexperiencesreallydoingmathematics,andtoshowyouhowchildrenthinkandlearn.Wefullyintendthiscoursetotransformyourrelationshipwithmath.Asteachersoffutureelementaryteachers,wecreatedorgatheredtheactivitiesforthistext,andthenwetriedthemoutwithourownstudentsandmodifiedthembasedontheirsuggestionsandinsights.Weknowthatsomeoftheproblemsaretough–youwillgetstucksometimes.Pleasedon’tletthatdiscourageyou.There’smuchvalueinwrestlingwithanidea. Allourbest,

John,Jason,Eric,Amy,Carol&Jen

Hey!Readthis.Itwillhelpyouunderstandthebook. Theonlywaytolearnmathematicsistodomathematics. PaulHalmosThisbookwaswrittentopreparefutureelementaryteachersforthemathematicalworkofteaching.Thefocusofthismoduleisdataanalysis–andthisdomainencompassesbothstatisticalandprobabilisticideas.Thistextisnotjustintendedtohelpyourelearnyourelementarymathematics;itisaboutteachingyoutothinklikeamathematiciananditisabouthelpingyoutothinklikeamathematicsteacher.TheNationalCouncilofTeachersofMathematics(NCTM,2000)writes:

Teachersneedseveraldifferentkindsofmathematicalknowledge–knowledgeaboutthewholedomain;deep,flexibleknowledgeaboutcurriculumgoalsandabouttheimportantideasthatarecentraltotheirgradelevel;knowledgeaboutthechallengesstudentsarelikelytoencounterinlearningtheseideas;knowledgeabouthowtheideascanberepresentedtoteachthemeffectively;andknowledgeabouthowstudents’understandingcanbeassessed(p.17).

Wearegoingtoworktowardthesegoals.(Readthemagain.Thisisatallorder.Inwhichareasdoyouneedthemostwork?)Throughoutthisbook,wewillaskyoutoconsiderquestionsthatmayariseinyourelementaryclassroom.

Whenyourollapairofdice,isaoneandathreeadifferentoutcomethanathreeandaone?Ifthechanceofrainis40%onSaturdayand20%onSunday,whatisthelikelihoodthatitwillrainthisweekend?HowmanypeopledoyouneedtosampleinordertoreasonablyfindouthowmanyhoursofTVsecondgraderswatchinaweek?Whatdoesitmeantosaythattheprobabilityofaheadwhenflippingacoinis½?

Asmathematicianswewillalsoconveytoyouthebeautyofoursubject.Weviewmathematicsasthestudyofpatternsandstructures.Wewanttoshowyouhowtoreasonlikeamathematician–andwewantyoutoshowthistoyourstudentstoo.Thiswayofreasoningisjustasimportantasanycontentyouteach.Whenyoustandbeforeyourclass,youarearepresentativeofthemathematicalcommunity.Wewillhelpyoutobeagoodone.Noonecandothisthinkingforyou.Mathematicsisn’tasubjectyoucanmemorize;itisaboutwaysofthinkingandknowing.Youneedtodoexamples,gatherdata,lookforpatterns,experiment,drawpictures,think,tryagain,makearguments,andthinksomemore.Thebigideasofdataanalysisandprobabilityarenotalwayseasy–buttheyarefundamentally

4

importantforyourstudentstounderstandandsotheyarefundamentallyimportantforyoutounderstand.EachsectionofthisbookbeginswithaClassActivity.Theactivityisdesignedforsmall-groupworkinclass.Someactivitiesmaytakeyourclassaslittleas30minutestocompleteanddiscuss.Othersmaytakeyoutwoormoreclassperiods.TheReadandStudy,ConnectionstotheElementaryGrades,andHomeworksectionsarepresentedwithinthecontextoftheactivityideas.Nosolutionsareprovidedtoactivitiesorhomeworkproblems–youwillhavetosolvethemyourselvesanddiscussthemwithyourclassThemathematicscontentinthisbookpreparesyoutoteachtheCommonCoreStateStandardsforMathematicsforgradesK-8.Thesearethestandardsthatyouwilllikelyfollowwhenyouareanelementaryteacher,sowewillhighlightaspectsofthemthroughoutthetext.Inorderforyoutoseehowthemathematicalworkyouaredoingappearsintheelementarygrades,wehavemadeexplicitconnectionstoBridgesinMathematicsfromTheMathLearningCenter.ThisistheonlineelementarygradesmathematicscurriculumadoptedbytheOshkoshAreaSchoolDistrict.Youwilloftenbeaskedtogotothesitebridges.mathlearningcenter.orgtoreadordoproblems.Yourinstructorwillprovideyouwithacodesothatyoucanaccessthesematerials.OnthisandthenextpageyouwillfindtheCommonCoreStateStandards(CCSS)forschoolmathematicsthatarerelatedtothetopicsinthisbook.TherearealsoStandardsforeachgraderelatedtonumberandoperation,geometry,andmeasurement–you’llstudythoseinothercourses.ReadthebelowStandardscarefully;theywillgiveyouanoverviewoftheelementarygradescurriculumindataanalysisandprobability.

5

Common Core State Standards for Measurement & Data Grade One:

• Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Grade Two: • Generate measurement data by measuring lengths of several objects to the nearest

whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

• Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Grade Three: • Draw a scaled picture graph and a scaled bar graph to represent a data set with several

categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

• Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.

Grade Four: • Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4,

1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

Grade Five: • Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4,

1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

6

Common Core State Standards for Statistics & Probability

Grade Six: Develop understanding of statistical variability.

• Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

• Understand that a set of data collected to answer a statistical question has a distribution

which can be described by its center, spread, and overall shape.

• Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Summarize and describe distributions.

• Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

• Summarize numerical data sets in relation to their context, such as by: o Reporting the number of observations. o Describing the nature of the attribute under investigation, including how it was

measured and its units of measurement. o Giving quantitative measures of center (median and/or mean) and variability

(interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

o Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.\

Grade Seven: Use random sampling to draw inferences about a population.

• Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

• Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

7

Draw informal comparative inferences about two populations. • Informally assess the degree of visual overlap of two numerical data distributions with

similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

• Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

Investigate chance processes and develop, use, and evaluate probability models. • Understand that the probability of a chance event is a number between 0 and 1 that

expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Grade Eight: Investigate patterns of association in bivariate data. • Construct and interpret scatter plots for bivariate measurement data to investigate patterns

of association between two quantities. • Describe patterns such as clustering, outliers, positive or negative association, linear

association, and nonlinear association. • Know that straight lines are widely used to model relationships between two quantitative

variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

• Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

• Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.

• Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

8

BigIdeasinDataAnalysisandProbabilityTableofContents

Tobeateacherrequiresextensiveandhighlyorganizedbodiesofknowledge.Shulman,1985,p.47

CHAPTER1:COUNTINGClassActivity1:Photographs………...…………………………………..…………….......……………….....p.13 CountingOrderedGroups TheMultiplicationPrincipleClassActivity2:Committees…….…………………………………..………...………………..............…...p.21 CountingUnorderedGroupsClassActivity3:PatternsinCommitteeNumbers…………………………..…..…..……………..…...p.25 ExplainingPatternsinPascal’sTriangle ClassActivity4:FromPhotostoCommittees…….……………….………..…….………………..…..….p.27 DealingwithDuplicatesCHAPTER2:PROBABILITYClassActivity5:DiceSums…………………………..………….……………....………………………………….p.32 TheLanguageofProbability ExperimentalandTheoreticalAnalysesClassActivity6:RatMazes….…………………………………………..……………………………………....….p.39 TreeDiagrams

CompoundEvents

ClassActivity7:TheMaternityWard…………………………………..…………………............……..…p.44 ThinkingaboutSampleSize

TheLawofLargeNumbersClassActivity8:ProbabilityChallenge!…………………………………..………………..…………….…...p.48 MisconceptionsaboutChance

9

CHAPTER3:STATISTICSandDEALINGWITHDATAClassActivity9:StudentWeightandRectangles……….....………….……….........................…p.54

TheLanguageofSamplingRandomSamples

ClassActivity10:SnowRemoval…...……..………………………….………..………………….………..….p.64 AnalyzingaSurvey StatisticallySpeakingClassActivity11:ClassSurvey……….………………….…….……….…….…………………………..……..…p.70 NumericalandCategoricalData

SurveyDesign ClassActivity12:NameGamesandIntheBalance…………..…………………..…………..…...……p.71 Mean,MedianandMode Outliers MeanasBalancingPoint MeanasRedistributionClassActivity13:MeasuringSpread……………...………...…………...….............................…..…p.81 Range,IQRandMeanAbsoluteDeviation ACriterionforFindingSuspectedOutliers

BoxplotsClassActivity14:MatchingGame..……………..………………………...…………....…………….…….....p.88 Distributions,SkewnessandSymmetry

PieCharts,andBarGraphsClassActivity15:OldFaithfulEruptions…………………...………………………..……………....……...p.99 FormingQuestionsfromData MisleadingDisplays CHAPTER4:RATIOS,RATES,andPROPORTIONS ClassActivity16:Let’sBeRational…………….……....………………….………….…………………...…p.104 Ratios BarDiagrams

10

ClassActivity17:TheWatermelonProblem……....………………….………..................……..…p.110 Percents PercentChange TheLanguageofPercentsClassActivity18:HowDoYouRate?……....………………….…….....................................……p.117 Rates UnitConversions ClassActivity19:Goldfish……....………………….……………………….………………..……………..……p.123 Capture/Recapture Proportions CHAPTER5:RELATIONSHIPSAMONGSETSOFDATAClassActivity20:ShoeSizeandHeight…………....…...……………….............................……….p.129 Scatterplots OutliersonaScatterplot ClassActivity21:LotsaboutLines…………………………………………………….…………………..……p.134 SlopeandIntercepts RecognizingLinearDataClassActivity22:ManateeData……………………….…………………………………………………….....p.139 TheIdeaofLinearRegression InterpolationandExtrapolation ClassActivity23:Correlation…………………….………….…...…………………………………………......p.146

TheIdeaofCorrelationCausationCautions

ClassActivity24:CaffeineandHeartRate…………………………………………………………..…...…p.152 TheLanguageofClinicalStudies ConfoundingVariablesClassactivity25:MusicandMath……..……………….………...……………………………………..…...p.156 ClinicalStudyAnalysisClassActivity26:ClassSurveyRevisited………..…………………..………………………………..…..…p.157

11

CHAPTER6:PROBABILITYREVISITED

ClassActivity27:CasinoRoyale..……………………………………………………………………..…………p.159 ExpectedValue ClassActivity28:BadBulbBinomial….....……………………………………………………………….....p.164 BinomialProbabilitiesClassActivity29:TheProblemofPoints……………………………………….…………………………...p.169 ABriefHistoryofProbabilityAPPENDICESProbabilitySimulationProject………………………………………………………………….………..p.173References……………………….……………………………..……………………………………………………...p.177Glossary.………….…………………………………………………………………………………………….…….…p.178

12

ChapterOne

Counting

13

ClassActivity1:Photographs

Tothecomplaint,'Therearenopeopleinthesephotographs,'Irespond,'Therearealwaystwopeople:thephotographerandtheviewer.'

AnselAdams

1) Howmanydifferentwayscanyoulineupfivepeopleforaphotographifallfivewillbeinthepicture?

2) Howmanydifferentwayscanyoulineup10peopleforaphoto?npeople?Explain

whyyourideamakessense.

3) Ifthereareatotaloffivepeople,howmanydifferentwayscanyoulinethemup3ata

timeforaphoto?Twoatatime?Oneatatime?

4) Iftherearentotalpeople,howmanydifferentwayscanyoulineupkofthemata

timeforapicture?(Youmayneedtocollectmoredatatohelpyouthinkaboutthis.)

14

ReadandStudy

Noteverythingthatcanbecountedcounts,andnoteverythingthatcountscanbecounted.

AlbertEinsteinItistimetolearnto‘count.’Basicallytherearetwotypesofcountingsituations.Thefirstislikethephotographsproblemwheredifferentorderingsofobjectsarecountedasdifferent.Inthesecondtypeofsituation,differentgroupsarecountedasdifferentbutorderingwithinagroupdoesn’tmatter.We’llexplorethatinthenextclassactivity.Bothtypesofcountingarebasedonafundamentalprinciple.Butbeforewetellyouaboutthatprinciple,spendabout10minutessolvingthefollowingproblemusingasmanydifferentrepresentations(pictures,charts,etc)aspossible.

AtDan’sIceCreamParloryoucanpickasugarconeoraregularcone.Thenyoucanchooseoneofthreesizes:large(3scoops),medium(2scoops),orsmall(1scoop).Finally,youmayselectanyofthefiveflavors:vanilla,chocolate,strawberry,chocolatechip,ormint.

Ifyoucannotmixflavors(i.e.,youcan’thaveonescoopofchocolateandonescoopofvanillaonyourcone),howmanydifferenticecreamconesarepossible?

We’llleaveyousomeblankspacetoworkontheproblem.Reallydoitbeforeyouturnthepage.Didyounoticetheitalicizedquestionsandrequestsinthepreviousparagraphs?Weaskthosetypesofquestionsthroughoutthetexttohelpyoutofocusonimportantideas.Also,theyarepartofyourhomework,sobesurethatyouanswerthemasyouread.

15

Okay,therearelotsofwaystorepresentasolutiontotheicecreamproblem.

1) Asadecisiontree:Chooseacone Chooseasize Chooseaflavor

Soweseethatthereare2setsof3setsof5=2×3×5=30differenticecreamcones.

mint

mintstrawberry

strawberry

vanilla

vanilla

chocolate

chocolate

chocolate chip

chocolate chip

mint

strawberryvanilla

chocolate

chocolate chip

L

M

S

mint

mintstrawberry

strawberry

vanilla

vanilla

chocolate

chocolate

chocolate chip

chocolate chip

mint

strawberryvanilla

chocolate

chocolate chip

L

M

S

16

2) Asatableofsomekind:

Small Medium LargeSugar 5 5 5Regular 5 5 5

Soweseethereare5+5+5+5+5+5=30differenticecreamcones. Wecouldseeitas2×3=6boxes,eachhaving5flavorsso6×5=30.

3) Inwords:Youcanpickanyoftwoconetypesandanyoffiveice-creamflavors,sothat’s10differenticecreamcones.Noweachofthose10comesin3sizes,sothatmakesatotalof30possibilities.

Noticethatineachcaseamultiplicativeprocessemerges.Thatbringsustothefundamentalprincipleofcounting:themultiplicationprinciple.Itsaysthis:Supposeyouwanttofindthetotalnumberwaysyoucanaccomplishsometask,andthattaskcanbebrokendownintoasequenceofsubtasks.IfthereareAdifferentwaystodothefirstsubtask,andforeachofthesewaysthereareBdifferentwaystothesecondsubtask,thenthetotalnumberofwaystodobothsubtasksinsuccessionisA×B.Forourproblem,inordertomakeanicecreamcone,wehadtoselectaconetype,asize,andaflavoroficecream.Sowecanthinktheentireprocessasasequenceofthreesubtasks.Thereare2differentwaystochooseaconetype,3differentwaystochooseasizeand5differentwaystochooseaflavor,sothatis2×3×5=30differentwaystochooseaconeandasizeandaflavor.Thedecisiontreeonthepreviouspageillustrateswhyitmakessensetomultiply2,3,and5tofindthetotalnumberofpossibleicecreamcones.Sincethereare3sizesand5flavors,thereare3groupsof5,or15waystochooseasizeandaflavor.Thenwhenwechooseoneofthe2kindsofcones,weseethereare2groupsof15.Inotherwords,thediagramillustratesthatthereare2groupsof3groupsof5,foratotalof2×3×5=30differentoptions.Youfoundthatthisprinciplewaspartofthestructureofyoursolutiontothephotographproblems.Ifyouthinkofthephotographerashavingthreechairsshemustfill,youcanthinkofhertaskasasequenceofthreesubtasks:fillthefirstchair,thenthesecondchair,andthenthethird.

17

Ifthereareatotaloffivepeopletoseat,thenshehas5choicesforwhositsinthefirstchair.Foreachofthese5choicesofwhositsinthefirstchair,shecanthenmake4choicesofwhositsinthenextchair.Thisgives20differentwaystoputpeopleinthefirsttwochairs.Thenforeachofthese20ways,therearethreeremainingchoicesforwhositsinthelastchair.Sotherewillbe5×4×3=60differentposes.Explainwhyitmakessensetomultiplythesenumberstogether.Listoutall60differentposesinawaythatshowsthatthereare5groupsof4groupsof3poses.ConnectionstotheElementaryGrades

Amajorresponsibilityofteachersistocreatealearningenvironmentinwhichstudents’useofmultiplerepresentationsisencouraged.

NCTMPrinciplesandStandardsSituationsinvolvingrepresentingandmodelingmultiplicativestructuresappearthroughouttheelementarycurriculum.Wewillremindyouofsomeofthosemodelsnow.Asyouread,comparethesemodelstotherepresentationsweusedwhensolvingtheicecreamproblemandthephotographsproblem.Groupingmodelofmultiplication:Thismodelhelpschildrentothinkoftheproductofsay4×3as4groupswith3ineach: *** *** *** ***

Hereisaproblemthatmighthelpchildrenimagineagroupingmodelfor43×22:Candycomesinbagsof22pieces.Ifyoubuy43bagsofcandy,howmanypiecesisthat?

RepeatedAdditionsModelofMultiplication:Similartoagroupingmodel,thisisamodelthathelpschildrentothinkofmultiplication(say4×3)as3+3+3+3.

Hereisaproblemthatmighthelpchildrenimaginearepeatedadditionsmodelfor43×22:Jenniearnedanallowanceof22centseachday.Howmuchwillshehaveafter43days?

18

ArrayModelofMultiplication:Thisisamodelthatleadschildrentothinkoftheanswertoamultiplicationproblemasarectangulargrid.Hereis4×3asanarray: * * * * * * * * * * * *

Hereisaproblemthatmighthelpchildrenimagineanarraymodelfor43×22:Iplanted43rowsofsunflowersandeachrowhas22sunflowers.HowmanysunflowersdidIplant?

AreaModelofMultiplication:Multiplication(4×3)canalsobemodeledastheareaofa4by3rectangle(notethatthisisreallyjustaspecialtypeofarray):

Hereisaproblemthatmighthelpchildrenimagineanareamodelfor43×22:Arectangularblackboardmeasured43inchesby22inches.Howmanysquareinchesofareaisthat?

TreeModelofMultiplication:Finally4×3couldberepresentedwithatreeshowingfoursetsofthreebranches:

Hereisaproblemthatmighthelpachildpictureatreemodelfortheproblem:4×3

(Notethatatreeiscumbersomeifthenumbersgettoobigsowe’llstickwiththesmallexamplehere.)Addiehad4choicesforashirttowear.Foreachshirt,shecouldchooseanyofthreepairsofpants.Howmanydifferentoutfitscouldshemakealltogether?

Themultiplicationprincipleofcountingisanexplicittopicforstudentsingrades4or5.

19

Homework

I'vemissedover9,000shotsinmycareer.I'velostalmost300games.I'vefailedoverandoverandoveragaininmylife.AndthatiswhyIsucceed.

MichaelJordan

1) DoalltheitalicizedthingsintheReadandStudysection.2) Inelementaryschool,childrenareencouragedtosolvemultiplicationruleproblems

bydrawingorphysicallymodelingallthepossibilities.Hereisaproblemforthirdgrade.Solveitthewayachildmightbydrawingallthepossibilities.Youhavethreecleanshirtsand2pairsofcleanpants.Howmanydifferentoutfitscanyoumake?

3) Alicenseplatesstartswiththreelettersandendsinthreedigits.

a) Howmanyplatescanbemadethatstartwithan“A”?b) Howmanycontainexactlytwo“A”s?c) Howmanystartwithavowelandhavenodigitorletterrepeated?

4) Supposeyouareallowedtoselectdifferentflavoredscoopsoficecreamonaconeat

Dan’sIceCreamParlor.Nowhowmanydifferenticecreamconesarepossible?Carefullyexplainyourthinkingandmakeclearanyassumptionsyouaremaking.

5) Youneedtoselectapresidentandavicepresidentforyour12-memberclub.How

manydifferentwayscanthatbedone?Explainhowthisproblemisrelatedtothephotographproblem.

6) Youareorderingamealfromfancyrestaurantthatgivesyoutwochoicesof

appetizers,threechoicesofmaincourses,and6choicesofdessert.Explainwhyitmakessensetomultiply2times3times6tofigureouthowmanydifferentmealsyoucanorder.Useatreediagramandalsoanarraymodeltoillustratethemultiplication.

7) Listallthedifferentwords(theydon’tneedtomeananything)canbemadeusingallthelettersinHAT.

8) HowmanydifferentwordscanbemadeusingallthelettersinBLANKET?Explainwhyyouranswermakessense.

9) Youareorderinga2-toppingpizzafromSam’sPizza.Samallowsyoutochooseanyof

5differenttoppings.Howmanydifferent2-toppingpizzasarepossible?Inwhatwaysdoesthissituationdifferfromthatofthephotographsituationfromtheclassactivity?Explain.

20

ClassActivity2:Committees

Togetsomethingdone,acommitteeshouldconsistofnomorethanthree(people),twoofwhomareabsent.

RobertCopeland

1) Listallthedifferentfour-membercommitteescanbemadewithagroupoffourpeople.(Note:acommitteeoffourwithAnn,Bob,CatandDonisthesamecommitteeasBob,Cat,AnnandDon.Changingtheorderofmembersdoesnotchangethecommittee.)

2) Listallthedifferentthree-membercommitteescanbemadewithagroupoffourpeople.

3) Listallthedifferenttwo-membercommitteescanbemadewithagroupoffourpeople.

4) Listallthedifferentone-membercommitteescanbemadewithagroupoffourpeople.

5) Listallthedifferentzero-membercommitteescanbemadewithagroupoffourpeople.

6) Supposeanotherperson(Ernie)joinsyourgroupandnowyouhavefivetotalpeople.

Howmanycommitteescanbemadecontainingzeromembers?1member?2members?3members?4members?5members?

21

ClassActivity2Continued:CommitteesNowwe’llorganizethisdatainabigtable.nisgoingtorepresenttototalnumberofpeople.kwillrepresentthecommitteesize.Thebodyofthetablewillshowthenumberofpossiblek-sizedcommitteesthatcanbemadewithntotalpeople.So,forexample,ifyouhave4totalpeopleandwanttomakecommitteesofsize2,therewillbe6possiblecommittees.Seehowmuchofthistableyoucancomplete. k(Numberofpeopleoneachcommittee)

n(totalpeople)

0 1 2 3 4 5 6

1

2

3

4

6

5

6

7

22

ReadandStudy Giveamanafishandheeatsforaday. Teachamantofishandheeatsforalifetime.

AuthorUnknownWehatetogivemuchawayatthispoint–soyou’llhavetobearwithuswhileweexplainwhywewon’tjusttellyouhowtocomputecommitteenumbersfromaformula.Bytheway,ifyourecallhowtodothis,butdon’tunderstandwhytheformulamakessense,thendon’tuseit.Anddon’tshowanyoneinyourclasshowtouseiteither.Rememberthatmathisnotaboutknowingaformula.Itisaboutmakingaformula,anditisaboutunderstandingwhyitmakessense.Ifyougetaformulaasa‘gift’theninsomesenseyoulosetheopportunitytoownit,toreallydosomemathematics,andtounderstandit.Wewon’ttakethatfromyou,andwehopethatyouwon’ttakethatopportunityawayfromyourfuturestudentseither.YoumayrecognizethetablethatyougeneratedinthelastclassactivityasaformofPascal’sTriangle.TousthisiswhatPascal’sTriangleis:anorganizedchartofthecommitteenumbers.AndwecanexplainmanyofthefeaturesinPascal’sTriangleintermsofthinkingaboutcommittees.Forexample,whydoesitmakesensethatwewouldseesymmetryineachlineofthetriangle? 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 . . . . . . . . . . . . . . . . . . . . . . . .Let’slookataspecificcaseofthatsymmetry:weseethatthefifthrowgoes 1 5 10 10 5 1Whatdoesthisrowrepresentintermsofcommittees?Stopandthinkaboutthis.Thenexplain.

23

Well,ifyouhave5totalpeople:A,B,C,DandE,thereis1emptycommittee(Doesthisbotheryou,bytheway?Inmathematicswedefineacommitteeintermsofitsmembers.Soanytwocommitteeswiththesamemembershipare,infact,thesamecommittee.Sothereisonlyonecommitteethatcontainsnomembersatall-andanyothercommitteewithnomembersisthatsamecommittee:theemptycommittee.)Thereare5one-personcommittees,10two-membercommittees,etc.Soinordertoexplainthesymmetry,weneedtoexplainwhyitmakessensethatwith5totalpeoplethereisthesamenumberofcommitteesofsizezeroasthereiscommitteesofsizefive,andwhythereisthesamenumberofcommitteesofsizeoneasthereiscommitteesofsizefour,andwhythereisthesamenumberofcommitteesofsize2ascommitteesofsize3.Thinkaboutthatlastcasenow.Lookatthegroupoffivetotalpeopleinthepicture.Whydoesitmakesensethattherewouldbethesamenumberofcommitteesofsize3ascommitteesofsize2?Explain.

Inthenextclassactivity,youwillhavetheopportunitytodiscussyourthinkingaboutthisquestioninclassandtoidentifyandexplainotherpatternsinPascal’sTriangleaswell.

Homework

Whenangry,counttofour;whenveryangry,swear. MarkTwain

1) DoalloftheitalicizedthingsintheReadandStudysection.

2) Explainhowtousethetabletoanswerthisquestion:Ifyouhave8totalpeople,

howmany5-membercommitteesarepossible?3) Explainwhyitmakessenseintermsofcommitteesthatthesecondcolumnofthe

tableisjustthepatternofnaturalnumbers.4) Spendtenminutesonthis:lookatthecommitteetableandidentifyatleastfour

morepatternsthatyouseethere.Writethemdownandbringthemtoclassnexttime.

24

ClassActivity3:PatternsinCommitteeNumbers

Artistheimposingofapatternonexperience,andouraestheticenjoymentisrecognitionofthepattern.

AlfredNorthWhiteheadAswehavediscussed,theCommitteeTableisfullofnicepatternsthatcanbeexplainedwhenimaginedintermsofcommittees.Yourjobasaclassistoidentifypatternsinthetable(theycanbetrivialorcomplex)andthentomakeargumentsforwhythepatternexistsbasedonthinkingaboutwhatthecommitteesthenumbersrepresent.

25

Homework

Andintheendit’snottheyearsinyourlifethatcount.It’sthelifeinyouryears. AbrahamLincoln

1) Supposethereare8totalpeople.Carefullyexplainwhyitmakessensethatthereis

thesamenumberofcommitteesofsize3astherearecommitteesofsize5.

2) AportionofPascal'sTriangleisshownbelow: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 . . . . . . . . . . . . . . .

a) ExplainwhyitmakessenseintermsofcommitteesthatthelastdiagonaloftheCommitteeTableisall1s.

b) Carefullyexplainwhyitmakessenseintermsofcommitteesthatthecircled4plusthecircled6equalsthecircled10.

c) Carefullyexplainwhyitmakessenseintermsofcommitteesthateachrowofthetrianglesumstoapowerof2.

3) Listallthedifferentthree-letter“words”(theydon’thavetomeananything)thatcan

bemadeusingallthelettersinBOA.NowdothesameforBOO.Seeifyoucanfigureouthowtodealwithrepeatedlettersinthiscase.

4) Listallthedifferentwords(theydon’thavetomeananything)youcanmakeusingallthelettersinPAST.NowlistallthewordsyoucanmakeusingthelettersinSASS.Seeifyoucanfigureouthowtodealwithrepeatedlettersinthiscase.

26

ClassActivity4:FromPhotostoCommittees Acommitteecanmakeadecisionthatisdumberthananyofitsmembers.

DavidCoblitzThepurposeofthisactivityistofindashortcutwaytocomputethenumberofcommitteesincaseswhereitmightbeinconvenienttousetheCommitteeTableandtounderstandwhythatmethodworks.Forexample,supposeyouneededtoknowhowmanydifferent5-cardhandscouldbedealtfromanordinarydeckof52cards.Thatislikeacommitteeproblemwherethetotalnumberofobjects(n)is52andthecommitteesize(k)is5.Bytheway,wewouldtypicallywritethatnumberofcommitteesas52C5.Bytheendoftoday,hopefullyyouwillbeabletofigurethatoutwithouthavingtomakeagiantCommitteeTabletodoso.Inordertodothis,we’llseeifwecanarelationshipbetweenphotosandcommittees…

1) Listallthephotographsthatcanbemadewithfourtotalpeopleifthreeofthemwillbeineachphoto.

2) Listallthecommitteesofsizethreethatcanbemadefromatotaloffourpeople.

3) Nowlookatthoseliststoseeifyoucanfigureouthowtoadjustthenumberofphotos

(whichiseasytocompute)togetthenumberofcommittees.(Youmayneedtotryacouplemorelisting-examples.Youdecide.)

27

ReadandStudy Thehardestarithmetictomasteristhatwhichallowsustocountourblessings.

EricHofferSoyouhavefoundthatthewaytogetfromPhotostoCommitteesistodivideouttheduplicatestuff.Thisisanimportantidea,sowearegoingtospendsometimeonit.Asusuallookingataspecificexamplecanhelp,solet’staketheexampleoffivetotalpeopleandlookfirstatallthephotosofsizethreethatcanbemade,andthenallthe3-membercommittees:Herearethefivepeople: A B C D EWealreadyknowtherewillbe5×4×3=60photos.Explainwhyweknowthatinadvanceoflistingthem.Photos: ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ACB ADB AEB ADC AEC AED BDC BEC BED CED BAC BAD BAE CAD CAE DAE CBD CBE DBE DCE BCA BDA BEA CDA CEA DEA CDB CEB DEB DEC CAB DAB EAB DAC EAC EAD DBC EBC EBD ECD CBA DBA EBA DCA ECA EDA DCB ECB EDB EDCNoticehowwehaveasystematicwayiflistingthings.Thatwaywearelesslikelytomissany.Describeoursystem.Nowhavealookathowthecommittees(inbold)showupinourlist: ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ACB ADB AEB ADC AEC AED BDC BEC BED CED BAC BAD BAE CAD CAE DAE CBD CBE DBE DCE BCA BDA BEA CDA CEA DEA CDB CEB DEB DEC CAB DAB EAB DAC EAC EAD DBC EBC EBD ECD CBA DBA EBA DCA ECA EDA DCB ECB EDB EDCDoyouseethateachcommitteeofsizethreecanbearrangedin6waystomakephotographs?Sothatmeansthatinthiscase,thenumberofcommittees×6=numberofphotos.Usingournewnotation,itlookslikethis: 5C3×6=5P3

28

Orwecouldwriteitthisway: 5C3=5P3÷6Butthisisonlytrueinthecaseofthree-membercommitteesandphotos.Whatwillweneedtodividebyinthecaseof2-membercommitteesandphotos?4?k?Thatistheproblemyousolvedtodayinclass.Butifyoustilldon’tfullyunderstandit,besuretoaskinyournextclassortalktoyourinstructor.Bepersistentaboutthis–itisakeyidea.

Homework

Therearenosecretstosuccess.Itistheresultofpreparation,hardwork,andlearningfromfailure.

ColinPowell

1)Supposewehaveaclassof20people.a) Howmanydifferentgroupsofsizefourcouldwemake?b) Howmanygroupsofsizefourcouldwemakethatdon’tincludeAnn?c) HowmanygroupsofsizefourcouldwemakethatincludeAnn?d) AnnandBobarenotspeakingtoeachother.Howmanygroupsofsizefour

couldbemadeifAnnandBobcannotbeinagrouptogether?

2) Howmanytotalgroups(allsizes)canbemadefrom8totalpeople?Explain.

3) Howmanydifferent“words”(theyneednotmeananything)canbemadea) usingallthelettersinORANGE?b) usingallthelettersinAPPLE?c) usingallthelettersinBANANA?

4) Makesureyouunderstandhowtofigureouteachofthefollowingproblems:

a) Howmanyphotoscanbemadewith7peopleifeveryonewantstobeinevery

photograph?b) Howmanygroupsofsize3canbemadefromagroupof10people?c) Howmanydifferentbridgehandscanbemadefromadeckofcards?(Thereare52

totalcardsandabridgehandcontains13cardsdealtatrandom.)d) Howmanyphotographscanbemadewith13peopleifonly6peoplewillbein

eachphotographatatime?

5) Calculatebyhand:

a)11C3 b)6P6 c)7P4 d)nC1 e)nPn

29

6) Supposewehave8totalpeopleandwanttotakeallpossiblephotographsthatcontain4people.Carefullyexplainwhyitmakessensethatthenumberofphotographs=8×7×6×5.Besuretoexplainthemultiplicationaswellasthenumbersinvolved.

7) Supposethereare8totalpeople.Carefullyexplainwhyitmakessensethatthe

(numberofphotographsofsize3)=3!×(numberofcommitteesofsize3).

8) Dothe“ChallengeProblem”aboutthedartboardinUnit1,Module3,Session1oftheBridgesinMathematicsGrade2HomeConnections.Whataresomedifferentwaysthatasecondgradermight“showtheirwork”insolvingthisproblem?

30

ChapterTwo

Probability

31

ClassActivity5:DiceSums

Theexcitementthatagamblerfeelswhenmakingabetisequaltotheamounthemightwintimestheprobabilityofwinningit.

BlaisePascalRolltwodiceatonceandifthesumis2,3,4,5,10,11or12,yourgroupwins.Ifthesumis6,7,8,or9,theinstructorwins.Isthisafairgame?Dothisproblemtwoways.First,actuallyplaythegamelotsoftimesandcollectdatatodecide.Then,doatheoreticalanalysisoftheproblemtodecide.(Youwillneedtobeginbydecidingwhatitmeansthatagameofchanceis“fair.”)

32

ReadandStudy Areasonableprobabilityistheonlycertainty.

E.W.Howe

Gamesofchancehavebeenaroundforatleastaslongaspeoplehaverecordedhistory,butitwasn’tuntilthe16thcenturythatmathematiciansfirstbegantostudytherulesthatmighthelpanswerquestionssuchas“Isthisgamefair?”or“HowlikelyamItodrawaroyalflushinagameofpoker?”Wecalltheareaofmathematicsthatcontainstheserulesprobabilitytheory.

Simplyput,probabilityisthestudyofchanceorrandomevents.Supposewerollastandardsix-sideddieonce.Wedon’tknowtheresultoftherollinadvance,sowesaythattheoutcomeisrandom.However,theoutcomeisnotcompletelyunpredictable.Ourdiehassixsidesandsoweknowtheoutcomewillbeoneofsixnumbers:1,2,3,4,5,or6.Wealsoknow(assumingwehaveafairdie)thatanyoneoftheseoutcomesisequallylikely;thatis,ifwerollthediemany,many,manytimesandrecordalltheoutcomes,wewillgetapproximatelythesamepercentageof‘1’saswedo‘2’saswedo‘3’sandsoforth.Supposewewouldliketobeabletoassignanumbertoexpresshowlikelyitisthatwegeta5ononeroll.Inotherwords,wewanttoknowtheprobabilityofgettinga5ononerollofastandard6-sideddie.Bytheway,theuseofgoodmathematicallanguageisahabit.Ifyouwanttousemathematicallanguagecorrectlywithyourelementarystudents,youmustpracticedoingsonow.Youwillfindunderlinedvocabularyandideasdefinedforyouintheglossary.Gothereandhavealookatthedefinitionof“probability”now.Inordertotalkabouttheanswertothisquestion,wemustagreeonawaytomeasureprobability.Fourmainruleshavebeenadopted:

1) Theprobabilityofanoutcomeisanumberfrom0to1.Ifanoutcomeiscertaintooccur(suchas,attheequatorthesunwillrisetomorrow),thenitsprobabilityis1.Ifanoutcomecannothappen(suchas,themoonwillfallintothePacificOceantonight),thenitsprobabilityis0.Ifanoutcomeisneithercertainnorimpossible,thentheprobabilitywillbesomenumberbetween0and1.

2) Thesumoftheprobabilitiesofallpossibleoutcomesofarandomexperimentis1.

3) Theprobabilitythataneventwilloccuris1minustheprobabilitythatitwillnotoccur.Thatis,aneventwilleitheroccuroritwillnotoccur,andthesumofthosetwoprobabilitiesmustbe1.

4) Iftwoeventsaredisjoint(theyhavenooutcomesincommon),theprobabilitythatoneortheotherwilloccuristhesumoftheprobabilitiesthateachwilloccurbyitself.

33

Sowemustthinkabouthowwecanassignanumberfrom0to1totheoutcome“obtaininga5whenrollingonedie”thatwewillcallitsprobability.Todosomathematiciansusethelanguageofsets.Herecometheofficialdefinitionsofrandomexperiment,outcome,event,andsamplespace.Arandomexperimentisanactivity(suchasrollingour6-sideddie)wheretheoutcome(1,2,3,4,5,or6)cannotbeknowninadvance.Asetofallpossibleoutcomes,{1,2,3,4,5,6},isasamplespaceforthatexperimentandaneventisanysubsetofthesamplespace(suchas“rollinga5”or“rollinganevennumber”).Noteasubtledistinctionherebetweenanoutcomeandanevent:anoutcomeisanelementofthesamplespace,whereasaneventisasubsetofthesamplespace.Soifthesamplespaceis{1,2,3,4,5,6}then‘3’iscalledanoutcome,whereas{3}iscalledanevent.{1,3,5}isanotherevent.Listafewmoreeventsforthissamplespaceusinggoodcurly-bracketsetnotation.Doestheemptysetmeetthedefinitionofan“event?”Whyorwhynot?Okay,nowbyrulenumber1,theprobabilityofeachoutcomeinthesamplespacewillbeanumberbetweenzeroandone,andbyrulenumber2,thesumofalltheprobabilitiesoftheoutcomesinthesamplespacemustequalone.Sohowdowedeterminetheprobabilityof“rollinga5?”Wellthatturnsouttobeprettyeasyinthecasewherealltheoutcomesareequallylikelytooccur.(Wewillassumethatourdieisevenlybalancedandaperfectcube-wecallsuchadiefair–andsowearejustaslikelytoobtainanyoneofthenumbersfrom1to6.)Whenoutcomesareequallylikely,theprobabilitythatanyoneoccursisjusttheratioof1tothetotalnumberofoutcomes.Inthecaseofrollingonefairdie,theprobabilityofrollinga‘5’is .Whenaneventismorecomplicated(suchasobtainingasumof3whenrollingtwodice),wecanusethesameapproachbutweneedtochooseoursamplespacewisely.Supposeweareinterestedinthesumobtainedwhenrollingtwofairdice.Ifwelistthepossibleoutcomesforthesum,wegetthesamplespaceof{2,3,4,5,6,7,8,9,10,11,12},whichcontainselevenoutcomes.Unliketheearlierexampleofrollingonediehowever,theseoutcomesarenotequallylikely.Forexample,thereareseveralwayswecouldobtainasumof7(1+6,2+5,etc.),butonlyonewaywecouldobtainasumof12(6+6).Whentheoutcomesarenotequallylikely,weneedadifferentwaytodeterminetheprobabilityofeachoutcome.Inthisexample,theprobabilityofeachsumisnot .

61

111

34

Onewaytodeterminetheprobabilitiesofeachsumistoconsiderasamplespaceoftherandomexperimentofrollingtwodicewheretheoutcomesareequallylikely.Forexample,wecanthinkoftheoutcomesinthissamplespaceasorderedpairs(x,y),wherexisthenumberonthefirstdieandyisthenumberontheseconddielikeyoumayhavedonewhenyouworkedontheclassactivity.Thissamplespaceisshowninthetablebelow.Theadvantageofusingthissamplespaceisthatalloutcomeshereareequallylikely.

1 2 3 4 5 6

1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Apossiblesamplespaceforarolloftwodice

Noticethatthereare36equallylikelyoutcomesinthissamplespaceandsotheprobabilityof

eachoutcomeis .Wecanusethissamplespacetocomputetheprobabilityoftheevent“obtainingasumof3”(that’stheevent{(1,2),(2,1)})bycountingthenumberofpossibleoutcomesthathaveasumof3.Wemustbecarefultomakecertainthatweactuallydocountallpossibleoutcomesthathaveasumof3.Forexample,inthesamplespace,theoutcome(1,2)whichgives1+2=3andtheoutcome(2,1)whichgives2+1=3aretwoseparateelements.Thatis,therearetwowaystogeneratethesumof3:1)wecanrolla1onthefirstdieandthena2ontheseconddieor2)wecanrolla2onthefirstdieandthena1ontheseconddie.Tohelpyourupperelementarystudentsthinkaboutthiswesuggesttwothings.First,havethemdoanexperimentwhereeveryonerollsapairofdiceforfiveminutesandtalliesthenumberoftimestheyrollasumof2andseparatelythenumberoftimestheyrollasumof3.Inthisway,theywillseethatthesumof3istwiceaslikelytooccur.Second,havethemeach

361

35

holdonereddieandonegreendie.Askthattheymakeasumof3.Thenaskthattheymakeasumofthreeadifferentwaysotheycanseethattheoutcome(1,2)giving1+2=3andtheoutcome(2,1)giving2+1=3aretwoseparateelementsinthesamplespace.Trythisyourself.Nowmakeasumof2andnoticethatthereisnootherwaytodothis.Rollinga1onthereddieanda2onthegreendieisadifferentoutcomefromrollinga2onthereddieanda1onthegreendie,butthereisonlyonewaytohavea1oneachdie.Okay,backtooursamplespacewith36outcomes.Usingprobabilityrule4,theprobabilityofrollingtwodicewithasumof3,writtenP(Sum=3),isthesumoftheprobabilitiesofeach

outcomewhichequals + = = .WecanalsocomputeP(Sum=3)astheratioofthenumberofoutcomesthathaveasumof3tothenumberoftotaloutcomessinceallthe

outcomesareequallylikely.SoP(Sum=3)= .

Thisprobabilitycanbeexpressedasafraction( ),asadecimal(0.056),asapercent(5.6%)orasodds(2:34,thechancesofhavingasumofthreetothechancesofhavingasumotherthanthree).Checkallthistobesureitmakessensetoyou.

Let’sconsideranotherexample:Whataremychancesofdrawingafacecardfromastandarddeckof52playingcards?Beforeyoureadfurther,takeaminutetoseeifyoucanfigureitout.

Hereisourthinking.Payattentiontothelanguageweuseaswellastothewaywesolvetheproblem.Astandarddeckofcardscontains52cards,13ofeachoffoursuits.Eachsuitcontains3facecards,thejack,thequeen,andtheking,soIhave4×3=12waysofdrawingafacecardand52waysofdrawinganycard.Sotheprobabilityofdrawingafacecardfroma

deckofcardsis .Thisprobabilitycanbeexpressedasafraction( ),asadecimal(0.231),asapercent(23.1%)orasodds(12:40,thechancesofdrawingafacecardtothechancesofnotdrawingafacecard).Inthisexample,theexperimentis“drawingacardfromadeckofcards,”theeventis“drawingafacecard,”andthesamplespacecontains52equallylikelyoutcomes(the52cardsinthedeck).

361

361

362

181

362

181

5212

133

36

ConnectionstotheElementaryGradesIngrades3–5,allstudentsshouldunderstandthatthemeasureofthelikelihoodofaneventcanberepresentedbyanumberfrom0to1.

NCTMPrinciplesandStandards,p.176InbothoftheReadandStudyscenarioswehavebeendiscussingwhatwecalltheoreticalprobabilities,thatisprobabilitiesthatareassignedbasedonassumptionsaboutthephysicaluniformityandsymmetryofanobject(suchasadieoradeckofcards).Thismayhaveleftyouwiththeimpressionthatallrandomexperimentscanbeanalyzedinatheoreticalmanner.Notso.Considertheexperimentof“flipping”alargemarshmallow.Nowitcouldlandonaflatcircularendoronitsside(thecurvedpartofthecylinder),sowe’llidentifythesamplespaceastheset:{end,side}.Childrenmaythinkthatthesetwooutcomeseachhaveaprobabilityof½sincetherearetwochoices,butsomeexperimentingwithactualmarshmallowswillshowthatthesearenotequallylikelyoutcomes.Furthermore,atheoreticalanalysisoftheprobabilitiesofeachoutcomewouldrequireknowledgeofphysicsandgeometry,andmightdependonthesurface,thetemperature,orotherfactors.Incaseslikethat,itisbettertoperformanexperimenttodeterminetheprobability.Inotherwords,itisbesttohavethechildrenflipthemarshmallowalargenumberoftimesandtousethedatatoestimatetheprobabilityofeachoutcome.

Homework

You'llalwaysmiss100%oftheshotsyoudon'ttake.WayneGretzky

1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.

2) Ifyoudecidedthatthegame“DiceSums”wasunfair,changetherulessothatthe

gameisfair.Istheremorethanonesetofrulesthatwouldgiveafairgame?Ifso,howmanydifferentsetsofrulescanyoumake?

3) Consideranewgamecalled“DiceDifferences.”Twodicearetossedandthesmallernumberissubtractedfromthelargernumber.PlayerIscoresonepointifthedifferenceisodd.PlayerIIscoresonepointifthedifferenceiseven.(Note:Zeroisanevennumber.)Isthisafairgame?Ifso,makeanargumentthatitis.Ifthegameisunfair,changetherulessothatthegameisfair.

4) Readthe“TheseBeansHaveGottoGo!”activityfromUnit2Module1Session1oftheBridgesinMathematicsGrade2HomeConnections,onpage29-34.

37

a. Howwouldyouplaceyourbeansifyouwereplayingthisgame(andwantedtowin).

b. Howwouldyouexpectatypical2ndgradertoplacetheirbeans?c. Whatanswerswouldyouexpectthatyourfuture2ndgradestudentsmight

writetoquestions3,4and5onpage34.

5) Abagcontainstwowhitemarbles,fourgreenmarbles,andsixredmarbles.Therandomexperimentistodrawamarblefromthebag.

a) Writeareasonablesamplespaceforthisexperiment.Usesetnotation.b) Whatistheprobabilityofdrawingawhitemarble?Aredmarble?Agreen

marble?Explainyouranswers.c) Whatistheprobabilityofdrawingablackmarble?Explain.d) Whatistheprobabilityofnotdrawingagreenmarble?Explain.e) Howmanymarblesofwhatcolorsmustbeaddedtothebagtomakethe

probabilityofdrawingagreenmarbleequalto0.5?

6) Abagcontainsseveralmarbles.Somearered,somearewhite,andtherestaregreen.

Theprobabilityofdrawingaredmarbleis andtheprobabilityofdrawingawhite

marbleis .

a) Whatistheprobabilityofdrawingagreenmarble?Explain.b) Whatisthesmallestnumberofmarblesthatcouldbeinthebag?Explain.c) Couldthebagcontain48marbles?Ifso,howmanyofeachcolor?Explain.d) Ifthebagcontainsfourredmarblesandeightwhitemarbles,howmany

greenmarblesdoesitcontain?Explain.

7) Decidewhetherthefollowinggameisfairorunfair:Playershavethreeyellowchips,eachwithanAsideandaBside,andonegreenchipwithanAsideandaBside.Flipallfourchips.PlayerIwinsifallthreeyellowchipsshowA,ifthegreenchipshowsA,orifallchipsshowA.Otherwise,PlayerIIwins.Ifthegameisfair,makeanargumentthatitis.Ifthegameisunfair,changetherulessothatthegameisfair.

8) The6outcomesofrollingafairdieareequallylikely;soarethe52outcomesofa

randomdrawofonecardfromadeckofcards.Nameatleastthreemorerandomexperimentsthatgenerateequallylikelyoutcomes.Nameatleastthreethatproduceoutcomesthatarenotequallylikely.

9) Considertherandomexperimentoftossingthreecoinsatonce.Writeasamplespaceforthisexperimentthathasequallylikelyoutcomes.Nowwriteanothersamplespaceforthissameexperimentthatdoesnothaveequallylikelyoutcomes.

61

31

38

ClassActivity6:RatMazesThetroublewiththeratraceisthatevenifyouwin,you'restillarat.

LilyTomlin

1) Supposethataratissentintoeachofthebelowmazesatthe‘start.’Iftheratcannotgobackwards,butotherwisemakesallthedecisionsatrandom,whatistheprobabilitythatshefindsacheeseineachcase?Explainyouranswersasyouwouldtoanupperelementaryschoolstudent.

2) Makearatmazetoillustrateeachoftheseproblems:a) Youflipacoinandthenrolladie.WhatistheprobabilitythatyougetaHead

andthenamultipleofthree?b) Youflipacointhreetimes,whatistheprobabilitythatyougetexactlytwo

Heads?

Start

Start

Start

39

ReadandStudy

Chancefavorsonlythepreparedmind. LouisPasteur

Manysituationsrequiresuccessivechoices–likeintheratmazeproblemswheretherathadtomakeatleasttwodecisionsaboutwhichpathtotaketogettothecheese.Hereisanotherexample:gettingdressedforschoolinthemorningrequiresseveraldecisions.Imustchooseoneoftwopairsofjeans(onlytwoareclean,abluepairandablackone),oneofthreesweaters(red,purple,blue),andoneoffourpairsofshoes(let’sjustcallthemA,B,C,&D)Let’ssaythatshoepairsBandDdon’tcoordinatewiththeblackjeans,butshoepairsA&Ccanbewornwitheitherpairofjeans.1)WhatistheprobabilitythatIwearbluejeans,aredsweater,andCshoes?2)Whatistheprobabilitythatmyoutfitcontainsthecolorblue?3)WhatistheprobabilitythatIweartheBpairsofshoes?Treediagrams(orratmazes)areusefulinfindinganswerstoquestionslikethese.Belowwehavemadeatreethatcountsallthepossibleoutfits(orpathsinthemaze).Inotherwords,thechoicesforgettingdressedlooklikethis.

Nowassumingthatallchoicesaremadeatrandom,thereareacouplewaystocomputetheprobabilitiesinquestions1-3above.Takeaminutetodosobeforereadingfurther.Mypersonalfavoriteisthesending-rats-into-the-mazetechnique.I’llsend,say,24ratsintothetopofthemaze.ThenIexpect12togoleftdownthebluejeanchoice,4ofthosetogodowneachsweatercolor,and1tocomeoutofeachoftheshoechoicesA,B,CandD.Ofthe12thatwillgorightdowntheblackjeanchoice,4willgodowneachsweaterrouteand2willemergefromeachshoetypeAandC.Nowwecanusethisinformationtoanswertheabovequestions.1)Since1outof24comeoutofthebluejeans,redsweater,Cshoespath,the

C

C

A

A

A

D

C B

D

C B

C

B A

A

D

A

blue sweater

blue sweater

red sweater red sweater

black jeans blue jeans

40

probabilityofthatis .2)Sincealltheblue-jeansratsandalsotheblue-sweaterratscontain

blue(that’s12+4), or istheprobabilitythatmyoutfitcontainsblue.3)Since3ratsend

upwithBshoes,theprobabilitythatIwearthoseshoesis or .Okay,nowyoumightbethinking,buthowdidsheknowtosend24ratsintothemaze?Doesthatnumbermatter?Tosee,youtryit.Firstuse48rats.Thentry100toseewhathappens.Reallydoit.Whatdidyoulearn?Anothersolutionistousetheprobabilityequivalentofthemultiplicationcountingprincipletofindtheprobabilitythatwereachtheendofeachbranch.Forexample,theprobabilitythatIwearbluejeans,aredsweater,andshoepairAis!

"×!

%× !&= !

"&.Whydowesayweare

usingthemultiplicationcountingprinciplehere?FindtheprobabilitiesofeachoftheoutfitsIcouldchoose.Doyouneedtocalculateeachprobabilityindividually?Orcanyoumakeanargumentthatalloftheoutfitswithbluejeansareequallylikely?Whatabouttheoutfitswithblackjeans?Finally,let’sthinkaboutthelasttwoquestionsweposedinthefirstparagraphusingthissecondapproach:Whatistheprobabilitythatmyoutfitcontainsthecolorblue?MyoutfitwillcontainblueifIwearbluejeansorifIwearthebluesweater.Halfofthepossibleoutfitsincludethebluejeansandonethirdoftheremaininghalfoftheoutfitsusethebluesweater.Soatotalof!

"+ !

%× !"= !

"+ !

*= "

%oftheoutfitscontainblue;theprobabilitythatIwearblue

is66 "%%.UsethisideatofindtheprobabilitythatIweartheBpairofshoes.

241

2416

32

243

81

41

ConnectionstotheElementaryGrades

Ingrades6–8allstudentsshouldcomputeprobabilitiesforsimplecompoundevents,usingsuchmethodsasorganizedlists,treediagrams,andareamodels.

NCTMPrinciplesandStandards,p.248Treediagramsarepartoftheelementarygradescurriculumbecausetheygivechildrenawayto“see”allofthepossibleoutcomesandtokeeptrackoftheprobabilitiesofeachchoicethatmightbemadeinacompoundevent.Bytheway,itisnotenoughtoknowjustonewaytodothingsanymore.Ifyouaregoingtobeateacher,youwillneedtomakesenseofthethinkingofyourstudentseveniftheirmethodsaredifferentfromyours.Practicingdifferentwaysofthinkingaboutandexplainingproblemswillmakeyouabetterandmoreflexibleteacher.

Homework

Onceyoulearntoquit,itbecomesahabit.VinceLombardi

1) DoalltheitalicizedthingsintheReadandStudysection.

2) Decideifeachofthefollowingstatementsistrueorfalse.Ifitistrue,explainwhy.Ifit

isfalse,rewritethestatementsothatitistrue.

a) Iftheprobabilityofaneventhappeningis ,thentheprobabilitythatitdoesnothappenis .

b) Theprobabilitythatanoutcomeinthesamplespaceoccursis1.c) Ifaneventcontainsmorethanoneoutcome,thentheprobabilityofthe

eventisthesumoftheprobabilitiesofeachoutcome.d) IfIhavea60%chanceofmakingafirstfreethrowanda75%chanceof

makingasecond,thentheprobabilitythatImissbothshotsis10%.

3) Makearatmazetomodeleachoftherandomexperiments.a) Ihaveabagcontainingthreechips.TheredchiphasasideAandasideB.

TheyellowchiphasasideBandasideC.ThebluechiphasbothsidesmarkedA.Idrawonechipatrandomfromthebagandtossit.

b) Threechipsaretossed.TheredchiphasasideAandasideB.TheyellowchiphasasideBandasideC.ThebluechiphasbothsidesmarkedA.

83

38

42

4) Supposethechipsin#3abovearealltossed.Useyoursecondratmazetodeterminetheprobabilityofgettingexactly

a)oneC b)oneB c)oneAd)twoCs e)twoBs f)twoAs

5) Supposethatthreechipsaretossedasin#3.PlayerIscoresapointifapairofAsturn

up,andplayerIIscoresapointifapairofBsoraCturnsup.Isthisafairgame?Ifso,explainwhy.Ifnot,changetherulestomakeitfair.

6) Abasketballplayerhasa70%freethrowshootingaverage.Theplayergoesupforaone-and-onefreethrowsituation(thismeansthattheplayershootsonefreethrow,andonlyifshemakesit,shegetstoattemptasecondshot).Whatistheprobabilitytheplayerwillmake0shots?1shot?2shots?

7) SupposethatItossafaircoinuptofourtimesoruntilIgetaHead(whichevercomesfirst).

a) Makeamazetomodelthisrandomexperiment.b) Writeoutthesamplespaceusinggoodnotation.c) WhatistheprobabilitythatyougetTTH?

8) Place2blackcountersand1whitecounterintoapaperbagandshakethebag.

Withoutlooking,draw1counter.Thendrawasecondcounterwithoutputtingthefirstcounterbackintothebag.Ifthe2countersdrawnarethesamecolor,youwin.Otherwise,youlose.Whatistheprobabilityofwinning?Doestheprobabilityofwinningchangeifyoureplacethefirstcounterbeforedrawingthesecondcounter?Whyorwhynot?Whatistheprobabilityofwinningifyoureplace?

9) Here’showyouplayourlottery:youwritedownanysixnumbersfromtheset{1,2,3,…35,36}.Thensixwinningnumbersaredrawnfromthatsetatrandom(withoutreplacement–soallofthemwillbedifferentnumbers).

a) Ifyoumatchallthewinnersinorder,youwin$5,000,000.00.Whatistheprobabilityofthat?

b) Ifyoumatchallthewinnersbutnotnecessarilyinorder,youwin$1,000,000.00.Whatistheprobabilityofthat?

c) HowaretheseproblemsrelatedtoourPhotographandCommitteeproblems?Explain.Bespecific.Howwouldyouexplainthistosomeone?

43

ClassActivity7:TheMaternityWard

WhenIwasakid,Iwassurroundedbygirls:oldersisters,oldergirlcousinsjustdownthestreet….EachyearIwouldwishforababybrother.Itneverhappened.

WallyLamb

Isitmorelikelythat70%ormoreofthebabiesbornonagivendayareboysinasmallhospitalorinalargehospital(ordoesn’titmatter)?Yourclassshoulddecideonasimulationtodoinordertohelpyoutoanswerthisquestion.

44

ReadandStudy“Ithinkyou’rebeggingthequestion,”saidHaydock,“andIcanseeloomingaheadoneofthoseterribleexercisesinprobabilitywheresixmenhavewhitehatsandsixmenhaveblackhatsandyouhavetoworkitoutbymathematicshowlikelyitisthatthehatswillgetmixedupandinwhatproportion.Ifyoustartthinkingaboutthingslikethat,youwouldgoroundthebend.Letmeassureyouofthat!”

AgathaChristieinTheMirrorCrack’dOurintuitionoftenmisleadsuswhenwethinkaboutprobabilities.Initially,youmayhavethoughtthatthechancesofhaving70%ormoreofthebabiesbornbeboyswouldbebetterinalargehospitalbecauseitismorelikelythatmorebabiesareborninalargehospitalononedayandthereforemorelikelytohavemoreboys.Itismorelikelythatthenumberofbabiesbornonagivendayislargerinalargehospitalthanitisinasmallone,butthisdoesnotmeanthatitismorelikelythatthepercentageofboybabieswillbe70%ormoreinthelargehospital.Infact,thelargerthehospital(andthereforethelargerthenumberofbabiesbornonagivenday),themorelikelyitisthatthepercentageofboysbornwillbe50%(theapproximateprobabilityofhavingababyboyinasinglebirth).Mathematically,wecallthisresultthelawoflargenumbers,whichstatesthatinrepeated,independenttrialsofarandomexperiment,(suchasbabiesbeingborninahospital),asnumberoftrials(births)increases,theexperimentalprobabilitiesobserved(theprobabilitythatababybornisaboy)willconvergetothetheoreticalprobability(inthiscase,50%).Inotherwords,thelargerthesample,themorelikelyyouaretogetclosetothetheoreticalresult.Sothemorebirthsthereareonagivenday(thelargerthehospital),themorelikelyitisthatthepercentageofboysbornisnear50%.Thefactthatthetrialsshouldbeindependentisimportant.(Besuretousetheglossaryfornewmathematicalterms;don’tassumethatthemathematicalmeaningofawordisthesameasthemeaningthewordmighthaveincommonusage.)Forourpurposesconsidertwoeventstobeindependentiftheoccurrenceornon-occurrenceofoneeventhasnoeffectontheprobabilityoftheothereventhappening.Inthehospitalproblem,thebirthofababyboytoonemotherhasnoeffectontheboy/girloutcomeofthenextbirthinthehospital.Theoutcomesofthetwobirthsareindependentofoneanother.Whatabouttherolloftwodice?Aretheoutcomesoneachdieindependent?Whataboutthreeflipsofacoin?Isthehead/tailoutcomeoneachflipindependent?Notalleventsareindependent.Considertheevent“itwillraintomorrow”andtheevent“theweddingwillbeheldoutsidetomorrow.”Theprobabilitythattheweddingwillbeheldoutsideisaffectedbywhetheritrains.Writedownanotherexampleoftwoeventsthatarenotindependent.

45

Onewaytoteachchildrenaboutthelawoflargenumbersistorunlotsofsimulationslikeyoudidinclasswiththehospitalproblem.Thatis,wemodeledthenumberofboyandgirlbirthsonagivendayinvarioussizehospitals,calculatedtheaveragepercentageofboybirthsforeachsizehospitalandthencomparedtheresults.Inordertohavethissimulationworkweneededamodel(acoinperhaps)thathastheapproximatelythesameprobabilityofsuccessasdoestheeventweareinvestigating.Andweneedtobecertainthatthenumberof‘births’weconsiderlargeislargeenoughforthelawoflargenumberstoapply.Apracticalwaytodesigncollectingdataonalargenumberoftrialsinyourclassroomistopooleveryone’sdata.Youwereaskedtorunasimulationintheclassactivity.Howdidyoudesignyoursimulation?Howmightyouhaveusedtherollofonefairdie?Howdidyoudecidethenumberofbirthsthatwouldrepresentasmallhospitalandthenumberofbirthsthatwouldrepresentalargehospital?Whatwastheeffectofpoolingalltheclasses’data?

Homework

IamalwaysdoingthatwhichIcannotdo,inorderthatImaylearnhowtodoit.PabloPicasso

1) DoalltheitalicizedthingsintheReadandStudysection.

2) Acoinhasbeentossedfivetimesandhascomeupheadseachtime.Whichofthe

followingstatementsaretrue?a) Thereisanequalchanceofcomingupheadsortailsonthenexttoss.b) Thecoinismorelikelytocomeupheadsonthenexttoss.c) Thecoinismorelikelytocomeuptailsonthenexttoss.d) Iquestionwhetherthecoinisfair.

3) Acoinhasbeentossedfivehundredtimesandhascomeupheadseachtime.Which

ofthefollowingstatementsaretrue?a) Thereisanequalchanceofcomingupheadsortailsonthenexttoss.b) Thecoinismorelikelytocomeupheadsonthenexttoss.c) Thecoinismorelikelytocomeuptailsonthenexttoss.d) Iquestionwhetherthecoinisfair.

4) Explainhowthelawoflargenumbersisrelatedtoyouranswerstothequestioninproblems2)and3)above.

5) Explainhowthelawoflargenumbersisrelatedtothedeterminationofexperimentalprobabilities.Whatdoesthislawtellyouaboutsimulationsyouwilldoinyourelementaryclassrooms?

46

6) Afaircoinhasbeentossedathousandtimesandthenumberofheadsandtails

recorded.Whichofthefollowingstatementsismostlikelytrue?a) Thenumberoftailsis601andthenumberofheadsis399.b) 47%ofthetosseswereheads.c) Therewere20moreheadstossedthantails.

7) Supposeweflipacoin10,000,000,000times(justsuppose)andnoticethat

5,801,595,601areheads.Whatcanyousayaboutthecoin?Explain.

8) Designasimulationtodeterminetheexperimentalprobabilityofdrawingaspadefromastandarddeckofcards.Howcanyoubereasonablycertainyourresultisclosetothetheoreticalprobability?Carryoutyoursimulationandcompareyourresulttothetheoreticalprobabilityofdrawingaspadefromadeckofcards.

47

ClassActivity8:ProbabilityChallenge!

Ifyoudon’triskanythingyouriskevenmore. EricaJong

DiscussasagroupwhethereachofthefollowingstatementsisTrueorFalse.Ineachcase,brieflyexplainyourgroup’schoice.

1) T F Ifthereisa20%chanceofrainonSaturdayanda40%chanceofrainon Sunday,thenthereisa60%chanceofgettingrainthisweekend.

2) T F Ifthereisa20%chanceofrainonSaturdayanda40%chanceofrainon Sunday,thenthereisa30%chanceofgettingrainthisweekend.

3) T F TheexactsequenceHHTHHHTTismorelikelythatthesequence HHHHHHHHwhentossingafaircoin8times.

4) T F Inafamilyof8children,itismorelikelythattherearehalfboysand halfgirlsthanitisthatthereareallboys.

5) T F IfJohnisaUWOstudentwhoistallandathletic,itismorelikelythatheisacollegebasketballplayerandabusinessmajorthanitisheisjustabusinessmajor.

6) T F IfJohnisaUWOstudentwhoistallandathletic,itismorelikelythat

heisacollegebasketballplayerthanitisthatheisabusinessmajor.

7) T F Youaremorelikelytodieinaterroristattackthantodieinacaraccident.

8) T F Youarelesslikelytowinthelotteryifyourpicksarethenumbers1,2, 3,4,5,6,thanyouareifyourpicksare2,31,15,19,9and5.

9) T F Ifyouhaveplayedandlostthelotteryfor300daysinarow,thenyou aremorelikelytowintomorrow.

48

ReadandStudy

Nothingdefineshumansbetterthantheirwillingnesstodoirrationalthingsinthepursuitofphenomenallyunlikelypayoffs.Thisistheprinciplebehindlotteries,dating,andreligion.

ScottAdamsHumansarenotverygoodatestimatingchance.Wefallvictimtolotsoffallacies(waysofthinkingthatarefaulty)intherealmofprobability.Forexample,intheclassactivityyoutalkedaboutthatfactthatthenumbers1,2,3,4,5,6wereaslikelytobewinningpicksforthelotteryasthenumbers2,31,15,19,9and5.Mostpeoplethinkthatsixconsecutivenumbersaremuchlesslikelytowinbecausesomehowtheyseemlessrandom.Thisisanerrorof"local"randomnessknownastherepresentativenessfallacy.Peoplewiththisfallacybelievethatevensmallsamplesshouldlooksufficientlyrandom.Forexample,manypeoplebelievethatthesequenceHHTHHHTTismorelikelythanthesequenceHHHHHHHHbecausethefirstsequencelookslikeamoretypicalresultthanthesecond,eventhoughbothsequencesof8cointosseshavethesameprobability.Thefallacyisconfusingaparticularoutcome(HHTHHHTT)withalargerclassofoutcomesthatitappearstorepresent(sequenceswithamixofheadsandtails).Somepeoplebelievethatafterastringoflossestheyare“due”towininordertobalanceoutwinsandlosses.That’salsoatypeofrepresentativenesscalledthegambler’sfallacy.Here’sonemoreflavoroftherepresentativenessfallacy.Considerthefollowingquestion:

Liemisanextremelyathleticlookingyoungmanwhodrivesafastcar.IsLiemmorelikelyapro-bowlprofessionalfootballplayeroranaccountant?

IfyousaidLiemwasmorelikelyapro-bowlprofessionalfootballplayer,youfellforthebaseratefallacy.Thecharacteristicsprovided(extremelyathleticlookingyoungmananddrivesafastcar)mayseemstereotypicallymorerepresentativeofprofessionalfootballplayersthanofaccountants,causingpeopletoignoretheratesatwhichprofootballplayersandaccountsoccurinthepopulation.Infactthattherearefarfewerprofessionalfootballplayersaroundthanthereareaccountants–soyoushouldbetthatLiemisthelatter.

49

KahnemanandTverskyusedthefollowingproblemtostudyanothertypeoffaultylogiccalledtheconjunctionfallacy.

Lindais31yearsold,single,outspoken,andverybright.Shemajoredinphilosophy.Asastudent,shewasdeeplyconcernedwithissuesofdiscriminationandsocialjustice,andalsoparticipatedinanti-nucleardemonstrations.IsitmorelikelythatLindaisabankteller,orbothabanktellerandafeminist?Why?

Answerthequestionabovebeforereadingthediscussionthatfollows.Itismorelikelythatsheisjustonething(abankteller)thanthatsheisboththings(abanktellerandafeminist).(Andisaconjunction–hencethenameofthefallacy.)However,thedescriptionofLindagivenintheproblemfitsthestereotypeofafeminist,whereasitdoesn'tfitthestereotypicalbankteller.BecauseitiseasytoimagineLindaasafeminist,peopleoftenmisjudgethatsheismorelikelytobebothabanktellerandafeministthanabankteller(feministornot).Hereisanotherwaypeoplereasonincorrectlyaboutchance:Peopletendtojudgetheprobabilityofcertaintypesofeventsbyhoweasyitistoremember,ortoimagine,theeventoccurring.Wecallthistheavailabilityfallacy.Whenanunusualeventispresentedvividlytoourminds,itbecomesmoreavailable;andinbecomingmoreavailable,itseemsmorelikely.IntheaftermathoftheeventsofSeptember11,2001,manypeoplechosetodriveratherthantoflywhenmakingalongtrip,becausetheyjudgeditmuchmorelikelythattheywouldbeinvolvedinaplanehijackingthaninanautomobileaccident.Availabilitycanbeverysubtleandpervasive.ManyparentsintheUSforexampleareafraidtolettheirchildrenplayoutsideorwalktoschoolbecausetheyfearstrangerabductions(dueinparttothefactthatstrangerabductionsalwaysmakenationalnewsandsoareheardbyandavailabletoallofus).Ironicallystrangerabductionisfarlesslikelytooccurthanchildrencontractingdiseases(liketypeIIdiabetesorproblemsrelatedtoobesity)duetosedentarylifestyles.Anditisfarmoredangerousformostchildrentorideinacartoschoolthanitisforthemtowalk.Infact,accordingtotheCentersforDiseaseControlandprevention,thefollowingarethetopfivemostlikelycausesofchildhoodinjury:caraccidents,homicide(almostalwaysatthehandsofsomeonethechildknows),childabuse,suicide,anddrowning.Whatdoparentsfearthemost?AccordingtosurveysconductedbytheMayoClinic,theyfearkidnapping,schoolsnipers,terrorists,dangerousstrangers,anddrugs(asreportedinTheNewYorkTimes,September19,2010).

50

Hereisanorganizationofthefallacieswehavediscussedinthissection: Representativeness

(BaseRateFallacyandGambler’sFallacyaretypesofrepresentativeness) ConjunctionFallacy AvailabilityOfcoursetherearemanyfaultywaysofthinkingaboutchancethatdon’thavefancynames.Forexample,considerthefirsttwoproblemsintheclassactivityaboutcomputingthechanceofrainontheweekendifthereisa20%chanceofrainonSaturdayanda40%chanceofrainonSunday.Basedonthisinformation,allwecanreallysayisthattheprobabilitythatitwillrainontheweekendissomewherebetween40%and60%.IfweassumethateverytimeitrainsonSaturday,italsorainsonSunday,thentheprobabilitythatitrainsontheweekendwouldbejust40%.Underwhatassumptionwouldtheprobabilitybe60%?Inbothoftheseextremecases“RainonSaturday”and“RainonSunday”wouldbedependentevents.Why?Ifinsteadweassumethat“rainonSaturday”and“rainonSunday”areindependentevents,thenwecouldcalculatethattheprobabilitythatitrainsontheweekendis52%.Makeatreediagramtoshowthatthiscalculationiscorrect.SourceofLindaProblem:AmosTversky&DanielKahneman,"JudgmentsofandbyRepresentativeness",inJudgmentUnderUncertainty:HeuristicsandBiases,Kahneman,PaulSlovic,andTversky,editors(1985),pp.84-98.

Homework

Therearethingswhichseemincredibletomostmenwhohavenotstudiedmathematics.

Aristotle

1) DoalltheitalicizedthingsintheReadandStudysection.

51

2) Billisintelligentbutunimaginative.Inschool,hewasstronginmathematicsbutweakinsocialstudiesandhumanities.TrueorFalse?ItismorelikelythatBillisanaccountantthatplaysjazzforahobbythanitisthatBillplaysjazzforahobby.Explainyourreasoninganddescribethetypeoffallacythisexamplerepresents,ifany.

3) SupposeMaryisauniversitystudentwholovestoplaythepiano.Isitmorelikelythatsheisamusicmajororaneducationmajor?Explainyourreasoninganddescribethetypeoffallacythisexamplerepresents,ifany.

4) TrueorFalse?Therearemorewordsthatendin“ING”thantherearewordswhose

secondtothelastletterisN.Explainyourreasoninganddescribethetypeoffallacythisexamplerepresents,ifany.

5) TrueorFalse?IfIflipacoin7times,IamjustaslikelytogetthesequenceTTTTTTTasI

amtogetthesequenceTTHTTHT.Doyouagreeordisagreewiththisstatement?Explainyourreasoninganddescribethetypeoffallacythisexamplerepresents,ifany.

6) TrueorFalse?IfIflipacoin7times,IamjustaslikelytogetexactlytwoheadsasIamtogetexactlynoheads.Explainyourreasoninganddescribethetypeoffallacythisexamplerepresents,ifany.

7) TrueorFalse?Ijustflipped7tailsinarowsonowIammorelikelytogetaheadtoeventhingsout.Explainyourreasoninganddescribethetypeoffallacythisexamplerepresents,ifany.

8) Apopulationofstudentshasanaverageheightof68inches.TrueorFalse?Itismorelikelythatonepersonchosenatrandomfromthepopulationistallerthan72inchesthanitisthatagrouptenpeoplechosenatrandomhasanaverageheightofmorethan72inches.Explainyourreasoninganddescribethetypeoffallacythatthisexamplerepresents,ifany.

9) Findsomeinnocentcollegestudenttoanswerquestions2)–8)above.Didtheyshow

anyofthemisconceptionsdescribedinthissection?Whichones?Bringtheresultstoclass.(Reallydothis.)

10) Abaseballteamfacingadouble-headerhasa70%chanceofwinningthefirstgameandan80%chanceofwinningthesecond.Assumewinningthesecondgameisindependentfromwinningthefirstgame.Whatistheprobabilitythattheywinexactlyoneofthosegames?

52

ChapterThree

StatisticsandDealingWithData

53

ClassActivity9:StudentWeightsandRectangles

Themathematicsisnotthere‘tilweputitthere. SirArthurEddington(MQS)

Whatistheaverageweightofallthestudentsregisteredforclassesatyourschoolthissemester?Inyourgroups,makeaplanforsomewaytogatherinformationtoanswerthisquestion.Bespecificaboutwhatyouwilldoandwhenyouwilldoit.Atthispoint,timeandcostisnoobject.(Waittodiscussthisquestionasaclassbeforeyouturnthepage.)

54

ClassActivity9(continued)Now,havealookatthepopulationof100rectanglesonthenextpage.Noticethateachonehasanarea(asize).Forexample,rectanglenumber74hassize6squareunitsbecauseitiscomposedof6smallsquares.Carefullyselectasampleoftenrectanglesthatyouthinkwillhaveanaveragesizethatisclosetoaveragesizeofalltherectanglesonthepage.Makesuretopersonallychooseeachoftheten.Thenfindtheaveragesizeofyourself-selectedten.Yourclasswillthenmakeafrequencygraphshowingeveryone’saverages.Next,eachpersonshouldselect10rectanglesatrandom.(Yourinstructorwillhaveaplanfordoingthis.)Thenfindtheaveragesizeofyourrandomly-selectedten.Yourclassshouldmakeanotherfrequencygraphshowingeveryone’saverages.Finally,estimate,basedonthefrequencygraphs,whatistheaveragesizeofall100rectanglesonthepage?Explainyouranswer.Whatdoesallthishavetodowiththestudent-weightproblem?

55

A Population of Rectangles

56

ReadandStudy

Heusesstatisticsasadrunkenmanuseslampposts–forsupportratherthanillumination.

AndrewLang Thisrectanglesizeproblemcontainsahugelessoninstatistics.Huge.Itisthis:humansarenogoodatselectingsamples.Ifyouwantunbiasedinformation,randomsamplesarethewaytogo.Inordertotalkmoreaboutthis,wearegoingtointroducethelanguageofsampling.Rememberthatwewantyoutolearnthesedefinitionsandtousethem.First,asamplereferstoasubsetofapopulation.Inourstudentweightproblem,thepopulationofinterestwasallstudentsregisteredforclassesatyourschoolthissemester.Inanidealworld,wejustwouldhaveeverysinglememberofthepopulationcometogetweighedandwewouldcalculatetheaverageweight.Unfortunately,thiswouldlikelyproveimpossible.Lots(most?)ofthestudentswouldnotshowuptobeweighed,andforcingthemtodosowouldbeunethical.Worse,thosewhowouldbewillingtoshowupmightbedifferentthanthosewhowouldrefuse.Peoplewhowereweight-consciousoroverweightmightbemorelikelytoavoidsteppingonyourscale.Sointheend,youwouldendupwithabiasedsample.Anothersolutionmightbetosendoutanonymoussurveystotheentirepopulationaskingeachpersontorecordhisorherweight.Butwouldyoureturnthatsurvey?Mostpeoplewouldnot(masssurveysmailedoutlikethattendtohaveverylowresponserates)andthosepeoplewhoreturnthemareagaindifferentfromthepopulation.Additionally,whenasurveyreliesonparticipantstoself-reporttheirowndata(likeweight),sometimesthatdataisn’taccurate.Okay,wellwemighttrytocalleveryoneorgodoor-to-doortocollectinformationonstudentweights.Thattypeofsamplingyieldsmuchhigherresponseratesbutcouldproveveryexpensiveandtakealotoftime(particularlyifyouhappentohavethousandsofstudentsinthepopulation).So,whatarewetodo?Hereisthestatistician’ssolution.Forgetaboutgatheringdata(information)fromeachandeverymemberofthepopulationandfocusyourtime,energyandresourcesinsteadongettinginformationfromjustarandomsampleofthepopulation.Thatwayyoucanaffordtogodoor-to-doorandgetdatafrommostmembersofthesample.Itstillwon’tbeperfect.Butitturnsoutthatthisistheverybestyoucandoinafreesociety.Whyshouldthesampleberandom?Whynotsimplycollectdataontheweightsofyourfriends,ormaybeeveryoneinyourdorm,orwhynotstandinfrontofthecampuslibraryandaskeveryonewhowalksbyfortheirweights?Becauseofthethreatof…samplingbias.

57

Asamplingprocedureisbiasedifittendstoover-sample(orunder-sample)populationmemberswithcertaincharacteristics.Thiswillmostsurelyhappenifyouself-selectyoursample–afterall,thestudentsyouknowhavedifferentcharacteristicsthanthepopulationofyourcampus.Whataresomeofthosedifferences?Stopandwritedownatleastthree.Butyoursamplewillalsobebiased,inamoresubtleway,ifyoustandinfrontofthelibraryandaskpeople“atrandom.”Thisisbecauseyouarenotreallyaskingrandompeople.Youareaskingthepeoplewhowalkbythelibraryatthattimeofday(maybeoversamplingpeoplewhostudyoftenorunder-samplingthosewhohaveaclassatthattime).Youareaskingpeoplewhohavealookatyouanddecidetheywanttoansweryourquestion(unwittinglyoversamplingpeoplelikeyourself).Giveanotherreasonwhythelibrarysamplemightbebiased.Samplingbiasissneaky.Itcreepsinbasedonwhenyousampleandwhereyousampleandhowyousample(aswellaswhoyousample).Watchforitasyouanalyzestudies.Sothesolutionistogeneratearandomsampleandthentouseyourresourcestogetthebestpossibleresponseratefromthem.Herearetwotypesofrandomsamplingthatareconsideredacceptable.Thefirstisthebest.Itiscalledsimplerandomsamplinganditismathematicallyequivalenttopullingnamesoutofahat.Everymemberofthepopulationhasthesamechanceofbeinginthesampleasanyothermemberofthepopulationandanysubsetofthepopulationhasthesamechanceofbeinginthesampleasanyothersubsetofthesamesize.Stop.Readthatlastsentenceagainandthinkaboutit.Ifyouhadyourregistrargeneratealistofallstudentsenrolledinclassesthissemesterandyouassignedeachstudentanumberandthenhadyourcalculatorgenerate100randomnumbersandusedthosenumberstogetnames,thenthatwouldbesimplerandomsampling.IfIlinedupyourclassinarowandthentookeverythirdpersontomakemysample,wouldthatbesimplerandomsampling?Explain.Thereareotherformsofacceptablesampling.Forexample,clustersamplingisrandomsamplinginstages.Forexample,ifyourandomlychosetenpagenumbersfromtheregistrar’slistofstudentsandthenrandomlychose10peoplefromeachofthosepages,thenthatwouldbeclustersampling.TheU.S.Censushasmadeuseofclustersamplingwhenithasattemptedtofindthosepeoplewhodidnotreturntheircensusforms.We’lltellthatstoryinalatersection,fornowletussaythatacensusmeansthatyougatherinformationfromeachandeverymemberofapopulation.TheU.S.Censusisbutoneexampleofanattemptatacensus.Ifyoutriedtogetweightinformationfromeverystudentregisteredatyourcampus,thenyoutoowouldbeattemptingacensus.Itisprettymuchimpossibletoconductacensuswithalargepopulationbecauseyounevergetdatafromeveryone.Ifyoucollectdatafromasubsetofthepopulation,youarereallyconductingasurvey.

58

Okay,herearetwomoretermstoknow:thenumericinformationthatyouwanttoknowaboutapopulation(ifyoucouldgetit)iscalledaparameter;thenumericinformationyougetfromasampleiscalledastatistic.Inourexample,theparameterwewereafterwastheaverageweightofallthestudentsregisteredforclassesatyourschool.Ifwecoulddoacensusofthatpopulation,wecouldfindthatparameter.However,wehavediscussedsomeofreasonswhywe’renotlikelytogetthatparameter.Instead,wewillsettleforperformingasurveyofasampleofthepopulation,andwhatwewillgetisastatistic.Hereisourilluminatingpicture:

Population Sample(fromthepopulation)

Askeveryone,it’scalledacensus. Askasubset,it’scalledasurvey.#computedfromthepopulation↔parameter. #computedfromthesample↔statistic.Thedifferencebetweentheparameterandthestatisticiscalledsamplingerror.Samplingerrorisanidea.Inpractice,whenconductingasurvey,youaren’tgoingtoknowwhatitisbecauseallyouwillhaveisastatistic.You’llprobablyneverknowtheexactparameter.Butwestillusethesewordstotalkabouttheideas.Onesourceofsamplingerrorwehavealreadydiscussed:samplebias.Anothersourceofsamplingerrorischanceerror.Chanceerroriserrorduetothediversityofthemembersofthepopulationandthefactthatyouarejustcollectingdatafromasubsetofthem.Ifthepopulationisdiverse(different)intheirweights,forexample,thenifyoupickarandomsampleofthem,youmayendupwithanaverageweightthatisabitdifferentfromthepopulationweight.Youmayeven,duetochancealone,getsomepeoplewhoareverythinorveryheavyinthesample.ThinkbacktoyourrandomsampleoftenrectanglesfromtheClassActivity.Theaveragesizeofyourrandomrectangleswaslikelycloseto,butnotexactly,theaveragesizeofall100rectanglesonthesheet.Thedifferencewasduetochanceerror.

59

Nowthinkbacktotheaveragesizeofyourtenself-selectedrectangles.Thedifferencebetweenthatnumberandtheaveragesizeofall100rectanglesonthesheetisduetobothchanceerrorandsamplebias.Howdoyoureducesamplebias?Makeyoursamplerandom.Howdoyoureducechanceerror?Makeyoursamplebigger.Sojusthowbigdoesasamplehavetobe?Itseemslikeyouwouldneedsomesignificantsamplingrate(likemaybe25%ofthepopulation),butitturnsoutthatgoodsurveyscanbedonewithquitesmallsamples.Forexample,nationalpollingtopredicttheoutcomeofthepresidentialelectionisoftendonewithonlyafewthousandpeople.That’safewthousandoutofmorethan100millionlikelyvoters.Samplingrate[(#inthesample)÷(#inthepopulation)]canbesosmallbecausepeoplearenotallthatdiversefromasamplingperspective.Thinkofitthisway:supposeyouaregoingtosampleabigpotofsouptoseehowittastes.Aslongasthatsoupisstirredupreallywell,andthechunksofmeatsandvegetablesintherearen’ttoobigortoodifferent,allyouneedisasmallbitetoknowtheflavor.Bytheway,whatwasthesamplingrateofyourrandomsamplefromtherectanglesheet?Figureitout.ConnectionstotheElementaryGrades

Ingrades3-5allstudentsshould–proposeandjustifyconclusionsandpredictionsthatarebasedondataanddesignstudiestofurtherinvestigatetheconclusionsandpredictions. NationalCouncilofTeachersofMathematics

PrinciplesandStandardsforSchoolMathematic,p.176Ideasofsamplingaresurprisinglycomplex–butgroundworkforthemcanbelaidintheelementaryschool.TheNCTMStandardsadvocatesthatstudentsingrades3-5should

“…begintounderstandthatmanydatasetsaresamplesoflargerpopulations.Theycanlookatseveralsamplesdrawnfromthesamepopulation,suchasdifferentclassroomsintheirschool,orcomparestatisticsabouttheirownsampletoknownparametersforalargerpopulation…theycanthinkabouttheissuesthataffecttherepresentativenessofasample–howwellitrepresentsthepopulationfromwhichitisdrawn–andbegintonoticehowsamplesfromthepopulationcanvary(p.181).”

Noticetheuseofthetechnicallanguage.Whatdotheymeanby“knownparametersforalargerpopulation?”Givesomeexamples.

60

Supposethatyourthirdgradershavecollecteddatafromtheirclassonthenumberoftimeslastweekthattheyeathotlunchatschoolandfoundthefollowingdatadisplayedbelowasafrequencyplot(eachXrepresents1childintheclass): X X X X X X X X X X X X X X X X X X X X 0 1 2 3 4 5 #ofHotLunchDayslastweekOfcoursetherearemanyquestionsyoumightaskyourstudentsaboutthesedata–buttherearesomequestionsparticularlyrelatedtosamplingthatshouldbeaskedanddiscussed.Questions1:Wasthereanythingspecialaboutlastweekthatmighthavemadethesedatadifferentthanifwe’daskedthesamequestionnextweekordoyouthinklastweekwasatypicalweek?Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?Question2:Howdoyouthinkthedatamighthavebeendifferentifwe’daskedthefifthgradersthesamequestion?

Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?Questions3:Woulditbeokaytolabelthisgraph“NumberofDaysEachWeekthatChildrenEatHotLunchatOurSchool?”Whyorwhynot?Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?

61

Noticethatthesequestionshelpchildrenthinkaboutwhetherthedataisrepresentativeofapopulationlargerthanthesampleof20third-gradersshownonthegraph,andithelpsthemtobegintothinkcriticallyaboutwhatthedatasaysandwhatitdoesnotsay.Samplingbecomesanexplicittopicinthemiddlegrades.

Homework

Weknowthatpollsarejustacollectionofstatisticsthatreflectwhatpeoplearethinkingin‘reality.’Andrealityhasawellknownliberalbias.

StephenColbert

1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.2) Explainthedifferencebetweenacensusandasurvey.

3) Explainthedifferencebetweenaparameterandastatistic.

4) TrueorFalse?Thebiggerthesamplethesmallerthesamplebias.Explainyour

thinking.

5) Sometimesawebsiteorpublicationwilladvertiseapollaskingitsviewersto‘callin’or‘clicktorespond’toquestionsonatopic.Inthiscasewesaythatthesampleisself-selected.Whattypesofbiasesarelikelyinthistypeofsurvey?Explain.

6) Itispossibletobiasresultsofasurveybyaskingquestionsusing“charged”language.ConsiderthesequestionsfromRepublicanNationalCommittee’s2010ObamaAgendaSurvey,andfromaDemocraticStateSenator’sdistrictsurvey.(Canyoutellwhichiswhich?)Explainhoweachquestionmightbeaskedinalessleadingmanner.

a) “Doyoubelievethatthebestwaytoincreasethequalityandeffectivenessof

publiceducationintheU.S.istorapidlyexpandfederalfundingwhileeliminatingperformancestandardsandaccountability?”

b) “DoyousupportthecreationofanationalhealthinsuranceplanthatwouldbeadministeredbybureaucratsinWashington,D.C.?”

c) “DoyoubelievethatBarackObama’snomineesforfederalcourtsshouldbeimmediatelyandunquestionablyapprovedfortheirlifetimeappointmentsbytheU.S.Senate?”

d) “Doyousupportoropposelegislationtorestoreindexingofthehomesteadtaxcredit,whichwouldboostoureconomyandhelppeopletostayintheirhomes?”

e) “DoyousupportoropposelimitinglocalcontrolandweakeningenvironmentalstandardstoencourageironoremininginWisconsin?”

62

7) TheColumbusCityCouncilwantedtoknowwhetherthe65,000citizensofthecityapprovedofaproposaltospend$1,000,000torevitalizethedowntown.Thefollowingstudywasconductedforthispurpose.Asurveywasmailedtoevery10thpersononthelistofregisteredvotersinthecity(itwasmailedto3500people)askingabouttheproposal.CitizenswereaskedtocompletethesurveyandreturnittotheCouncilbymailinaself-addressed,stampedenvelope.3100peopleresponded;ofthese,75%supportedtheexpenditureand25%didnot.

a) Thepopulationforthisstudyis

A)allregisteredvotersinthecity B)allcitizensofthecityofColumbus C)the3500peoplewhoweresentasurvey D)the3100peoplewhoreturnedasurvey E)noneoftheabove

b) Whatwasthesamplingrateforthissurvey?

c) The"75%"reportedaboveisa A)parameter B)population C)sample D)statistic E)noneoftheabove

d) Thissurveysuffersprimarilyfrom A)samplingbias B)chanceerror C)havingasamplesizethatistoosmall D)non-responsebias

e) TrueorFalse?Theresultsofthissurveymaybeunreliablebecauseregisteredvotersmaynotberepresentativeofallthecitizens.Explainyourthinking.

f) TrueorFalse?Themethodofsamplinginthissurveycouldbestbedescribed

assimplerandomsampling.Explainyourthinking.

g) TrueorFalse?Samplebiascouldbereducedbytakingalargersampleofregisteredvoters.Explainyourthinking.

63

ClassActivity10:SnowRemoval

Errorsusinginadequatedataaremuchlessthanthoseusingnodataatall.CharlesBabbage

Inordertostudyhowsatisfiedthe120,000citizensofGreenBay,WIarewithsnowremovalfromcitystreets,Iconductedthefollowingstudy.IstoodinfrontoftheGreenBayWalmartonaMondaymorninginFebruaryfrom9:00untilnoonandIaskedeverythirdpersonwhowalkedinwhethertheyweresatisfiedwithsnowremovalfromstreetsinGreenBay.Onehundredandeightpeoplesaidtheyweresatisfied.Twenty-fivepeoplesaidtheywerenot,and18refusedtoanswermyquestion.

Answerthebelowquestioninyourgroups.

1) Describethepopulationforthestudy.

2) DidIconductacensusorasurvey?Explain.

3) Whatwasthesamplingrate?(Writethisinseveraldifferentways.)

4) Whatwastheresponserate?(Writethisinseveraldifferentways.)

5) TrueorFalse?Thisstudysuffersprimarilyfromnon-responsebias.Explain.

6) TrueorFalse?Thisstudysuffersprimarilyfromsamplingbias.Explain.

7) TrueorFalse?Theresultsofmystudymaynotbevalidbecausecityemployeesmayhavebeenpartofthesample.Explain.

8) TrueorFalse?Amainproblemwithmystudyisthatmysamplesizeistoosmall.

Explain.

9) TrueorFalse?PeopleshouldconcludefrommystudythatGreenBaycitizensaregenerallyhappywithsnowremovalfromcitystreets.

10) Describeatleastthreewaystoimprovemystudy.

64

ReadandStudy

Youcanuseallthequantitativedatayoucanget,butyoustillhavetodistrustitanduseyourownintelligenceandjudgment.

AlvinToffler

“Fouroutoffivedentistssurveyedrecommendsugarlessgumfortheirpatientswhochewgum.”You’veheardthatclaim.Itisperhapsthemostwellrememberedstatistical‘fact’inthehistoryofmarketingresearch.Butwhatdoesittellyou?Themakersofsugarlessgumhopethisstatisticconvincesyoutopurchasetheirproduct.Butbeforeyourunoutforapackofsugarlessgum,youshouldaskafewquestionsaboutthis‘fact.’Howwerethedentistsinthesamplechosen?Whatwastheresponserate?Whatwasthesamplesize?Whatwastheexactquestiontowhichthedentistswereaskedtorespond?Andevenifitturnsoutthatthesurveyusedexemplarymethodology,notethattheresultdoesnotsuggestyouthatyoushouldtakeupchewingsugarlessgum.Itonlysuggeststhatifyoualreadychewgum,thesedentistssuggestyouchewsugarlessgum.Allsurveysarenotcreatedequal.Infactsomearedownrightmisleading.Itisimportantforyoutothinkcriticallyaboutinformationbasedonsurveydata.Surveysaremeanttoprovideinformationaboutapopulation.Theycanhelpcompaniesunderstandhowtomarketproducts;theycanhelpgovernmentalagenciesunderstandconstituents;theycanhelpcampaignspredictpublicreactionstoideasorevents;theycanhelpsecondgradeteachersunderstandwhichmoviestheirstudentsprefer.Hereisanexampleofonerealsurvey.TheUnitedStateCensusBureauusessurveystounderstandthepopulationofourcountry.(Bytheway,asteachersyoucangetgooddataforyourstudentstoanalyzefromtheU.S.CensusBureauwebsite.Keepthatinmind.)Forexample,theyconductasurveyofU.S.housingunitseveryotheryearinordertomakedecisionsaboutfederalhousingprojects.Intheirwords,theiraimisto“[p]rovideacurrentandongoingseriesofdataonthesize,composition,andstateofhousingintheUnitedStatesandchangesinthehousingstockovertime.”Hereisafigureshowingsomeoftheresultsofthe2007AmericanHousingSurvey(AHS).Takeafewminutestostudyit.

65

U.S.CensusBureau,CurrentHousingReports,SeriesH150/07AmericanHousingSurveyfortheUnitedStates:2007.U.S.GovernmentPrintingOffice,Washington,DC.

66

Theseresultsdrivenationalpolicy,sowehopethattheAHSiswelldesigned.Let’shavealookatthemethodology.ReadthefollowingparagraphbasedontheinformationfromtheUSCensusBureauwebsite:

TheAHShassampledthesame53,000addressesineachodd-numberedyearsince1985.Thoseaddresseswerechoseninthefollowingmanner.

“FirsttheUnitedStateswasdividedintoareasmadeupofcountiesorgroupsofcountiesandindependentcitiesknownasprimarysamplingunits(PSUs).AsampleofthesePSUswasselected.ThenasampleofhousingunitswasselectedwithinthesePSUs”(U.S.CensusBureau,2007,p.B-1).

TheAHSincludesbothoccupiedandvacanthousingunitsandallpeopleinthosehousingunits.Dataiscollectedusingcomputer-assistedtelephoneandpersonal-visitinterviews.Whenahousingunitwasvacant,informationwasgatheredfromneighbors,rentalagents,orlandlords.TheAHSisavoluntarysurveymeaningthathouseholdscandeclinetoanswerquestions.Interviewsforthedatapresentedabovewereconductedfrommid-ApriltoSeptember2007.

Okay,shouldwebelievetheAHSresults?Let’stalkthroughananalysisofthesurveydesign.First,whatisthepopulationforthissurveyandwhatisthesample?Accordingtothe2001census,thereareabout200,000,000housingunitstotalintheUnitedStates–thatistheapproximatepopulation.Thesamplecontainedabout53,000ofthesehomes.Thatgivesasamplingrateofabout0.0027%.Thismightseemsmall,butthesamplewascarefullyselectedusingclustersamplingandsothesamplingerrorwillbefairlysmalltoo.(Whywasthisclustersampling?Explainwhybasedonthedefinition.)Inotherwords,thereisreasontobelievethatthehouseholdsinthesamplearerepresentativeofhouseholdsintheUnitedStates.Let’sgoonestepfurtherinthinkingaboutthequalityofthedata.Itcouldbethatthesamplewaswell-chosen,butthatmanyhouseholdsgavefaultyinformation(orsimplyrefusedtogiveinformationatall).LuckilytheCensusBureauusedthebestpossiblemethodforcollectingdata–theycalledfirst,andiftheycouldnotreachahouseholdbyphone,theysentsomeonetotheaddresstoconductaninterviewwitheithertheoccupants,orinthecaseofanemptyunit,aneighbororlandlordorrealestateagent.Thislikelyproducedafairlyhighresponserate.(Infact,theBureaureportsan89%responserate.Intheothercaseseithernoonewasathomeevenafterrepeatedvisits,theoccupantsrefusedtobeinterviewed,ortheaddresscouldnotbelocated.)So,ourtakeisthatthemethodologyfortheAHSissoundandtheresultsarebelievable.Wewantyoutolearntothinklikethiswheneveryouneedtojudgethebelievabilityofaclaimbasedondata.

67

Homework

USATodayhascomeoutwithanewsurvey--apparently3outofevery4peoplemakeup75%ofthepopulation.

DavidLetterman

1) DoalltheitalicizedthingsintheReadandStudysection.2) Spendafewminuteslookingatthefigureonoccupiedhomesfromthe2007AHS.

Writeaparagraphhighlightingthethingsthatyoufindmostinterestingornotableabouttheseresults.

3) Inordertofindouthowitsreadersfeltaboutitscoverageofthe2008Presidential

Election,alocalnewspaper(withacirculationofabout10,000)ranafront-pagerequestintheSundaypaperthatreadersgototheirwebsiteandcompleteanonlineanonymoussurveythatcontainedthefollowingtwoquestions:

I. Whatisyourpoliticalaffiliation?____Democrat____Republican____Independent____Other

II. Howwouldyourateourcoverageofthe2008presidentialelection?____slantedleft____slantedright____unbiased____noopinion

Sixhundredreadersresponded.Ofthose,70%ofDemocrats,40%ofRepublicans,66%ofIndependents,and20%of“Other”saidthatthecoveragewasunbiased.

a) Analyzethesamplingmethod.Isthesampleofrespondentslikelytoberepresentativeofthepopulationthepaperwantstorepresent?Givespecificreasonswhyorwhynot.

b) Whatdoesthedatatellusaboutthepaper’scoverageofthe2008PresidentialElection?Explain.

c) Describesomewaystoimprovethissurvey.

4) GototheUnitedStatesCensusBureauwebsite(http://www.census.gov/schools/)andfindtheTeachersandStudentslink.Spend15minutesbrowsingtheactivitiesthere.ThendesignamathematicslessonforGrade3usingtheStateFactsforStudentslink.

68

5) AresearcherwantstopredicttheoutcomeoftheFrostbiteFallsmayoralelection.Inordertodothis,shemailedasurveytoeverytenthpersononalistofall40,980registeredvotersinthecity.Onethousandfifty-sixpeoplereturnedthesurvey.34%ofrespondentssaidtheysupportedSnidelyWhiplash;48%saidtheysupportedDudleyDoRight;theremaining18%wereundecided.

I. Thepopulationforthisstudyis

A)allmayoral-racevotersinthecityofFrostbiteFalls B)allcitizensofthecityofFrostbiteFalls C)thepeoplewhoweresentasurvey D)thepeoplewhoreturnedasurvey E)noneoftheabove

II. Theresponseratewas___________.III. Thesamplingratewas___________.

IV. The"48%"reportedaboveisa

A)parameter B)population C)sample D)statistic E)noneoftheabove

V. Thissurveysuffersprimarilyfrom

A)selectionbias B)nonresponsebias C)chanceerror D)havingasamplesizethatistoosmall

VI. TrueorFalse?Theresultsofthissurveymaybeunreliablebecausemembersofthecitygovernmentmayhavebeenincludedinthesample.Explainyourthinking.

VII. TrueorFalse?Themethodofsamplinginthissurveycouldbestbedescribed

assimplerandomsampling.Explainyourthinking.

VIII. TrueorFalse?Biascouldbereducedbytakingevery5thpersononthelistofregisteredvotersratherthanevery10th.Explainyourthinking.

69

ClassActivity11:ClassSurveyEveryonetakessurveys.Whoevermakesastatementabouthumanbehaviorhasengagedinasurveyofsomesort.

AndrewGreeleySupposeyourgrouphasbeenhiredbytheAmericanFederationofTeachers.Youarechargedwithdesigningasix-itemsurveytobegiventothestudentsinyourclassforthepurposeofbetterunderstandingsomeaspectofteacherpreparationatyourschool.Thesurveyshouldbewrittenaroundatheme(cost,content,methods,qualityofinstruction,opportunitiesforworkwithchildren,job-outlook–oranyotherrelevantthemechosenbyyourgroup).Threeofyourquestionsshouldrequestcategoricalinformation,andthreeshouldrequestnumericalinformation.Youaregoingtodisplayandanalyzethisdatainafutureactivitysowritecarefully-wordedquestions,andifyouprovideresponseoptionstosomeitems,thinkcarefullyaboutthosetoo.Onceyoursurveyisturnedintotheinstructor,heorshewillreproduceitverbatimforyourclasstotake;youwillnothavetheopportunitytoclarifyitemsatthetimethesurveyisgiven.Inwhatwaysisyourclassrepresentativeofpre-serviceteachersatyourschool?Inwhatwaysmightyounotberepresentativeofallpre-serviceteachersatyourschool?

70

ClassActivity12a:NameGames

Ifeellikeafugitivefromthelawofaverages. WilliamH.Mauldin

1) Eachpersonintheclassshouldwritehisorherfirstnameonstickynotessothatthere

isoneletteroneachoftheperson’snotes.(Soforexample,IwouldwriteJononenote,Eonanother,andNonthethird.)Putyournamesinstacksontheboardlikewedidintheexamplebelow.Nowyourclassneedstorearrangeyourclasses’stickynotes(orevenpartsofstickynotes)sothateverystackendsupwiththesamenumberofletters.Dothatnow.

AmiJen Mo Omar…

Themean(oraverage)ofasetofnumericalobservationsisthesumofalltheirvaluesdividedbythenumberofobservations.Explainwhyitmakessensethatyourmethodjustcomputedthemeannamelengthforyourclass.

2) Nowwewillmakeadifferenttypeofgraph,abargraph.Eachpersonshouldgetonestickynotetoplaceonthegraphbasedonthelengthofhisorherfirstname.ForexampleiftherewereeightpeopleintheclasswithnamesMo,Ami,Jen,Omar,Karen,Steve,Sienna,andJuanitathebargraphwouldlooksomethinglikethis:

frequencies

____________________________________ 234567 NameLengths

Figureoutawaytorearrangethenotestohelpyoutofindthemeannamelengthonagraphlikethis.Whydoesitmakesensethatyourmethodjustcomputedthemeannamelengthforyourclass?Howisthismethoddifferentfromthemethodyouusedinquestion1?Discussthis.

(Thisactivitycontinuesonthenextpage.)

71

ClassActivity12a(continued):

3) Themedianofasetofnumericalobservationsisthemiddleobservationwhenthedataisplacedinnumericalorder.Inthecasewheretherearetwo“middle”observations,themedianistheaverageofthem.Decidehowyoucoulduseeachtypeofgraphabovetohelpyoufindthemediannamelength.

4) Whichislikelytobelargerforyourclass?Themeannumberofsiblingsorthemediannumber?Explain.Then,collectthedataandseeifyouarecorrect.

72

ClassActivity12b:IntheBalance Sodivinelyistheworldorganizedthateveryoneofus,inourplaceandtime,isin

balancewitheverythingelse.JohannWolfgangvonGoethe

Yourtaskistoputweightsonthenumberedpegsofabalancescaletokeepthesystembalanced.

1) Findseveraldifferentarrangementsof3weightssothatthesystemwillbalance.Makeageneralconjectureaboutallthearrangementsthatwillbalancethesystem.

2) Ifyouhadoneweight1totherightandtwoweights2totheleft,predictwhereyouwouldneedtoplaceafourthweighttobalancethesystem.

3) Findseveralwaystobalancefourweightswhereonlyoneweightisontheleftside.Makeageneralconjectureaboutallthearrangementsofthistypethatwillbalancethesystem.

4) Predictsomearrangementsthatwillbalancefor5weights.Makeageneralconjecture

aboutallthearrangementsof5weightsthatwillbalancethesystem.

5) Supposethereare4catswithmeanweightof10pounds.Youknowtheweightsof3ofthecats.Thoseweightsare:4pounds,7pounds,and15pounds.Howcanyouusethebalancetofigureouttheweightofthe4thcat?(Don’tdoacalculation.Figureitoutwiththebalanceonly.)

6) Usethebalancetofigureoutthisproblem:Youhavetakenfourmathquizzes.Your

scoreswere82,67,85,and72.Whatdoyouneedtoscoreonthe5thquizforyouraveragequizscoretobe75?(Don’tdoacalculation.Figureitoutwiththebalanceonly.)

73

ReadandStudy

Theaverageadultlaughs15timesaday;theaveragechild,morethan400times. MarthaBeck

Therearethreestatisticsthatarecommonlyusedtodescribethecenterofadistributionofnumericaldata:mean,medianandmode.Themeaniscomputedbysummingthevaluesanddividingbythenumberofthem.Butabetterwayforchildrentothinkaboutmeanisusingtheideaof“eveningoff.”Themeanistheaveragevalue,thevalueeachpersonwouldgetifallthedatawassharedevenly.Forexample,supposethepicturebelowshowsthecookiesthatbelongtoeachchild.

Keesha’scookies:

Andi’scookies:

Tim’scookies:

Myosia’scookie:

Carlos’cookie: Asyousawintheclassactivity,inordertofindthemeannumberofcookies,thechildrencouldimaginesharingcookies(orevenpartsofcookies)sothattheyeachchildhasexactlythesamenumber.Bytheway,isthisdatadisplayliketheoneinproblem1)ortheoneinproblem2)fromtheClassActivity?Explain.

74

Keesha’scookies:

Andi’scookies:

Tim’scookies:

Myosia’scookies:

Carlos’cookies: Sothechildrenseethatthemeannumberofcookiesis3.Inthiscasewedidn’tneedtobreakupcookiesinordertosharethem,butyoucertainlyshoulddoexampleslikethatwithyourstudents.Whatwehaveillustratedhereistheredistributionconceptofthemean:thatthemeanvalueofasetofnumbersiswhateachobservationwouldgetifthedatawasredistributedtothateachobservationhadthesameamount.Thisconceptisessentiallythesameasthepartitiveconceptofdivision:takingatotalamountandsharingitequallyintoanumberofgroups,andseeinghowmucheachgroupgets.Inclassactivity12b,wealsodiscoveredthebalancepointconceptofthemean,namely,thatthemeanisthepointinadistributionwherethesumofthedistancesthatthedataisabovethemeanbalanceswiththesumofthedistancesthatthedataisbelowthemean.Themodeofasetofdataisthevaluethatoccursmostfrequently.Whatisthemodeforthecookiedata?Sometimesadatasetwillhavemanymodesbecauselotsofvalueswilloccurwiththesamemaximumfrequency.Theideaofmedianistheideaofmiddleorcentervalue.Sowesawthatthenumberofcookiesheldbythefivepeopleis4,3,7,1,and1.Ifweputthesevaluesinnumericalorder,weget:

75

1 1 3 4 8Whydoweneedtoputtheminnumericalorderfirst?Howwouldyouexplainthistoachild?Weseethatthemiddlevalueis3(we’llwriteM=3.)Inthecaseofanevennumberofvalues,wehavenoonemiddlevalue,andsothemedianistheaverageofthetwomiddlevalues.Findthemean,mode,andmedianofmybowlingscores:

112 114 120 123 127 127 130 167 220Whydoesitmakesensethatthemeanislargerthanthemedianforthesedata?Explain.WewilldefineQ1(thefirstquartile)asthemedianofthedatapositionedstrictlybeforethemedianwhenthedataisplacedinnumericalorder.Forexampleusingmybowlingscores,127isthemedian.

112 114 120 123|127| 127 130 167 220Sothescores112,114,120and123arepositionedbeforethemedian.Themedianofthosescoresis(114+120)/2=117.SoQ1=117.Q3(thethirdquartile)isthemedianofthedatapositionedstrictlyafterthemedianwhenthedataisplacedinnumericalorder:127,130,167and220.Q3=148.5.Nowwecanreportsomethingwecallafive-numbersummaryforthesescores:Minimum=112, Q1=117, M=127, Q3=148.5, andMaximum=220

76

Bythewaychildrenmaywanttoknowifduplicatesinthedatacanbethrownout.Whatwouldyousaytothem?Why?Whataboutvaluesofzero–couldtheybediscarded?Whatwouldyousaytothechildrenaboutthat?Weusedasmalldatasetforthepurposeofillustratingtheseideas.Wenotethatforlargedatasets,thesestatistics(mean,median,mode,quartiles,andpercentiles)aretypicallycomputedusingacalculatororcomputer.We’llgiveyoutheopportunitytousetechnology-ifyouhaveit-andworkwithalargerdatasetinalaterclassactivity.Asateacher,youwillneedtodiscusswithparentsthemeaningoftheirchild’sscoreonastandardizedexam,solet’sspendafewminutesontheideasthatyoumightfindusefulinthatcontext.Typicallythesescoresarereportedaspercentiles.Ifachildinyourclassscoresinthenthpercentilethismeansthat“npercent”ofthechildrenwhotookthetestscoredatorbelowthatscore.Forexample,ifachildscoresinthe80thpercentileonatest,thatmeansthat80%ofthechildrenwhotookthetestscoredatorbelowherscore.Anotherwaytothinkaboutthisistoimagineliningup100childrenaccordingtotheirtestscores,thischildwouldbenumber80fromthebottomintheline(ornumber20fromthetop).Inthiscontextthemedianisthe50thpercentile.WhatpercentileisQ3?ConnectionstotheElementaryGrades

Ingrades3-5allstudentsshouldusemeasuresofcenter,focusingonthemedian,andunderstandwhateachdoesanddoesnotindicateaboutthedataset.

NationalCouncilofTeachersofMathematics PrinciplesandStandardsforSchoolMathematics,p.176

Childreningrades3-5caneasilyunderstandandcomputemedianandmode,buttypicallytheideaofmeanisamoredifficulttopic.Thisisnotbecauseitishardtocalculateamean;ratheritisbecauseitismoredifficulttounderstandtheideaofamean.Belowisanexampleofanactivityappropriateforchildreningrade4thatrepresentsanopportunityfortheteachertodiscussmeanasaredistributionprocessratherthanasacomputation.Explainhowyoucouldusestickynotestohavechildrenuseevening-offtofindthemeanofthebelowdatawithouteverdoingthestandardcomputation.Reallydoit.

77

Thefamilysizesforchildreninourclassareshownbelow.Makeabargraphforthesedataandthenfindthemeannumberoffamilymembers.

Family sizeJenson 8Lewis 3McDuff 7EarlesSeaman

45

Ortiz 4PetersToddMiene

653

Weknowthatthestandardcomputation(addthevalues,thendividebythenumberofvalues)isquickerand‘easier’todo.Butasanelementarymathematicsteacher,yourjobistohelpyourchildrenunderstandbigideas–andnottogivethemshortcutsthatallowthemtogetrightanswerswithoutunderstandingtheideas.Childrenmustlearnthatmathematicsmakessense,andthatthinking(insteadofmemorizing)isthewaywedomathematics.

Homework

Wecannotteachpeopleanything;wecanonlyhelpthemdiscoveritwithinthemselves.

GalileoGalilei

1) DoalltheitalicizedthingsintheReadandStudysection.2) DoalltheitalicizedthingsintheConnectionssection.

3) Lookatthe“FulcrumInvestigation”inUnit8,Module1,Session3ofthe4thgrade

BridgesinMathematicscurriculum.Inthisactivity,studentsaregivenapencil,a12inchrulerandseveral“tiles”(weightsorpennies)allofthesameweight.Byplacing6tileson“0”ontheruler,theycansimulatea60-poundfourthgradestudent(Sarah)sittingontheendofaseesaw.Anotherweightisplacedontheotherendoftheseesaw(at12inches).Ifthefulcrumisinthemiddleofthepencil(at6”),thenitwouldtake6tiles(representing60pounds)attheotherendto“lift”orbalanceSara.Usinganactualpencil,ruler,andweights(suchaspennies),predictthenexperimenthow

78

muchweightittakestobalanceSarahifthefulcrumisplacedatthe4inchmark,the5inchmark,the7inchmark,andthe8inchmark.

a. Answerquestions1-6inthe“FulcrumInvestigation”.b. HowcanyouusetheconceptoftheMeantoanswerquestions5and6inthis

investigation?c. A“ChallengeQuestion”inthenextsessionasks“IfSarahcan’tmovethe

fulcrumfromthemiddleoftheseesaw,whatcanshedotobalancewiththe40-poundfirstgrader?Whydoesthismethodwork?”Answerthisquestion.Howdoyouthinkfourthgraderswouldbeabletothisout?Howcanyouusetheconceptofthemeantofigurethisout?

d. Repeatpartc,butnowsupposeshewastobalancewiththe100-pound7thgraderwithoutmovingthefulcrum.

4) Hereisadatasetofthenumberofyearsmembersofthemathdepartmenthavebeen

employedatmyschool:7,25,7,15,8,23,27,6,1,3,28,22,24,9,15,14,18,8.Computethemean,medianandmodeforthesedata.Whatdothesedatatellyouaboutthegroupofmathematicsfacultyatthisschool?

5) DecidewhethereachofthefollowingisTrueorFalse.Ineachcase,arguethatyouare

right.

a) T F Iftenpeopletookanexamthenitispossiblethatallbut oneofthemscoredlessthanthemean.

b) T F Iftenpeopletookanexamthenitispossiblethatallbut oneofthemscoredlessthanthemedian.c) T F Iftenpeopletooka100pointexam,itispossiblethatthe meanandmedianare90pointsapart.d) T F Iftenpeopletookathousandpointexam,itispossiblethe themeanandthemedianare90pointsapart.

e) T F Themeanincomeofyourtownislargerthanthemedian income.

6) Supposethattenpeoplegraduatedwithadegreeincomputersciencefromyour

schoolandnowtheirmeanannualincomeis$75,000.Ifmoneyisyouronlyconcern,doesthisconvinceyoutomajorincomputerscience?Explain.

7) Supposethattenpeoplegraduatedwithadegreeincomputersciencefromyour

schoolandnowtheirmedianannualincomeis$75,000.Ifmoneyisyouronlyconcern,

79

doesthisconvinceyoutomajorincomputerscience?Isitmoreconvincingorlessconvincingthanthescenariointhepreviousproblem?Explain.

8) Supposethatonehundredpeoplegraduatedwithadegreeincomputersciencefrom

yourschoolandnowtheirmeanannualincomeis$75,000.Ifmoneyisyouronlyconcern,doesthisconvinceyoutomajorincomputerscience?Explain.Isthismoreorlessconvincingthanthescenariosintheprevioustwoproblems?Explain.

9) Annneedsameanof80onherfiveexamsinordertoearnaBinherclass.Herexam

scoressofarare78,90,64and83.WhatdoessheneedtogetonherfifthexamtoearntheB?Dothisprobleminatleasttwodifferentways.Onewayshouldusetheredistributionconceptofthemean.Anothershouldusethebalancepointconceptofthemean.

10) Awaytothinkaboutthemedianofasetofnumbersisthatitisthevaluethatsplits

thedistributionintotwoequal‘counts.’Considerthisfrequencygraphof18scoresonaquiz:

X X

X X X X X X X

X X X X X X X X X 1 2 3 4 5 6 7 8 9 10

ScoreonaQuiz(outoftenpossiblepoints)

a) Findthemedianscoreandexplain,asyouwouldtoastudent,whyitmakessense.b) Wesaythatthisdistributionisskewedleftbecauseithasanasymmetrictail(or

evenoutliers)totheleft.Inskewedleftdata,doyouexpectthemeantobehigherorlowerthanthemedian?Explainyourthinking.

c) Nowimaginethatthedistributionaboveshows18equalweightsarrangedonaseesaw.Ifyouwantedtheseesawtobalance,youwouldneedtoplacethefulcrumatthemean.Inotherwords,youcanthinkaboutmeanasthebalancingpointofthedistribution.Findthemeanandseeifitlookslikethebalancingpoint.Whydoesitmakesensethatthemeanissmallerthanthemedianforthesedata?

11) Achildinyourfourth-gradeclassscoreda200outof500onastandardized

mathematicsassessmentinwhichtheaveragescorewas240.Thisstudentisreportedasscoringinthe35thpercentile.Youneedtoexplaintotheparentsexactlywhatthismeans.Whatdoyoutellthem?Areyouconcernedaboutthischild’sperformance?Explainyourthinking.

80

ClassActivity13:MeasuringtheSpread

It’snotthatGooddoesn’ttriumphoverEvil,it’sthatthepointspreadistoosmall. BobThaves

Herearetheexamscoresfortwostudents,AndreasandBelle:Andreas:45,57,69,69,80,94 Belle: 54,70,71,72,72,77Whichofthesestudentshadbetterscores?Explainhowyoudecidedbeforeyoureadfurther.Noticethatthesestudents’scoresareverysimilarbasedonthemeasuresofcenter(meanandmedian)–buttheyarenotsosimilarbasedonthespreadoftheirscores.Spreadisanideathatiscapturedbydifferentstatistics.Herearethreecommonmeasuresofspread:

1) Range=maximum-minimum.Computetherangeforeachstudent’sscores.Whatisitthatrangemeasures?Beasspecificasyoucan.

2) Inter-QuartileRange(IQR)=Q3–Q1.ComputetheIQRofeachstudent’sscores.What

isitthattheIQRmeasures?Beasspecificasyoucan.

3) MeanAbsoluteDeviation(d)= [|(x1–m)|+|(x2–m)|+|(x3–m)|+…+|(xn–m)|].Inthisformulanisthenumberofvalues,misthemean,andeach“x”isaspecificvalue.Recallthat,forexample|-3|means“theabsolutevalueof-3.”ComputethemeanabsolutedeviationofAndreas’scoresandthenBelle’sasagroup.Whatisitthatthemeanabsolutedeviationmeasures?Beasspecificasyoucan.

(Thisactivitycontinuesonthenextpage.)

n1

81

ClassActivity13continued:Nowwehavethemachineryweneedtoidentifyanoutlierinthedata.First,adefinition:Anoutlierisaspecificobservationthatlieswelloutsidetheoverallpatternofthedata.Youmightask,whatdoesitmeantobe“welloutside”theoverallpatternofthedata?Gladyouasked.Wehaveacriterionforthat.Itiscalledthe1.5×IQRcriterion:avalueisconsideredanoutlierifitfallsmorethan1.5×IQRaboveQ3orifitfallsmorethan1.5×IQRbelowQ1.Let’susethiscriteriontotestAndreas’scoresforoutliers:45,57,69|69,80,94M=(69+69)/2=69andsitsinthepositionheldbythebarintheabovedataset,sothescores45,57and69arepositionedbelowthemedian(M).Q1=57andlikewise,69,80and94arepositionedabovethemediansoQ3=80.

IQR=Q3-Q1=80–57=23 1.5×IQR=34.5 Soanvalueisanoutlierifitismorethan34.5pointsaboveQ3ormorethan34.5pointsbelowQ1.Inotherwords,outliersinAndreas’dataarebiggerthan80+34.5=114.5orsmallerthan57–34.5=22.5.Sincetherearenoscoresthatbigorthatsmallinhisdata,hehasnooutliersbasedonthecriterion.CheckBelle’sscoresforoutliersusingourcriterion.HowcanitbethatBellehasanoutlieramongherscoreswhenAndreasdoesnot?Explain.

82

ReadandStudy

Americaisanoutlierintheworldofdemocracieswhenitcomestothestructureandconductofelections.

ThomasMannIntheClassActivityyouworkedwiththreedifferentstatisticsthatdescribethespreadofthedata.Therangeshowsyoutheentirespreadofthedataset.TheIQRisameasureofthespreadofthemiddlehalfofthedata.Themeanabsolutedeviationmeasureshowspreadoutthedatais(onaverage)arounditsmean.Wewillusetwoprimarystatisticsfordeterminingthecenterofadistributionofdata:meanandmedian,andtwoprimarymeasuresofspreadofthedata:meanabsolutedeviationandinterquartilerange(IQR).So,howdotheyfittogetherandwhywouldwechoosetoreportoneoveranother?Ourfirstconsiderationisthis:doesthedatacontainoutliers?See,weknowthatanoutliercanhaveabigaffectonthemean(butnotonthemedian).Forexample,considerthesetestscoreswithmean75.44andmedian78:

67,68,68,71,78,79,80,83,85Beforeyougoanyfarther,checkthesescoresforoutliersusingthe1.5×IQRcriterion.Now,supposethatthescoreof67wasreplacedbyascoreof12.(Noticethat’sanoutlier.)Thischangehasnoeffectonthemedian(whichisstill78)butitdoesaffectthevalueofthemean.Computethenewmeantocheck.Wesaythatthemedianis“resistanttooutliers”andthatthemeanis“affectedbyoutliers.”Ifyourdatacontainsanoutlier(orseveraloutliers)thataffectthemean,thenthemedianisoftenthebetter(morehonest?)measureofcenterforyoutoreportforthosedata.Whichofthemeasuresofspreadisresistanttooutliersandwhichisaffectedbythem?Takeafewminutestofigureitout.

83

Nowwearegoingtointroduceadisplayofthefive-number-summaryofadataset:theboxplot.Recallthatthefive-number-summaryconsistsoftheminimum,Q1,themedian,Q3,andthemaximumvaluewhenthedataisarrangedinnumericorder.Hereissomefictitiousdataonthenumberofyearsmathematicsfacultymembershavebeeninourdepartment.Takeaminutetofindthemedianandthequartilesforthesedata. 1,3,6,7,7,8,8,9,14,15,15,18,22,23,24,25,27,28Hereisanexampleofaboxplotdisplayofthedataonnumberofyearsmathematicsfacultymembershavebeeninourdepartment.

NumberofYearsofService

Noticethataboxplotisnothingmorethanapictureofthefive-numbersummary.The“box”partshowsQ1,M,andQ3,andthe“arms”stretchoutthereachtheminandmaxoneitherside.Whenmakingaboxplot,besurethatyournumberlinehasaconsistentscale,thatyoulabelyourdisplay,andthatyouprovideinformationaboutthenumberofvalues(inthiscaseN=18)shownontheplot.

Homework

Ohwhocantelltherangeofjoyorsettheboundsofbeauty? SaraTeasdale

1) DoalltheitalicizedthingsintheReadandStudysection.

84

2) Answerallofthequestionsinthe“Mean,Mode&Range”probleminUnit4,Module4,Session2intheBridgesinMathematicsGrade4StudentBook.Then,asafutureteacher,considerthefollowingquestions:Whymightachildanswer17inchesonquestion6?Whyachildanswer36inches?Whymightachildanswer42inches?Whymightachildanswer52inches?

3) Consideragainthislinegraphof18scoresonaquiz:

X XX X X

X X X X X X X X X X X X X

1 2 3 4 5 6 7 8 9 10 ScoreonaQuiz(outoften)

a) Whatistherangeofthesedata?b) WhatistheIQR?c) TrueorFalse?(Dothiswithoutperformingacalculation.)Themeanabsolute

deviationofthesedataisabout5.Explainyourthinking.d) Areeitherofthescores1or2anoutlier?Useourcriteriontocheck.e) Whatmeasureofcenterandwhatmeasureofspreadwouldyoureportfor

thesedataandwhy?f) Makeaboxplotofthesedata.g) Howcouldyouusetheboxplottoquicklyspotoutliersinthedata?Explain.

4) Herearethosebowlingscoresonceagain:

112 114 120 123 127 127 130 167 220

a) Which(ifany)ofthesescoresareoutliersbasedonourcriterion?b) NowgototheintroductiontothistextandreadtheCommonCoreState

StandardsforGrade6.Noticethattheyaskthatchildren“summarizenumericaldatasetsinrelationtotheircontext.”Dothethingstheysuggestforthesetofbowlingscores.

i. Reportingthenumberofobservations.ii. Describingthenatureoftheattributeunderinvestigation,including

howitwasmeasuredanditsunitsofmeasurement.iii. Givingquantitativemeasuresofcenter(medianand/ormean)and

variability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.

iv. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.”

85

5) EditorsofEntertainmentWeeklyrankedeverysingleepisodeevermadeofStarTrek:TheNextGenerationfrombest(ranked#1)toworst(ranked#178).Thentheycompiledtherankingsbasedontheseasoninwhichtheepisodeaired.Belowareboxplotsshowingtherankingsofepisodesineachofthesevenseasons(fromWorkshopStatistics,p.89):

StarTrekRankingsplottedbySeason

a) Whichwasthebestoverallseasonandwhichwastheworstbasedontheserankings?Makeanargumentineachcase.

b) Doanyoftheseseasonshaverankingsthatareoutliers(forthatseason)?Explain.

c) Whichseason(s)hasadistributionofrankingsthatisskewedleft?Explain.d) Createaseriesofgoodquestionsyoucouldaskupperelementarystudents

abouttheseboxplots;thenanswerthemyourself.

6) Thisisameanabsolutedeviationcontest.Youmayuseonlynumbersintheset{1,2,3,4,5}(Youcanusethesamenumberasmanytimesasyoulike).

a) Selectfournumberswiththelowestmeanabsolutedeviation.(Actually

computeit).Arethosetheonlyfournumbersthatgivethelowestmeanabsolutedeviation?

b) Selectfournumberswiththehighestmeanabsolutedeviation.(Actuallycomputeit).Arethosetheonlyfournumbersthatgivethehighestmeanabsolutedeviation?

86

7) Whichislikelybigger?

a) Themeannew-housecostinSeattle,WAorthemediannew-housecost?Explain.

b) Themeanorthemedianinaskewed-leftdistribution?Explain.c) Therangeofasetofnumbersorthemeanabsolutedeviationoftheset?

Explain.d) Thesecondquartileorthemedian?Explain.e) Now,giveanexampleofasetof8numberswhosemeanis1000biggerthan

itsmedianorexplainwhyitisimpossibletodoso.

87

ClassActivity14:TheMatchingGame

Letthepunishmentmatchtheoffense. Cicero

Onthefollowingpageyouwillfindbargraphsdisplayingdistributionsforeightdifferentsetsofexamscores.Thenumericalsummariesforthosedatasetsarelistedbelow.

DataSet Mean Median Mean

AbsoluteDeviation

A

44.5 44.0 7.7

B

37.0 36.0 4.0

C

32.5 31.5 12.0

D

40.0 41.5 25.5

E

11.5 3.0 20.0

F

40.0 38.5 11.0

G

33.5 34.5 10.0

H

20.5 20.5 10.0

1) Thesamenumberofstudentstookeachexam.Howmanywasthat?Explainhowyouknow.

2) Matcheachsummarywithadistribution.Makesuretowriteexplanationsforyourchoices.

3) Oneofthebargraphsismissing.Figureoutwhichdatasethasthemissingbargraph,thencreateapossiblebargraphforthatdataset.

4) Whatarethingsthatanupperelementarystudentcouldlearnbyworkingonthisactivity?Bespecific.

88

Insertbargraphsfromp.23ofoldmanualhere.

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80

Frequency

C2

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80

Frequency

C3

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80

Frequency

C4

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80

Frequency

C5

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80

Frequency

C6

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80

Frequency

C7

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80

Frequency

C8

89

ReadandStudy

Anapprenticecarpentermaywantonlyahammerandsaw,butamastercraftsmanemploysmanyprecisiontools.

RobertL.KruseInthissectionwearegoingtotalkspecificallyaboutsomewaystodisplaydataandaboutfeaturesofdatadistributions.First,adistributionofdataissimplyadisplaythatshowsthevaluesofthevaluesandtheirfrequencies(orrelativefrequencies).Oftenwhenwerefertoadistribution,wemeanthepatternofthedatawhenitismadeintoabargraphoralinegraphshowingvaluesofthedatasetonthehorizontalaxis,andeitherfrequencyofvaluesorrelativefrequency(percentsofvalues)ontheverticalaxis.AllthreeofthedistributionsyouareabouttoseewerefoundattheQuantitativeEnvironmentalLearningProjectwebsite(http://www.seattlecentral.edu/qelp/Data_MathTopics.html).ForexamplethebargraphbelowshowsthedistributionofozoneconcentrationinSanDiegointhesummerof1998asmeasuredbytheCaliforniaAirResourcesBoard.TheEPAhasestablishedthestandardthatozonelevelsover0.12partspermillimeter(ppm)areunsafe.

Approximatelyhowmanydaysaredisplayedalltogether?InhowmanydayswastheozonelevelinSanDiegoclassifiedasunsafebytheEPA?Istheozonedistributionskewedrightorskewedleftorneither?Explain.

0

5

10

15

20

25

FrequenceyinDays

Ozone(ppm)

Summer1998SanDiegoOzone

90

Nowtakeaminutetomakesenseofthereservoirdatabelow.

Abouthowmanyreservoirsareshownonthisbargraph?Whatpercentofreservoirsarefilledtoatleast90%capacity?Doyouexpectthemeanofthereservoirdatatobehigherthanthemedianortheotherwayaround?Explain.ThegraphsincludesreservoirsinbothOregonandArizona.WheredoyouthinkArizona’sreservoirsarelikelyappearingonthebargraph?Why?

0510152025303540

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

>100

>105

Frequency

PercentofCapacity

ReservoirLevels12/31/98

91

Finally,belowwehaveonemoregraphshowingafairlysymmetricdistributionofthenumberofhurricanesoccurringineachtwo-weekperiodbeginningJune1.

Onwhichdatesarehurricanesmostlikelybasedonthesedata?

Inabargraphtheverticalaxisrepresentsacount(frequency)orapercent(relativefrequency)andthehorizontalaxiscouldrepresentvaluesofacategoricalvariable(likecolors),valuesofanumericdiscretevariable(likeshoesize),orvaluesofacontinuousvariable(likeheights).

Ifyouhavecollecteddataonthenumberofseatswonbyvariouspartiesinparliamentaryelections,youcouldmakeabargraphofthatdatabylistingpartiesalongthehorizontalaxisandmakingthebar-heightsrepresenteitherfrequencyorrelativefrequencyofyoursamples’responses.Thisbargraph(datafromwikipedia.org)showsboththeresultsofa2004election,andthoseof1999election.Takeamomenttomakesenseofthesedata.

020406080100120140160

2 4 6 8 10 12 14 16 18 20 22 24 26

Num

berofHurricane

WeeksBeginningJune1

USAtlanticHurricanes:1851-2000

92

ConnectionstotheElementaryGrades

Instructionalprogramsfromprekindergartenthroughgrade12shouldenablestudentstoformulatequestionsthatcanbeaddressedwithdataandcollect,organize,anddisplayrelevantdatatoanswerthem.

NationalCouncilofTeachersofMathematicsPrinciplesandStandardsforSchoolMathematics,p.176

Childreninelementaryschoollearntomakeandinterpretpictograms,frequencyplots(sometimescalledlinegraphs),bargraphs,andpiecharts.Onordertohelpyoudistinguishamongthesetypesofdisplays,let’sbeginwithanexampleofeach.ThedatabelowshowhowchildreninMr.Jensen’sclasstraveltoschool.

Student Transportation Student Transportation Student TransportationAbby Bus Sasha Foot Xia FootWinton Bicycle Lucy Bus Kim FootDexter Car Joe Car Ben CarTrevor Car Aiden Bus Leah CarAudrey Car Justin Foot Jeffery FootMavis Foot Campbell Car Queztal CarKate Foot Sonja Car Cheyenne BicycleLim Bicycle Jen Bus Steve Bicycle

0

50

100

150

200

250

300

EUL PES EFA EDD ELDR EPP UEN Other

#ofSeats

Group

EuropeanParliamentElections

1999

2004

93

First,hereisapictogramofthedata. TransportationPictogramforMr.Jensen’sClass

Bus

Bike

Car

Foot

=1childonfoot =1childdriventoschool =1biker =1busriderApictogramrepresentseachvalueasameaningfulicon.Thedisplaycanbeorientedeitherverticallyorhorizontally(ascanlineplotsandbargraphs).Noticehowthegraphiscarefullylabeledandthatakeyisprovidedfortheicons.Makesuretohelpchildrengetinthehabitoflabelinganddefiningtheirsymbols.Alsostressthatitisimportantthatalltheiconsbethesamesizeorthegraphcouldbemisleading.Apictogramcanbeunderstoodbychildrenasyoungasthoseinkindergarten.Afrequencyplot(sometimescalledalineplotifthehorizontalaxisisanumberline)ofthesamedataisshownbelow.Notethatthisplotisaprecursortothebargraphinthesensethatitdisplaysfrequencyofresponseinamoreabstractmannerthandoesthepictogram. TransportationFrequencyPlotforMr.Jensen’sClass X X X X X X X X X X X X X X X X X X X X X X X X Bus Bike Car Foot

94

CreateabargraphforthedatafromMr.Jensen’sclass.CreateapiegraphforthedatafromMr.Jensen’sclass.Explainhowyoudecidedhowbigtomakeeachsector.

Homework

Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butalackofwill.

VinceLombardi

1) DoalloftheitalicizedthingsintheReadandStudysection.2) DoalltheproblemsintheConnectionssection.Whataretheexactpercentagesfor

thepiechart?Howcanyoucomputetheanglestothenearestdegree?Doitandthenexplainyourwork.

95

3) HereisatableofdatashowingthepetsownedbychildreninMs.Green’sthirdgradeclass.

Student PetsintheHouseAnna NoneJames 1catand1dogCleo 3dogsViolet 1catAndre 6fishJose 7fish,1dog,2catsAidos NoneYvonne NoneTyrise 1hamsterEllen 2dogsTomas 1dogYani 3fishand2catsOwen NoneAudrey 1cat

a) Makeafrequencyplotofthesedatawithchildren’snamesonthehorizontalaxis.b) Makealineplotwith‘numberofpets’onthehorizontalaxis.c) Makeabargraphofthesedatawith‘numberofpets’onthehorizontalaxis.d) Makeapiechartofthesedata.e) Whatproblemsmightchildrenencounterintryingtorepresentthesedatainthe

aboveways?4) Answerthequestionsinthe“FavoriteBooks”probleminUnit2Module4Session1in

theGrade3BridgesinMathematicscurriculum.Howcanyouusethisproblemtodiscussthesimilaritiesanddifferencesbetweenpicturegraphsandbargraphs?

5) Answerthequestionsinthe“GiftWrapFundraiser”probleminUnit2Module4Session2intheGrade3BridgesinMathematicscurriculum.Here’ssomeadditionalquestionstothinkaboutasafutureteacher:

a) Explainwhyachildmightgivetheanswerof“5”toquestion2abouthowmanystudentssold7rollsofgiftwrap.

b) Explainwhyachildmightgivetheanswerof“7”toquestion4abouthowmanyrollsofgiftwrapSarahsold.

c) Whyistheanswertoquestion5NOTthetotalnumberofXsonthelineplot?d) MakeafrequencyplotforthesamedatawhereeachXrepresentsarollofgift

wrap.Thenexplaintousethegraphtoanswerquestions1-5.

96

6) ThisgraphshowslengthsofthewingsofhousefliesfromtheQuantitativeEnvironmentalLearningProject.(OriginaldatafromSokal,R.R.andP.E.Hunter.1955.AmorphometricanalysisofDDT-resistantandnon-resistanthouseflystrainsAnn.Entomol.Soc.Amer.48:499-507)

a) Howwouldyoudescribetheshapeofthisdistributionintermsofskewnessandsymmetry?

b) Approximatelyhowmanyfliesweremeasuredforthisstudy?Explain.c) TrueorFalse?Themeanofthesedataisequaltothemedian.Explain.d) TrueorFalse?About90%offlieshavewingssmallerthan51.5(×0.1mm).

Explain.e) Makeupagoodquestiontoaskanelementaryschoolchildaboutthisdataset.

7) Hereisagraphshowing100yearsofdataonWorldwideEarthquakesbiggerthan7.0

ontheRichterScalefromtheQuantitativeEnvironmentalLearningProject(datafromtheUSGeologicalSurvey)Soforexample,therewere4yearswhenthenumberofglobalquakes(above7.0)wasabout35.

02468101214161820

37.5 39.5 41.5 43.5 45.5 47.5 49.5 51.5 53.5 55.5

Frequency

Length(x.1mm)

HouseflyWingLengths

97

a) Whatdoesthisgraphtellyou?Bespecific.b) Estimatethemedianannualnumberofquakesabove7.0.c) Howwouldyoudescribethisshapeofthedistributionintermsofskewnessor

symmetry?Whatdoesthatshapetellyouaboutearthquakes?

0

5

10

15

20

25

30

10 15 20 25 30 35 40 45

frequency

AnnualNumberofQuakes>7.0

WorldwideEarthquakes1900-1999

98

ClassActivity15:OldFaithfulEruptions

Itisbettertoknowsomeofthequestionsthanalloftheanswers. JamesThurber

OldFaithfulgeyserinYellowstoneNationalParkunleashesanimpressivecolumnofwaterandsteameveryhourorso.Belowyouwillfindasampleofwaittimes(readhorizontally)betweeneruptions(inminutes)forthegeyser.Thejobforyourgroupistoanalyzethesedatausinganytechniquesthatmakesensetoyou.Youarewelcometousetechnologyifavailable.Feelfreetoexplorewhateverquestionsariseinyourgroupyoulookatthedata–andtomakewhateverdisplaysmakesensetohelpyoutoanswerthosequestions.Afterabout30minutesofgroupwork,youwillpresentyourquestions,graphs,andfindingstoyourclass. 80 71 57 80 75 77 60 86 77 56 81 50 89 54 90 73 60 83 65 82 84 54 85 58 79 57 88 68 76 78 74 85 75 65 76 58 91 50 87 48 93 54 86 53 78 52 83 60 87 49 80 60 92 43 89 60 84 69 74 71 108 50 77 57 80 61 82 48 81 73 62 79 54 80 73 81 62 81 71 79 81 74 59 81 66 87 53 80 50 87 51 82 58 81 49 92 50 88 62 93 56 89 51 79 58 82 52 88 52 78 69 75 77 53 80 55 87 53 85 61 93 54 76 80 81 59 86 78 71 77 76 94 75 50 83 82 72 77 75 65 79 72 78 77 79 75 78 64 80 49 88 54 85 51 96 50 80 78 81 72 75 78 87 69 55 83 49 82 57 84 57 84 73 78 57 79 57 90 62 87 78 52 98 48 78 79 65 84 50 83 60 80 50 88 50 84 74 76 65 89 49 88 51 78 85 65 75 77 69 92 68 87 61 81 55 93 53 84 70 73 93 50 87 77 74 72 82 74 80 49 91 53 86 49 79 89 87 76 59 80 89 45 93 72 71 54 79 74 65 78 57 87 72 84 47 84 57 87 68 86 75 73 53 82 93 77 54 96 48 89 63 84 76 62 83 50 85 78 78 81 78 76 74 81 66 84 48 93 47 87 51 78 54 87 52 85 58 88 79 ThesedatacomefromAHandbookofSmallDataSetseditedbyD.J.Hand,F.Daly,A.D.Lunn,K.J.McConwayandE.OstrowskifromChapmanandHall.Thesedatacanbefoundathttp://www.stat.ncsu.edu/sas/sicl/data/geyser.dat.

99

ReadandStudy

Therearethreekindsoflies:lies,damnedlies,andstatistics. BenjaminDisraeli

Itisimportantnotonlythatchildrenlearntodisplaydatabutalsothattheylearntomakesenseofchartsandgraphstheyfindinbooksandothermedia,andtobecriticalofmisleadingdisplays.Considertheexamplesbelow:Amyclaimsthatthechildreninherkindergartenclasslikecatsmorethandogs.Shehasmadethefollowinggraphofherclassmates’petpreferencestosupportherclaim.Whyisthispictogrammisleading?Explain.Whatwouldyousaytothischild?

Childrenwholikecats

Childrenwholikedogs TuffTruckCompanyclaimsthattheirtrucksarethemostreliableandshowsthebelowgraphtomaketheircase.Whyisthisbargraphmisleading?Explain.(Bytheway,thisisareal-lifeexample.Onlythenameshavebeenchanged.)

Infact,criticalanalysisofdatadisplaysispartofthecurriculumforchildreninupperelementaryschool.

Perc entoftruc ks s tillontheroad,fiveyears afterpurc has e

93%

94%

95%

96%

97%

98%

99%

TuffTruck B randB B randC

100

Homework

Smokingisoneoftheleadingcausesofstatistics. FletcherKnebel

1) AnswertheitalicizedquestionintheReadandStudysection.2) Reviewquestions:Carefullydefinethetermsand,ifappropriate,describethe

differencerequested.Someexamplesmighthelp,butexamplesalonearenotenough.

a) Whatismeantbythetermdescriptivestatistics?b) Whatisthedifferencebetweenanumericalvariableandacategoricalvariable?c) Whatisthedifferencebetweenadiscretevariableandacontinuousvariable?d) Whatisanoutlier?e) Whatdowemeanwhenwestayastatisticisresistanttooutliers?

3) Astemandleafplot(orsimplystemplot)isalistingofallthedatatypicallyarranged

sothetensplacemakesthestemandtheonesarethe‘leaves.’Forexample,herearescoresonaGeometryFinalExamarrangedasastemplot:

ExamsScoresinGeometry

2|03|2,74|5|6,8,9,96|2,4,5,5,7,87|0,0,2,4,5,6,78|0,1,1,2,4,49|2,4,510|0

Wewouldreadthescoresas20,32,37,56,58,59,59,62etc.

a) Describetheshapeofthedistributionofscoresintermsofskewnessandsymmetry.

b) Whatisthemedianscore?c) Givesomeexamplesofdatasetsforwhichastemplotwouldbeimpractical.d) Makeaboxplotofthesedata.

4) Considerthisgraphonthenextpagemakingthecasethatwearehavinganational

‘crimewave!’Whatdoyouthink?Doesthegraphprovidecompellingevidence?Explain.

101

Reportedburg laries intheUnitedS tates ,2001-2006

S ource: S tatis ticalAbs tractoftheUnitedS tates ,2008

2.11

2.12

2.13

2.14

2.15

2.16

2.17

2.18

2.19

2001 2002 2003 2004 2005 2006

Millions

ofrep

ortedbu

rlaries

102

ChapterFour

Ratios,Rates,andProportions

103

ClassActivity16:Let’sBeRational

Thosewhorefusetodoarithmeticaredoomedtotalknonsense.JohnMcCarthy,artificialintelligencepioneer

Belowaresomeproblemsinvolvingratios.Inyourgroup,firstleteveryonespendafewminutesbythemselvesthinkingabouteachproblemandhowtheywouldsolveit.Thenshareyoursolutionmethodsinyourgroup.Trytofindseveralwaysthatchildrencouldthinkabouttheseproblems.Foreachproblem,thinkofawayyoucouldillustratetheproblemwithpicturesoradiagram.

1) SupposetheratioofboystogirlsinMr.Gauss’fifth-gradeclassis5:3.Howmanystudentsareintheclass?

2) SupposetheratioofboystogirlsinMr.Gauss’fifth-gradeclassis5:3,andthatthereare12moreboysthangirls.Howmanystudentsareintheclass?

3) CalvinandHannaheachhaveacollectionoftoycars.TheratioofthenumberofCalvin’scarstothenumberofHannah’scarsis3:7.IfCalvingives1/6ofhiscarstoHannah,whatwillbethenewratioofthenumberofCalvin’scarstoHannah’s?

4) Beforetheywentshoppingtogether,JackhadthesameamountofmoneyasKaren.AfterJackspent$34andKarenspent$16,thetwonoticedthatJackhad¼theamountofmoneythatKarenhad.Howmuchmoneydideachhavetostart?

104

ReadandStudy

TheratioofWe'stoI'sisthebestindicatorofthedevelopmentofateam.LewisB.Ergen

Aratioisacomparisonoftwoquantitiesthathavethesameunit.SpecificallyaratiobetweentwoquantitiesisA:BifthereisaunitsothatthefirstquantitymeasuresAunitsandthesecondquantitymeasuresBunits.Let’sconsideranexampleindetailtoreallyunderstandthisdefinition.SupposeJoeis6feettall,andhisyoungerbrotherBenis4feettall.ThenwecansaytheratioofJoe’sheighttoBen’sheightis6:4.Apowerfulwaytovisualizeratiosisthroughabardiagram,inthetwoquantitiesbeingcomparedareeachrepresentedasabarwithlengthmeasuredinthesameunits.SotoillustratearatioofA:B,wewouldmakeabaroflengthAunitsandabaroflengthBunits,andthenwecan“see”theratioasavisualcomparisonofthelengthofthetwobars.Forexample,hereisabardiagramtorepresenttheratio6:4.Joe’sHeightBen’sHeightNoticehowwemadeeachunitthesamesize,toemphasizethatwearemeasuringbothquantitieswiththesameunit.Let’slookfurtheratthedefinitionwegaveforaratio.Whenwesaythat“theratioofJoe’sheighttoBen’sheightis6:4”,thetwoquantitieswearecomparingareJoe’sheightandBen’sheight,andthatthereissomeunitsothatJoe’sheightis6unitsandBen’sheightis4units.Sowhatisthatunit?Inthiscase,theunitis1foot,sowecanthinkofeachsmallrectangleinourdiagramaboveasbeingonefoot.Butwhatifwechooseadifferentunitofmeasurement?Let’ssayweuse“1inch”asourunitofmeasurement.ThenwhatistheratioofJoe’sheighttoBen’sheight?Usinginches,Joeis72inchestall,andBenis48inchestall,sotheratioofJoe’sheighttoBen’sheightis72:48.Wecouldmakeabardiagramillustratingtheratio72:48bydividingeachoftheunitsinthediagramaboveinto12smallerequalsizedunits.WritetheratioofJoe’sheighttoBen’sheightiftheunitofmeasurementis1yard.

105

WritetheratioofJoe’sheighttoBen’sheightiftheunitofmeasurementis“Ben’sheight”?WritetheratioofJoe’sheighttoBen’sheightiftheunitofmeasurementis“Joe’sheight”?IftheratioofJoe’sheighttoBen’sheightis3:2,thenwhatistheunitofmeasurementbeingused?

Noticehowwecanhavemanydifferentwaysofwritingthesameratio.Inthislastexample,theratioofJoe’sheightcanbewrittenas6:4,or72:48,or1.5:1,or1:2/3.Wesaythattworatiosareequivalentifoneisobtainedfromtheotherbymultiplyingallmeasurementsbythesamenon-zeronumber.Showthateachratiosintheprevioussentenceisequivalentto6:4byfindingthenumbertomultiplythemeasurementsby.Let’slookatthefirstproblemfromtheClassActivity.SupposetheratioofboystogirlsinMr.Gauss’fifth-gradeclassis5:3.Howmanystudentsareintheclass?Therearemanypossibleanswerstothisquestion.Itdependsonwhattheunitisinthegivenratioof5:3.Inthebardiagramshownbelowillustratingtheratio5:3,eachunitshowncouldrepresent1student,inwhichcasetherewouldonlybe8studentsintheclass(5boysand3girls).Butifeachunitis2people,thenthereare16students(10boysand6girls).Ifeachunitis3people,thenthereare24students(15boysand9girls),andsoon.NumberofBoysNumberofGirlsWithoutanyfurtherinformation,theonlythingwecansayisthatthereissomemultipleof8studentsinMr.Gauss’class.However,inquestion2,weweregivensomefurtherinformationthattherewere12moreboysthangirls.Withthis,wecanthenfigureoutthattheremustbe30boysand18girlsintheclass,foratotalof48students(poorMr.Gauss).

106

12 people

Youmighthavefiguredthisoutbythinkingofratiosthatareequivalentto5:3wherethedifferencebetweenthetwonumbersis12.Forexample

5/3=10/6=15/9=20/12=25/15=30/18Sotheratio5:3isequivalentto30:18,andiftheunitwere1person,thiswouldbe30boysand18girls,whichistherequired12moreboysthangirls.Youmayhavereasonedthatthedifferencebetweenboysandgirlsintheratio5:3is5‒3=2units.Sincethisdifferenceis12people,2unitsmustequal12people,soeachunitis6people.Thebardiagramabovegivesapowerfulwayofvisualizingthis.Wecanseethetwoextraunitsthattheboyshave,andidentifythesetwounitswiththe12moreboysthanthegirls.NumberofBoysNumberofGirlsSoitturnsouttheratioweweregivenforthenumberofboystogirls,namely5:3,wasusingaunitof6people.Oncewefigureoutthisinformation,wecouldaddittoourbardiagram,labelingeachunitasbeing6people.Thenwecaneasilyseethe30boysand18girls.NumberofBoysNumberofGirlsTheseexampleshighlightaBigIdeaaboutratios:ratiosareunitless,soaratiobyitselfdoesnottellyoutheactualquantitiesinvolved.Ifyouwanttodeterminetheactualquantities,youwillneedtoknoworfigureoutexactlywhatunitis.Inthehomeworkwewillgiveyouseveralmoreproblemswhereyouwillneedtofigureoutwhattheunitisinordertosolvetheproblem.Ratioscanbeexpressedincolonnotation,asfractions,asdecimals,aspercents,orinwords.Let’sgobacktoJoeandBen,whereJoeis6feettallandBenis4feettall.Thecomparisonbetweentheirheightscanbeexpressedinmanyequivalentways:

• TheratioofJoe’sheighttoBen’sheightis6:4

6 6 6 6 6

6 6 6

107

• Joe’sheightis6/4ofBen’sheight.• Joe’sheightis3/2ofBen’sheight• Joe’sheightis1.5timesBen’sheight• Joe’sheightis150%ofBen’sheight

Wecouldalsochangetheorderofthecomparison,andcompareBen’sheighttoJoe’sheight:

• TheratioofBen’sheighttoJoe’sheightis4:6.• Ben’sheightis4/6ofJoe’sheight.• Ben’sheightis2/3ofJoe’sheight• Ben’sheightisabout.67timesJoe’sheight• Ben’sheightisabout67%ofJoe’sheight

Homework

Successisfollowingthepatternoflifeoneenjoysmost. AlCapp

1) DoalltheitalicizedthingsandunfinishedexamplesintheReadandStudysection.

2) Readthe“MoreMultiplicativeComparisonsattheGiantsDoor”activityinUnit1,

Module3,Session3oftheBridgesinMathematicsGrade4StudentBook.Answerallofthequestions1-7inthatactivity.Thenalsoanswerthese:

a) Thehammeris astallasthedog.b) Therakeis timesastallasthesaw.c) Thesawis astallastherake.d) Theshovelis astallasthewoodendoor.e) Thewoodendooristimesastallastheshovel.

3) Inasampletheratioofmentowomenisabout5:8.Ifthesamplecontains200

people,howmanymenandhowmanywomenareinthesample?Explainyourthinking.

4) Readthe“Facts&Fish”probleminUnit6,Module1,Session5oftheBridgesinMathematicsGrade1HomeConnections.Answer#6inthatactivity.Inwhatwaydoesthisprobleminvolveratios?Whataresomewaysafirstgradechildmight“showtheirwork”insolvingthisproblem?

5) Considerthisproblem:Miamakes12outofevery20shotsshetakesfromthefreethrowline.Howmanyshotsdoessheexpecttomakefromthelineifshetakes50total

108

shots?Supposethechildreninyoursixth-gradeclassexplainedtheirthinkingabouttheproblemistheseways.Ineachcase,makesenseofthechild’sthinkinganddecidewhetherthechildiscorrect.

a) Joe’ssolution:“12outof20isthesameas6outof10.SoIdid6times5toget

30outof50.”b) Omar’ssolution:“Itook12outofthefirst20andthen12moreoutofasecond

20andthat’s24.Thenweneedtenmoreshots,andshe’dmake6ofthoseandsothat’s30shotsthatshe’dmake.”

c) Gabrielle’ssolution:“20times2.5makes50.SoItook12times2.5andthatis30shotsshe’dmake.”

d) Terrell’ssolution:“12/20is60%,andthatis60outof100or30outof50.”

6) It’stimetomaketheconnectionbetweenratiosandfractionsexplicit:a) Ifthefractionofacakethatisremainingis3/5,interpretthemeaningofthe

ratio3:5.b) Iftheratioofboystogirlsinclassis3:5,interpretthemeaningofthefraction

3/5.c) Ingeneral,whatisthemeaningofthefraction3/5?Explainpreciselyhowthis

fractionisrelatedtotheratio3:5.

7) Discussthewaysinwhichproportionalreasoning(thinkingaboutequivalentratios)canhelpstudentsunderstandconvertingfractionstodecimals.

8) Showhowtosolvethefollowingproblemsbyusingabardiagram.

a) Theratioofapplejuicetocranberryjuiceinajuicecocktailis8:3.Ifapitchercontains24cupsofapplejuice,howmuchjuicecocktailisthereinthepitcher?

b) Atthebeginningoftheweek,theratioofKatie’smoneytoSasha’smoneyas4:7.ThenKatiespenthalfofhermoney,andSashaspent$60,sothatattheendoftheweek,SashahadtwiceasmuchmoneyasKatie.HowmuchmoneydidKatiestartwith?

c) Aboxofcookieshasthreedifferentkindsofcookies.Theratioofoatmealcookiestosugarcookiesis2:3,andtheratioofsugarcookiestochocolatechipis6:13.Whatistheratioofoatmealcookiestochocolatechip?

109

ClassActivity17:TheWatermelonProblem

TheysaythatninetypercentofTVisjunk.But,ninetypercentofeverythingisjunk.GeneRoddenberry

A100-poundwatermelon,initially99%water(byweight),wasleftsittinginthesun.Someofthewaterevaporates.Afterwards,itisonly98%water.Howmuchdoesitweighnow?

110

ReadandStudy

Eightypercentofsuccessisshowingup.WoodyAllen

Let’stalkaboutratiosthatareexpressedaspercents,asthisisperhapsthemostcommonwayinwhichratiosareexpressed.Apercentisaratiowherethesecondnumberis100.Thefollowingareequivalent:

A% A:100A/100

Let’sconsideranexample.SupposeAbbyhas$4andBobhas$5.Wecanillustratethissituationwithabardiagramwithaunitof$1.Abby’sMoneyBob’sMoneyTakeafewminutestoanswerthefollowingquestions:

• Abby’smoneyiswhatpercentofBob’smoney?

• WhatpercentofAbby’smoneydoesBobhave?

• AbbyhaswhatpercentlessmoneythanBob?

• BobhaswhatpercentmoremoneythanAbby?

Sincepercentsareratioswherethesecondnumberis100,wecouldanswerthesequestionsbyfindingequivalentratios.SinceweknowtheratioofAbby’smoneytoBob’smoneyis4:5,thiscanalsobeexpressedas“theratioofAbby’smoneytoBob’smoneyis80:100”,soAbby’smoneyis80percentofBob’smoney.Similarly,wecanreversetheorderofcomparison,andsaythat“theratioofBob’smoneytoAbby’smoneyis5:4”,whichisequivalentto“theratioofBob’smoneytoAbby’smoneyis125:100”,soBob’smoneyis125percentofAbby’smoney.

111

Thebardiagramcanhelpustovisualizethesecomparisons.Whenasked:“Abby’smoneyiswhatpercentofBob’smoney?”weareaskedtocompareAbby’smoneytoBob’smoney.WecanseethatAbbyhas4units,andBobhas5units,soAbbyonlyhas4/5,or80%ofBob’smoney.Inthefraction4/5,thewholethathasbeensplitinto5equalpartsisthewholeofBob’smoney.Noticehowthewholeinthefraction(Bob’smoney)isthesamewholewearefindingthepercentof.Since“Bob’sMoney”isthewholeinthissituation,wearethinkingofthewholeofBob’smoneybeingthe100%,whichwecouldalsoindicateonthebardiagram.Since5unitsis100%Bob’smoney,theneachunitis20%ofBob’smoney,soAbby’smoneyis80%ofBob’smoney.Abby’sMoneyBob’sMoneySimilarly,ifwechangetheorderofthecomparisonandcompareBob’smoneytoAbby’s,wecanseethatBobhas5unitstoAbby’s4,sohehas5/4,or125%ofAbby’smoney.Inthefraction5/4,thewholethathasbeensplitinto4equalpartsisthewholeofAbby’smoney,andthisisthesamewholewearefindingthepercentofwhenwesaythatBobhas125%ofAbby’smoney.Inthediagram,ifwethinkof100%asrepresentingthewholeofAbby’smoney,theneachunitis25%ofAbby’smoney.SinceBobhas5units,Bobhas125%ofAbby’smoney.Abby’sMoneyBob’sMoneyIntheothertwoquestionsweaskedabove,weareaskingyoutopayattentiontothedifferencebetweenthetwoquantitiesbeingcompared.SinceAbbyhas4unitsandBobhas5units,AbbyhasonefewerunitthanBob.Thisonefewerunitis1/5or20%ofBob’smoney,andsowecansaythatAbbyhas20%lessmoneythanBob.Nowhereisacoolthing.ThefactthatAbbyhas20%lessmoneythanBobdoesnotmeanthatBobhas20%moremoneythanAbby.Whynot?Reallythinkaboutthis.Thenexplain.

100 percent of Bob’s Money

20% 20% 20% 20% 20%

100 percent of Abby’s Money

25% 25% 25% 25%

112

ThefactthatAbbyhas1unitfewerthanBobdoesmeanthatBobhas1unitmorethanAbby.However,wenowwanttocomparethis1unitdifferencetoAbby’smoney.This1unitmoreis1/4or25%ofAbby’smoney,sowesaythatBobhas25%moremoneythanAbby.Ingeneral,tocomputeafractionalchange,orapercentchange,wealwayscomparetheamountofchangetotheoriginalamount.Hereisanothersetofexamplestoconsider.Takesometimenowtoanswerthesequestions.Foreach,makeabardiagramcomparingthetwonumbersofcredits,andthencomparetheamountofincreaseordecreaseinthebarstotheoriginalamount.Weleftyousomespacetomakeyourdiagrams.

• LastsemesterKarentook8credits.Thissemestersheistaking12credits. Bywhatpercentdidhercreditloadincrease?

• LastsemesterLorentook12credits.Thissemestersheistaking16credits. Bywhatpercentdidhercreditloadincrease?• LastsemesterMoetook16credits.Thissemesterheistaking12credits. Bywhatpercentdidhiscreditloaddecrease?

Havealookattheanswerstothethreequestionsabove.Noticehowineachexample,theamountofthechangewas4credits.However,thissameamount,4credits,correspondsintheseexamplesto50%,33%,or25%,dependingonwhatthis4creditsisbeingcomparedto.Thisisareallybigideaaboutfractionsandpercents:thatthesamefixedquantitycanbeexpressedasdifferentpercents,dependingonwhatwholethatquantityisbeingcomparedto.Here’sanotherBigIdea:sincepercentsareratios,percentsareunitless.Byworkingonthewatermelonproblemwehopethatyounowseehowimportantitistorealizethisfact.Weknowitisverytemptingtothinkofapercentasbeingaunit,afterall,itsuresoundslikeone.Whenwehearaphrasesuchas“30percent”,thatsuresoundsasif“percent”isaunit,justlikeinthephrase“30dollars”,adollarisaunit,orin“30feet”,afootisaunit.Butthedistinctionbetweena“percent”andunitssuchas“dollars”or“feet”ishuge.Considerthis:30

113

feetisalwaystwiceaslongas15feet.And30dollarsisalwaystwiceasmuchas15dollars.But30percentmaynotbetwiceasmuchas15percent.Readthatlastsentenceagain.Forexample,compare30percentofthepopulationofWisconsinwith15percentofthepopulationofCalifornia.Giveanotherexamplewhen30percentisnottwiceasmuchas15percent.Here’sanotherillustration:50feetisalways10feetlongerthan40feet.And50dollarsisalways10dollarsmorethan40dollars.However,50percentisnot10percentmorethan40percent.Infact,ifallofthepercentagesinvolvedwerebasedonthesamewhole,then50percentwouldbe25percentmorethan40percent.Readthatlastsentenceagain,andfighttounderstandit.Itmaybehardtobelievethat50percentcouldbe25percentmorethan40percent,butwehopeyouwillbeabletoseethisafterthinkingthroughsomemoreexamples,andinthehomework,wewillaskyoutogiveyourownexampleofthis.ConnectionstotheElementaryGrades

Saywhatyoumeanandmeanwhatyousay.Anonymous

“Thenyoushouldsaywhatyoumean,"theMarchHarewenton.

"Ido,"Alicehastilyreplied;"atleast--atleastImeanwhatIsay--that'sthesamething,youknow.""Notthesamethingabit!"saidtheHatter."Youmightjustaswellsaythat"IseewhatIeat"isthesamethingas"IeatwhatIsee"!”

LewisCarrollWewillclosethissectionwithsomethoughtsaboutlanguage.Whenratiosareexpressedinwordsorwithcolonnotation,thetwoquantitiesbeingcomparedareusuallystatedexplicitly.Forexample,ifwesay“theratioofAbby’smoneytoErin’smoneyis4:5”weknowwearecomparingthequantityof“Abby’smoney”tothequantityof“Erin’smoney”.Especiallywhenratiosareexpressedaspercents,thendependingonthelanguageused,thetwoquantitiesbeingcomparedmightnotbeexpressedexplicitly.Whenwesaythat“Abby’smoneyis80percentofErin’smoney”thismeansthatwearecomparing“Abby’smoney”to“Erin’smoney”andthat“Erin’smoney”hasbeenexpressedasthenumber100.However,itiscommontonotexpressthingssoexplicitly.Forexample,thewatermelonproblemstatedthatthewatermelonis99%water(byweight).Noticethatthisstatementdoesn’texplicitlystate“percentof”tomakeitclearwhatisbeingconsideredthewhole100.

114

Inparticular,thephrase“99percentwater”doesnotmean“99percentofthewater”.Weneedtointerpretthisstatementandfigureoutthatismeansthatthewater’sweightis99percentofthewatermelon’sweight.Wehopethisallillustratestheimportanceofpayingparticularattentiontolanguageandbeingpreciseinouruseoflanguage.Furthermore,itmeansthatcannotreducemathematicalproblemsolvingtomerelytranslating“English”into“Math”asmanyteachersandstudentsaretemptedtodo,withoutcarefulthought.AbigpartoftheCommonCoremathematicalpracticestandardof“attendingtoprecision”ispayingattentiontotheprecisemeaningsofthelanguagethatweuse.Formulatingwhatwemeanusingpreciselanguageisapowerfultoolinproblemsolvingandanimportantmathematicalpractice.

Homework

I'vegotatheorythatifyougive100percentallofthetime,somehowthingswillworkoutintheend.

LarryBird

1) DoalltheitalicizedthingsandunfinishedexamplesintheReadandStudysection.

2) Let’sreconsiderMr.Gauss’sclass,wheretheratioofboystogirlsintheclassis5:3.Answerthefollowingquestions:

a) Whatpercentoftheclassisgirls?b) Whatpercentofthenumberofboysisthenumberofgirls?c) Whatpercentoftheclassisboys?d) Whatpercentofthenumberofgirlsisthenumberofboys?

3) Wayne’ssnowboardis90cmlong.Luke’ssnowboardis120cmlong.Luke’sboardis

whatpercentlongerthanWayne’s?Wayne’sboardiswhatpercentshorterthanLuke’s?Makebardiagramstoillustrateyouranswers.

4) SupposeeachyearatESU,thereare2000freshmen.Lastyear25%ofthefreshmenatESUliveoffcampus.Thisyear30%offreshmenliveoffcampus.Whatisthepercentincreaseinthenumberoffreshmenlivingoffcampus?Makeabardiagramtosupportyouranswer.

5) Lastyear,CRIresearchconductedatelephonesurveyof500adultAmericansandfoundthat45%wereinfavorofthepresident.Thisyeartheyagainsurveyed500adultAmericansandfoundthatthenumberofpeoplethatwereinfavorofthepresidentfellby30%.Whatpercentofthepeopletheysurveyedthisyearwereinfavorofthepresident?Makeabardiagramtosupportyouranswer.

115

6) Lastyear8%ofthecityofIllvillehaddysentery.Thisyear10%ofthecityhaddysentery.BywhatpercentdidthenumberofpeopleinIllvillewithdysenteryincrease?Justifyyouranswer.

7) Whatassumptiondidyoumakeinordertosolvethepreviousproblem?Isthisarealisticassumption?Canyoufigureouthowtherecouldbenochange(inotherwords,a0percentchange)inthenumberofpeopleinIllvillewithdysenteryandstillhavethepercentwithdysenteryincreasefrom8percentto10percentofthecity?

8) Acmesellscansofmixednuts.Inthepast,5%ofthenutswerecashews,butnow

Acmeisadvertisingthattheirnewandimprovedcanofmixednutshas40%morecashews.Whatpercentofthecanofnewandimprovedmixednutsiscashews?Whatassumptionsareyoumakingabouttheoldandnewcansinyoursolution?

9) Makeupyourownexampleofaproblemwhere50%is25%morethan40%.

10) Makeupyourownexampleofaproblemwhere60%isthesamequantityas30%.

11) Fiveyearsago,agovernorcutauniversity’sbudgetby20%.Thisyear,sheincreasedthebudgetby5%,andannouncedthatshegavebackaquarterofthebudgetcut.Isthisaccurate?Explain.

12) BobbuysandsellsrarestampsonEbayforfunandprofit.Usually.YesterdayBobsoldtwostamps.Eachsoldfor$60.However,hehadpaiddifferentamountsforthestampsoriginally,sothatthefirststampendedupbeingsoldata25%profit,andthesecondwassoldata25%loss.DidBobendupmakingmoney,losingmoney,ordidhebreakeven?Justifyyouranswer.

116

ClassActivity18:HowDoYouRate?

Nothingtravelsfasterthanthespeedoflight,withthepossibleexceptionofbadnews,whichobeysitsownspeciallaws.

DouglasAdams

1) ItwouldtakeBiff60minutestomowalawn,andittakesCharlie30minutestomowthatsamelawn.Iftheyworktogether,howlongwouldittakethemtomowthatlawn?

2) SupposeyoucommuteeachdaybetweenAbbotsfordandBelleview.Inthemorning,youaverage60mph.Onthereturntriphome,youhitrushhourandonlyaverage40milesperhour.Whatisyouraveragespeedfortheentirecommute?

3) Acaristravelling60milesperhour.Howfastisthisinfeetpersecond?

117

ReadandStudy

Thefutureissomethingwhicheveryonereachesattherateofsixtyminutesanhour,whateverhedoes,whoeverheis.

C.S.LewisArateisacomparisonoftwoquantities,eachmadewithadifferentspecifiedunit.Intheprevioussectionwedefinedaratioasacomparisonoftwoquantitiesthathavethesameunit.Soarateislikearatio,exceptthatinarate,thetwoquantitiesbeingcomparedhavedifferentunits.Theresultofthisisthatwhileratiosare“unitless”,ratesdohaveunits.Aspeed,suchashowfastacaristraveling,isagoodexampleofarate,andsomefamiliarunitsforspeedsaremilesperhour,orfeetpersecond,etc.WeassumethatyoufoundthefirsttwoproblemsintheClassActivityquitetricky.Thereasonwhytheyaretrickyisthatwehavegottenusedtoratesthatareexpressedwherethesecondquantityinthecomparisonisone.Whenwehaveaspeedof60milesperhour,thisisaratecomparing60mileswith1hour.Thismeansforevery60mileswewouldtravel1hour.Thisrateisequivalenttotravelling120milesin2hours,or30milesin½hour,or1milein1/60ofanhour.Thereareaninfinitenumberofratesthatareequivalenttothisspeed,butourstandardwayofexpressingthisrateis60milesin1hour,wherethesecondquantityinthecomparisonis1.Sointhecommutingproblem,therateswearegivenarebothcomparingdistancesto1hour.Sowearefooledintothinkingthatbothpartsofthecommuteare1hour.However,wearen’ttravelingthesameamountoftimeinthemorningasintheafternoon.(Whynot?)Instead,wearetravellingthesamedistanceinthemorningasintheafternoon.Sotomakesenseoftheproblem,weshouldinsteadthinkofequivalentrateswherethedistancesarethesame.Forexample,inthemorning,wearetravelingatarateof120milesin2hours,andintheafternoonwearetravelingatarateof120milesin3hours.Sonowwecanthinkiftheentirecommutewere120milesthereand120milesback,itwouldtakeus5hourstogoatotalof240miles,whichisequivalentto48milesperhour.Orwecouldjustthinkabouthowlongitwouldtaketotravelonemileofourcommute.Inthemorningitwouldtake1minutetotravelonemile,andintheafternoonitwouldtake1.5minutestotravelonemile.Socombinedthatis2.5minutesfor2miles,whichisequivalentto48milesperhour.Therearemanygoodwaystothinkaboutthisproblem,aslongaswerealizethatitisthedistancestraveledthatarethesamefrommorningtoafternoon,notthetimestraveled.Similarly,inthelawn-mowingproblem,therateswearegivenare60minutesperlawnand30minutesperlawn.Bothratesareexpressedwiththesameamountoflawn(namely1lawn).However,BiffandCharliewhenworkingtogetherwillnotmowthesameamountoflawn.(Whynot?)Instead,theywillmowthesameamountoftime.Soweshouldfindequivalentratesforeachwherethetimeisthesame.(Identifyingwhatischangingandwhatisstayingthesameisabigthemeinmanyofthemorechallengingproblemswe’velookedat

118

recently.Forinstance,insolvingthewatermelonproblem,itwasimportanttorealizethattheamountofwaterwaschanging,buttheamountofrindwasstayingthesame).Therearemanywaystoexpressequivalentrates.Onewe’veseenalreadyistochangethenumberofeachunitproportionally.Forexample60milesperhouristhesameas120milesin2hours.Wecanalsochangetheorderofcomparison.60milesperhouristhesameas1hourper60miles.Yetanotherwaytofindequivalentratesistoexpressoneorbothofthequantitiesindifferentunits.Forexample,60milesperhouristhesameas60milesper60minutes.Considerthefollowingequivalentrates:

1mile1hour =

4miles4hours

Toobtainthesecondratefromthefirst,youmightimaginethatthefirstratehasbeenmultipliedbythefactorof4/4.

1mile1hour×

44 =

4miles4hours

Nowconsiderthispairofequivalentrates:

4miles4hours =

21120feet4hours

Howhasthesecondratebeenobtainedfromthefirst?Reallythinkaboutthis!Onewayistothinkofthisasasimplesubstitution.Since1mileis5280feet,4milesis21120feet,andsowehavejustsubstitutedonequantityforanequivalentquantity.Anotherwayistothinkofthefirstratebeingmultipliedbyarateof5280feetpermile,sothat

4miles4hours×

5280feet1mile

=21120feet4hours

Whydoesitmakesensethatyoucangetanequivalentratebymultiplyingby4/4?Whydoesitmakesensethatyoucangetanequivalentratebymultiplyingby3"45feet

!mile?

119

ConnectionstotheElementaryGradesAnunderstandingof,andafacilitywithrates,ratios,andproportionalityareimportanttoolsstudentsuseinlearningmanyareasoftheschoolcurriculum.BelowyouwillfindtheCommonCorestateStandardsforMathematicsinthedomainofratioandproportion.Readthem.

Common Core State Standards for Ratios and Proportional Relationships Grade Six:

• Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

• Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

• Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by

reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

o Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

o Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

o Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

o Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

120

Homework

Ifpeopledonotbelievethatmathematicsissimple,itisonlybecausetheydonotrealizehowcomplicatedlifeis.

JohnLouisvonNeumann

1) DoalloftheitalicizedthingsintheReadandStudysection.

2) IntheConnectionssection,intheCommonCoreStateStandardsboxthefollowingproblemwasposed:“Ifittook7hourstomow4lawns,thenatthatrate,howmanylawnscouldbemowedin35hours?Atwhatratewerelawnsbeingmowed?Answerthosequestions.Howcanyouuseabardiagramtoillustratethissituation?

3) Threeapplescost$2.Howmuchshould7applescost?Explainwhythisproblemdemandsproportionalreasoning.WheredoesthisproblemfitwithintheCommonCoreStateStandards?

4) Doproblem#3in“Estimate&ReasonwithWater”Unit8,Module2,Session3fromBridgesinMathematicsGrade3StudentBook.

a) Answerquestionsaandbintheproblem.b) Makealistofalltheratesinvolvedinthisproblem,andillustrateeachrate

withabardiagram.c) SupposeJhongalsohasa100gallonwadingpool.Makeabardiagramthat

illustratesthatitwilltake81/3minutestofillupthepoolwiththegardenhose.

5) Thespeedofabulletasitleavesamodernfirearmisabout1000meterspersecond.

Howfastisthisinmilesperhour?a) Showhowyoucansolvethisproblembysettingupasinglecalculationthat

multipliesappropriateratestogether.Inyourcalculation,howmanydifferentratesareinvolved?Whichoftheseratesareequivalenttothespeedofthebullet?

b) Showhowyoucansolvethisproblembyfindingasequenceofrateswhereeachrateisequivalenttothespeedofthebullet.

6) GasinGermanycostsabout1.50eurosperliter.Howmuchisthisindollarsper

gallon?

7) Inaprevioushomework,yousolvedthefollowingproblem:Theratioofapplejuicetocranberryjuiceinajuicecocktailis8:3.Ifapitchercontains24cupsofapplejuice,howmuchjuicecocktailisthereinthepitcher?Byusingunitsof“cupsofapplejuice”,“cupsofcranberryjuice”and“cupsofjuicecocktail”,showhowtosolvethisproblemsettingupasinglecalculationthatmultipliesappropriateratestogether.

121

8) Thetemperatureatwhichwaterfreezesis0degreesCelsius,and32degreesFahrenheit.Thetemperatureatwhichwaterboilsis100degreesCelsius,and212Fahrenheit.Usingthisinformation,ifathermometerreads15degreesCelsius,whatitthistemperatureinFahrenheit?Ifathermometerreads18degreesFahrenheit,whatisthetemperatureinCelsius?Inwhatwaydoesthisprobleminvolverates?

9) Here'sanoldpuzzlethatwelike:Ifahenandahalflayaneggandahalfinadayandahalf,howlongdoesittakeonehentolayanegg?

10) If3peoplecanmake3chairsin3days,howlongdoesittakeonepersontomakeonechair?(Doesthismakeyoureconsideryouranswertothehenpuzzle?).

122

ClassActivity19:Goldfish

Fishingisboring,unlessyoucatchanactualfish,andthenitisdisgusting. DaveBarry

Wewouldliketocountthenumberofgoldfishinalargepond.Hereisourplan:

Step1:We’llcaptureasampleofgoldfishfromthepondandtagthem.Callthatnumbern1.Step2:We’llreleasethefishbackintothepondandletthemswimaround.Step3:Afewdayslater,we’llreturnandcaptureanothersampleofgoldfish.Callthatnumbern2.Step4:We’llfindpercentofgoldfishinoursecondsamplethataretagged.We’llcallthatp%.

Weclaimthatthisisenoughinformationtoestimatethepopulationofgoldfishinthepond.Canyoufigureouthowtodoitandexplainwhyyourideamakessense?Underwhatcircumstances(whattypesofenvironments,samplingmethods,orpopulationsofanimals)willyourmethodproducereliableresults?

123

ReadandStudy

Thereissafetyinnumbers.Authorunknown

Themethodyouusedtocountfishinthepondiscalledcapture-recapture,anditisbasedontheassumptionthatbothsamplesarerepresentativeofthepopulation.Themethodisn’tverygoodifthatisnotthecase.Forexample,thismethodwouldn’tworkforcountinganimalsthatmigrateawayfromtheareaoranimalsthatdon’tgetsufficientlymixedinamongtheothers.Likewise,itwouldn’tworkwellifthesampleswerechoseninabiasedway.

Youmightbeexpectingacapture-recapture“formula”inthissection.Butyouwon’tneedonebecauseyouaregoingtounderstandthemethod,andthatismuchbetterthanknowingaformula.Ifyouunderstandtheidea,youwon’tgetconfusedbychangesinwordingornotationinaproblem.Sohereitis:

Ifthesamplesarebothrepresentativeofthefishinthepond,thentheproportionoftaggedfishinthewholepondshouldbeaboutthesameastheproportionoftaggedfishinthesecondsample.Butyouknowtheproportionoftaggedfishinthesecondsample.Youalsoknowthetotalnumberoftaggedfishinthepond.Fromthisinformation,youcancomputeanestimateforthetotalnumberoffishinthepond.Thinkthisthroughusingthedatayoucollectedinclass,andbesuretotalkwithyourgrouporinstructorifyoudon’tseehowtothinkaboutthis.

Capture-recaptureisn’tusedonlyforcountingfish;infact,thissamplingmethodhassometimesbeenusedto‘fix’theU.S.Censusdata.Aswementionedinaprevioussection,itistoughtoconductacensus,andtheU.S.Censusisparticularlyproblematic.Foronething,weareacountryofapproximately300millionpeople(wethink).Foranotherthing,thereisasignificantbunchofpeoplewhoarehomelessorillormistrustfulofthegovernment,andmanyofthesepeopleeitherdonotreceivecensusformsordonotreturnthem.Inpastyearsthegovernmenthasattemptedtocorrectforthesetypesofundercountingbyestimatingthenumberofpeople“missed”inacertaincityorregionusingthemethodofcapture-recapture.Sincethatisthesamemethodthatyouusedtoestimatethenumberoffishinthepond,we’lltalkaboutitnow.Hereistheidea:Aftercensusformsarereturnedforaregion,ateamofcensusworkerswillgodoor-to-door(andinthecasesofthehomeless,alsototheplacestheymightbefound)inthatregion.Theywillaskeveryonewhethertheyhavereturnedacensusform.Andusethatpercenttoadjustthecount.Let’sdoaspecificexample.SupposethecensusbureauwantstocountthepopulationofdowntownNorthGary,Indianaandtheyhave2100censusformsreturnedfromthatregion.Ateamofcensusworkersusesclustersamplingtoselectseveralcityblocksfromtheregion.Theythengotothoseblocksandtrytofindeveryonewholivesthereandtalktothem.

124

Supposetheyfindthatonly70%ofthepeopletheytalktohadreturnedtheform.Theywillassumethatforevery70peoplewhoreturnedtheform,30didnotandtheywillusethistocorrecttheirestimateofthepopulationofdowntownNorthGaryfrom2100upto3000.Workthroughthatcomputationtobesurethatwe’recorrect.ConnectionstotheElementaryGrades

Facilitywithproportionalitydevelopsthroughworkinmanyareasofthecurriculum,includingratioandproportion,percent,similarity,scaling,linearequations,slope,relative-frequencyhistograms,andprobability.Theunderstandingofproportionalityshouldalsoemergethroughproblemsolvingandreasoning,anditisimportantinconnectingmathematicaltopicsandinconnectingmathematicsandotherdomainssuchasscienceandart.

NationalCouncilofTeachersofMathematics PrinciplesandStandardsforSchoolMathematics,page212Youhavenoticedthatthetypeofmathematicalthinkingrequiredtounderstandcapture-recaptureinvolvesratios.Proportionalreasoningisthinkingaboutequivalentratios.Ifwehavetwoequivalentratiosandwritethisdownusingfractions,wegetanequationofthetypea/b=c/d.Inschoolalgebra,anequationofthistypeifoftencalledaproportion.Solvingproblemsbysettingupaproportionisacommontechniquetaughtinschool.However,wehavefoundthatmanystudentswillwritedownaproportionwithoutthinkingaboutwhatthetwofractionstheyhavewrittenaremeanttorepresent,orwhetherthetwofractionstheyhavewrittenoughttobeequal.(PerhapsyouexperiencedthisyourselfwhenworkingontheWatermelonProblem.)Asateacher,ifyourstudentsaregoingtouseaproportiontohelpsolveaproblem,insistthattheyunderstandtheratiostheyareusing,andbeabletojustifywhythetworatiosshouldbeequal.Manystudentsinschoollearnatechniquecalled“crossmultiplication”tosolveproportions.Intheauthors’experience,thiscanhaveseveralbigunintendedconsequences.First,thistechniquebypassesandseeminglyviolatesafundamentalconceptofequations,namelythattopreserveanequality,whateverisdonetoonesideoughttobedonetotheothersideaswell.Second,studentscometothinkof“crossmultiplication”assomekindofmysteriousseparatekindofoperation.Third,manystudentsthenapplythismysteriousoperationwhenevertheymultiplyfractions,resultinginmistakessuchas%

&× "3= 4

!3.

The“crossmultiplication”techniqueissimplyashortcutthatallowsonetoskiponestepinsolvinganequation.Butinskippingthatonestep,wearguethatwearealsoshortcuttingan

125

understandingofequalityandofratios.Inthehomework,wewillaskyoutofindotherwaysofthinkingaboutandexplaininghowtosolveproportionsthatdonotbypasstheseimportantconcepts.

Homework

Lifeshrinksorexpandsinproportiontoone’scourage.AnaisNin

1) DoalltheitalicizedthingsintheReadandStudysection.

2) ExplainhowtheUSCensususesthecapture-recapturemethod.Whatisthe

population?Whatistheinitialsample?Therecapturedsample?

3) ThissummerinMinnesotatheDNRcapturedandtagged240small-mouthbassandthenreleasedthembackintoLakeWissota.Twoweekslater,theyreturnedtothelakeandcaught360small-mouthbass.Ofthose,81weretagged.

a) Estimatethenumberofsmall-mouthbassinthelakeandbrieflyexplainthe

ideabehindthisprocedure.b) Whatassumptionsmustbemadetousecapture-recaptureforthisexample?

4) Whatdoescapture-recapturehavetodowithproportionalreasoning?Explain.

5) Makeupyourowncapture-recaptureproblemandthensolveit.

6) Considerthewatermelonproblemagain,

a) Explainwhytheproportion99/100=98/xisnotanappropriateproportionthatcanbeusedtosolvethewatermelonproblem.Trytogiveatleasttworeasonswhythisproportiondoesn’tmakesenseforthisproblem.

b) Writeanappropriateproportionthatcouldbeusedtosolvethewatermelonproblem.

7) Solvethefollowingproblems.Inwhichofthesecouldyousetupaproportiontofind

thesolution?Inwhichoftheseisitnotappropriatetowriteandsolveaproportion?Explain.

a) WhenBobrunsat4milesanhour,ittakeshim20minutestorunthepatharoundthepark.Ifheraninsteadat5milesanhour,howlongwouldittakehimtorunthatpatharoundthepark?

b) IfittakesClara10minutestoassemble3chairs,howlongwillittakehertoassemble5chairs?

126

c) Ifittakes4people3hourstopaintafence,howlongwouldittake6peopletopaintthatfence?

d) Acellphoneplanhasafixedmonthlyfeeandthenaper-minuteusecharge.IfBobtalkedfor100minutesandwascharged$40,howmuchwouldhisphonebillhavebeenifhehadtalkedfor200minutes?

e) Theareaofacirclewithradius10feetisabout314squarefeet.Whatistheareaofacircleofradius20feet?

8) Showtwowaystoexplainhowsolvetheproportion!3

&= 63

7forx.Bothofyour

explanationsshouldsupportaconceptualunderstandingofratiosandequality.

9) SueandJuliewererunningaroundatrack.

a) SupposeSueandJulieranequallyfast,andSuestartedfirst.WhenSuehadrun9laps,Juliehadrun3laps.WhenJuliecompleted15laps,howmanylapshadSuerun?

b) SupposeSuerantwiceasfastasJulie,andSuestartedfirst.WhenSuehadrun9laps,Juliehadrun3laps.WhenJuliecompleted15laps,howmanylapshadSuerun?

c) RewritethefirstsentenceofthepreviousproblemssotheanswertohowmanylapsSuehadrunwouldbethesolutiontotheproportion8

%= 7

!3.

127

ChapterFive

RelationsAmongSetsofData

128

ClassActivity20:ShoeSizeandHeight

Arithmeticisbeingabletocountuptotwentywithouttakingoffyourshoes. MickyMouse

Yourclassisgoingtomakeahugelivingscatterplotonthefloorofyourclassroomtohelpyoupredictaperson’sshoesizebasedonhisorherheight.Youwillusemaskingtapeforyouraxesandstickynotesforlabeling.Then,everyonewillstandonthedatapointrepresentingtheirheightandshoesizetoobservethepatternoftherelationship.Firstsomedefinitions:Anexplanatoryvariableisonethatisusedtoexplainvariationsinaresponsevariable.Itisnotrequiredthatonevariablecausetheother,justthatonebeusedtopredictvariationintheother.Whenmakingascatterplotitisstandardpracticetousetheexplanatoryvariableonthex-axisandtheresponsevariableonthey-axis.Youwillneedtodiscussissuesthatmightariseinworkingwiththesedata.Howwillyoudealwiththefactthatwomen’sshoesaresizeddifferentlythanmen’sshoes?Whatscaleandunitsshouldbeusedonthex-axisandwhatscaleandunitsshouldbeusedonthey-axis?(Whenmakingascatterplot,itishelpfultospreadthedataoutasmuchaspossible.)Answerthesequestionsasaclassandmakeyourplot.Whenyouaredone,returntoyourgroups,butleaveastickynoteonthefloorwhereyoustood.

1) Arethereanyoutliersinthedata?Lookupthedefinitionanduseittomakeanargument.

2) Isitpossiblefortheretobeanoutlieronascatterplotthatisnotanoutlierineither

theexplanatoryortheresponsevariable?Explain.

3) Isitpossibletohavedatapointthatisanoutlierinboththeexplanatoryandresponsevariablesyetitisnotanoutlieronthescatterplot?Explain.

4) Comeupwithsomeideasabouthowtousetheplottopredictaperson’sshoesize

basedonherheight.Ifamanis6’8’’tall,whatdoyoupredicthisshoesizetobebasedonthepatternoftherelationship?

5) Placeastringortapeacrossthedatasothatthestringcapturesasbestyoucanthe

relationship.Findtheequationofthatline.(Makesureeveryoneinyourgroupunderstandshowtodothis.)Usetheequationtopredicttheshoe-sizeofa6’8”man.Howconfidentareyouinthisprediction?Explainyourthinking.

129

ReadandStudy

Itisacapitalmistaketotheorizebeforeonehasdata. SirArthurConanDoyleinScandalinBohemia

Sometimesthepurposeofdatacollectionistoexaminearelationshipbetweentwovariables.Thereareseveralwaystodisplayarelationshiplikethat.Wecould,insomecases,makeside-by-sideboxplots(likewedidintheStarTrekactivitytocomparethequalityoftheseasons)orwemightmakeside-by-sidebargraphs(likethisonebelowthatcomparesstreamtemperaturesundertwoconditions:underaclosedcanopyofvegetationandinopenskywithnoshieldingvegetation).

1999datafromtheCenterforUrbanWaterResourcesManagement(CUWRM)andtheCenterforStreamsideStudies(CSS)attheUniversityofWashingtoninSeattle

We’llhaveyouconsidersomeofthosedisplaysaspartofyourhomework.Inthissectionwewillfocusonthescatterplot.AscatterplotisagraphofnumericdatausingaCartesiancoordinatesystem(anx-axisanday-axis)sothateachcase(orvalue)isrepresentedbyapointhavinganx-coordinateanday-coordinate.Intheclassactivityyoumadeascatterplotcomparingheightandshoesizeforstudentsinyourclass.Eachperson(case)wasrepresentedontheplotfirstasthemselvesandthenasstickynote.Thisallowedyoutostudythepatternoftherelationshipandtomakepredictions.Youlikelyfoundarelativelylinearpatterninyourheight-shoedata,butothertypesofpatternsarepossible.

0

5

10

15

20

25

30

35

40

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

frequency

tempertature(°C)

WAStateStreamTemperatures

closedcanopy

opencanopy

130

Forexample,havealookattheCarbonDioxidelevelmeasuredeachmonthintheatmosphereatthevolcanoMaunaLoainHawaii.Thisdatasetshowsbothalineartrendandaperiodictrend.Theperiodicbehaviorcanbeexplainedbytheyearlycyclicbehaviorofgrowthanddecayofvegetation.Howwouldyouexplainthelineartrend?Whatarethetwovariablesbeingcomparedinthesedata?

Komhyr,W.D.,Harris,T.B.,Waterman,L.S.,Chin,J.F.S.andThoning,K.W.(1989),AtmosphericcarbondioxideatMaunaLoaObservatory1.NOAAGlobalmonitoringforclimaticchangemeasurementswithanondispersiveinfraredanalyzer,1974-1985;J.Geop.Res.,v.94,no.D6,pp.8533-8547.Graphfromhttp://celebrating200years.noaa.gov/datasets/mauna/image3b.html

Homework

Ingreatattemptsitisgloriouseventofail. VinceLombardi

1) DoalltheitalicizedthingsintheReadandStudysection.

131

2) Whenanincreaseintheexplanatoryvariablegenerallycorrespondstoanincreaseintheresponsevariable,wesaythattherelationshipisapositiveassociation.Whenanincreaseintheexplanatoryvariablegenerallycorrespondstoadecreaseintheresponsevariable,wesaytherelationshipisanegativeassociation.

a) Istheassociationbetweenheightandshoesizepositiveornegative(orneither)?

Explain.b) IstheassociationbetweentimeandcarbondioxideoverMaunaLoapositiveor

negative(orneither)?Makeacase.c) Whatkindofassociationisitifadecreaseintheexplanatoryvariable

correspondstoanincreaseintheresponsevariable?Explain.d) TrueorFalse?Wewouldexpectthevariablesof‘averagenumberofcigarettes

smokedeachday’and‘longevity’(lifespan)tohaveapositiveassociation.e) Makeuptwonewvariablesthatyouwouldexpecttohaveapositiveassociation.

3) Belowisasetofdatashowingthepositionofaglacierovertime.(Where“0”wasits

positionin1985.)

a) Makeascatterplotusingtimeastheexplanatoryvariableandpositionastheresponsevariable.

b) Descibethepatternoftherelationship.c) Usethepatterntopredicttheglacier’spositionin1997.d) Istheassociationnegativeorpositive(orneither).

RainbowGlaciershrinkage,NorthCascades,WA

datacourtesyofMauriPelto,NorthCascadesGlacierClimateProject,NicholsCollege

positionofglacier'ssnoutrelativeto1985positionnegativevalues=glacierretreats/shortens

year position(meters)1985 01986 -111987 -221988 -331989 -441990 -551991 -601992 -75

132

1993 -961994 -1161995 -1371996 -1611997 1998 -2011999 -2412000 -246

133

ClassActivity21:LotsAboutLines

Ithinkit’sthedutyofthecomediantofindoutwherethelineisdrawnandcrossitdeliberately.

GeorgeCarlinThisactivityisdesignedtoprovideyouwithareviewoflinearrelationships.Theremaybepeopleinyourgroupwhohaven’tthoughtaboutthesethingsinafewyears,soifthisisfamiliartoyou,youwillhavethechancetopracticebeingagoodteacher.

1) Theslopeofalineistherateofthechangeinyperoneunitchangeinx.Explainwhy

theformula makessenseasawaytocomputetheslopeofaline.What

picturesmightyoudrawtoillustratethis?2) Findtheequationofthelinethatpassesthroughthepoints(5,2)and(-1,12).Make

sureeveryoneinyourgroupunderstandsthis.

3) Sketchtheline .Thenusethegraphtodiscusshowyouwouldexplainthemeaningofthe andthe-12inthecontextofthegraph.

4) Postageforapackagecosts$4.50forthefirstpoundand$1.50foreachadditional

pound.a) Modelthepostagepriceasafunctionofweight.Inwhatwaysisthisrelationship

linear?Inwhatwaysisitnotreallyanidealline?b) Carefullyexplaineach‘number’inyourlineequation.c) Howmuchwillitcosttomaila12lbpackage?Figureitoutinthreedifferentways.

5) Howcanyourecognizelineardatafromatable?Explorethesethreesetsofdata

collectedonpopulationsofthreeexperimentalcoloniesonthreeplanetstoseewhatpatternsinthetableshowyoulineardata.Ifoneoftheseturnsouttobefairlylinear,findareasonableequationforthelinethatdescribestherelationshipbetweentimeandpopulation.Whatdoyouexpectthepopulationofthatcolonytobewhent=15?

ColonyI ColonyII ColonyIII

t(decades)Population t(decades)Population t(decades)Population 1 56 1 167 1 15 3 70 3 467 3 39 5 88 5 766 5 88 7 110 71064 7 162 9 139 91369 9261 11 174 111667 11 385

12

12

xxyym =

1231= xy

31

134

ReadandStudy

WheneverIseeaslipperyslope,myinstinctistobuildaterrace. JohnMcCarthy

Therearemanyrelationshipsthatcanbestbemodeledbylines:totalearningsifanemployeeispaidafixedamountperhour;shoesizeandheight;andtotaldistancetraveledwhendrivingaconstantspeed,tonameafew.Inordertogetreadytousealinetodescribearelationshiponascatterplot,wearegoingtoremindyouaboutlinearequationsingeneral.Ifyourememberallofthis,youcanfocusasyoureadonthinkingabouthowyouwillhelpyourfuturestudentstounderstandtheseideas.Okay,soimagineanyoldlinedrawnonaplane(thispieceofpaper).Recallthataplaneandalinearemathematicalobjects,andassuchareidealobjectsandnotrealobjects.Soimagethatthisplaneisinfiniteandsoistheline:itstretchesperfectlystraightandforeverinbothdirections.

Mathematicianshavecreatedready-madenotationforustousetodescribealinelikethis.Firstthinkofourplaneashavingasetofaxesonit:wecallthehorizontalaxisthe“x-axis”andtheverticalaxisthe“y-axis.”Thisgivesourplaneastructure.Nowpointsintheplanecanbedescribedbasedonanx-coordinateanday-coordinateasmeasuredfromtheorigin(theplacewheretheaxescross).Soifapointonthelineislocatedback3unitsandup5,thenithasthename(-3,5)anditlookslikethisontheplane.

135

Infactthislinecanbedescribedasawholesetofpointsthatallsatisfyanequationoftheform y=mx+bwheremandbareconstants(numberswecallparameters).Youneedonlytwopiecesofinformationtomaketheequationofaline:m(theslope)andb(they-intercept).Andifyoudon’thavethesedirectly,youcangetthemfromanytwopointsontheline.Here’show:

(-3, 5)

136

Supposethat(-3,5)and(4,1)arepointsonthelinebasedonthewaywe’veputdownouraxes.Sothatmeansthatwhenwemovefrom-3to4onthehorizontalaxis(that’s7units),wemovedown4unitsontheverticalaxis.Nowslope(m)isdefinedtobethechangeinyforaone-unitchangeinx,soforeveryoneunitwegorightwegodown ofaunit:m= .Noticethatnegativeslopescorrespondtonegativeassociationsandpositiveslopestopositiveassociations.Sofarourequationis:y= x+bTogetthevalueofb,we’llnowreuseapoint.Since(4,1)isontheline,itsatisfiesourequation,so 1= ×4+bAndsowefindthatb=1+ orb= + = .Dothiscalculationyourself.Thefinalequationforourlineisy= x+ .Youmightbewonderingwhetherwecouldhaveusedthepoint(-3,5)insteadof(4,1)tofindourb.Trythatnowandseethatyoudogetthesamevalueofb.Reallydoit.Sofar,allofthishasbeenprettyabstract.IntheHomeworkandinthenextsectionwe’lllookatsomemeaningfulsituationsinvolvinglinearrelationships.ConnectionstotheElementaryGrades

Ingrades3-5allstudentsshouldinvestigatehowachangeinonevariablerelatestoachangeinasecondvariable.

NationalCouncilofTeachersofMathematicsPrinciplesandStandardsforSchoolMathematics

74

74

74

74

716

77

716

723

74

723

137

Ifaslopeisdefinedtobetherateofchangeintheyvalueforaone-unitchangeinthexvalue,thenwhataretheunitsofslope?Well,ofcoursethatdependsontheunitsonthexandyvariables,solet’slookataspecificexample.Supposexistimeinminutesandyrepresentsthedepthofrisingwaterinatankininches,thenslopegivesachangeinwaterdepthforaunitoftime.Inotherwords,itisarateanditsunitsareinches/minute.Iftherelationshipbetweenwaterdepthandtimeislinear,thenthatrateisconstant(theslopeisthesameforallvaluesofx).Ifthetankisfillingslowerandslower,thentherelationshipwillnotbelinear(thegraphwilllookcurved),andtheratewillchangewithtime(andsowilltheslopeofthegraph).

Homework

Inchesmakechampions. VinceLombardi

1) DoalltheitalicizedthingsintheReadandStudysection.

2) DotheproblemsanditalicizedthingsintheConnectionssection.Whatcouldyour

studentslearnaboutslopesbydoingtheEverydayMathematicsproblems?Makeanequationexpressingprofitasafunctionoftimeforeachgirl.

3) Thedepthofwaterinatankbeginsat3feetandisfillingatarateof2feet/hour.Makeanequationforthedepthofthewaterasafunctionoftime.Nowgraphtherelationshipwherexismeasuredinhoursandyismeasuredinfeet.Howdoesthe3feetshowuponthegraph?The2feet/hour?

4) Hereisatableshowingthepricesforfudgeatacandystore:

#ofounces Price4 $3.208 $6.0016 $11.0032 $20.0048 $29.0064 $38.00

138

a) Isthepriceoffudgealinearfunctionofweight?Explainhowyoucantell

fromthetablealone.b) Nowgraphthisdata.Whatdoyounotice?Findanequationforthepriceof

fudgeasafunctionofweightwhenyoupurchasemorethan16ounces.Whatexactlydoestheslopetellyouinthiscase?

c) Istheassociationbetweenweightandpricepositiveornegative?Explain.

5) Giveexamplesofapairofvariablesthatyouwouldexpecttobelinearlyrelated.Explainyouthinking.Istheassociationbetweenyourvariablespositiveornegative?Why?

6) Findtheequationofthelinethatpassesthroughthepoints(2,-5)and(4,-1).Explainyourworkasyouwouldtoastudent.

7) Graphthelinethathasslopem= andpassesthroughthepoint(-1,-5).Explain

yourworkasyouwouldtoastudent.

ClassActivity22:ManateeData

BarbaraManatee,youaretheoneforme. SungbyLarrytheCucumber,VeggieTales

Themanateeisontheendangeredspecieslistandsoitspopulationiscarefullytracked.AresearcherbelievesthatmanateedeathsinFloridacouldbecausedbyboating.Inordertomakeacase,shefoundthefollowingdataonboatregistrations(FloridaDepartmentofMotorVehicleswebsitegivesdatafor2000-2005.Therestareestimates.)andknownmanateedeaths(fromtheFloridaFishandWildlifeResearchInstitutewebsite)forthepastseveralyears.Makeascatterplotofthesedata.1) Istherearelationshipbetweenthenumberofboatregistrationsandthenumberof

manateedeaths?Explain.

2) UseyourscatterplottopredictthenumberofManateedeathswhenthenumberofboatersis900,000.Youjustinterpolated(lookthattermupintheglossary).Howmanydeathswilltherebewhenthenumberofregistrationsreaches2,000,000?Youjustextrapolated.

3) Basedonthesedata,wouldyoubewillingtoconcludethatboatingiscausingthedeaths

ofmanatees?Explainyourthinking.

4) Findtheequationofalinethatfitsthedata.Interpretthemeaningoftheslopeandy-interceptofyourlineinthecontextofthesedata.

32

139

Year

FLBoatRegistrations(thousands)

No.ofManateeDeaths

1991 598 1741992 628 1631993 643 1451994 712 1931995 730 2011996 785 4151997 800 2421998 820 2321999 852 2692000 880 2722001 943 3252002 961 3052003 978 2762004 983 3962005 1010 417

140

ReadandStudy

Factsarestubbornthings,butstatisticsaremorepliable. MarkTwain

Inthissectionwewilldiscusstheideaoffittingabestlinetodata.Typicallyinpracticeyoursoftwarewillactuallyperformthefit(calleda‘leastsquaresregression’orsimplya‘linearregression’);whatyouneedtodoisunderstandwhatitallmeans.Firstthepurposeoffittingaline(oranotherfunction)tomodeldataistobeabletomakepredictionsaboutvaluesoftheresponsevariable.Forexamplelet’sthinkaboutthisdatatableandthebelowplotpresentedbytheQuantitativeEnvironmentalLearningProjectatthewebsite:http://www.seattlecentral.edu/qelp/sets/028/028.html.

Source:USGeologicSurveyWaterResourcesInvestigationsReport89–4004WaterQualityandSupplyonCortinaRancheria,ColusaCounty,CaliforniaTable1:Characteristicsofselectedstreamsalongthewestsideofthe

SacramentoValley

GaugingStationEstimatedmeanannualrainfall

(inches)

DrainageArea(mi2)

Meanannualflow(ft3/sec)

averageannualrunoffperunitarea(in./area)

MiddleForkCottonwoodCreek

nearOno40 244 254 14.1

RedBankCreeknearRedBluff 24 93.5 44.3 6.4

ElderCreekatGerber 30 136 96.2 9.6

ThomesCreekatPaskenta 45 194 304 21.2

GrindstoneCreeknearElkCreek 47 156 193 16.8

StoneCorralCreeknearSites 21 38.2 6.1 2.2

BearCreeknearRumsey 27 100 44.3 6.0

Itmakessensethatacommunitywouldwanttopredicttheamountofwaterrunoffbasedonrainfalltobetterunderstandflooding,nutrientlossinthesoil,ordownstreampollutantsfromfarms.Thefirstthingtodothenmightbetomakeascatterplotofthesedatatoexaminethepattern.

141

Havealookatthedatapoints.Noticethateachpointrepresentsameasurementfromastreamorcreek.Alsonoticethatthissetofdatashowsafairlylinearpatternandthattherearenooutliersamongthesedata.Iftherewasanoutlieronthescatterplot,shoulditbeincludedwhenfittingabestlinetothedata?Explainyourthinking.Whatdoesitmeantoforalinetobea‘bestfit?’Gladyouasked.Mathematicallyabestfitline(orregressionline)isonethatminimizesthesumofthesquaresoftheverticaldistancesofthepointstotheline.Hmm.Readthatdefinitionagain–itisatoughone.Wewillspendthenextparagraphsexplainingjustwhatthatmeansandthisisabigideasowewantyoutoworkhardtounderstandit.Imaginethatthesoftwareprogramonyourcalculatororcomputerputsdownalineontheabovescatterplot(anyline)likebelow.Thenitcomputesthedistancethateachpointisfromtheline(measuredvertically)andsquareseachofthosedistances.Thosesquaresarerepresentedgeometricallyonourgraph.Finallythecomputersumstheareasofallthesquaresandreturnsanumber.

0

5

10

15

20

25

0 10 20 30 40 50

AnnualRunnoff(inches)

AnnualRainfall(inches)

SacramentoValleySelectedDrainages

142

Nowimaginethecomputerdoesallthisagainwithadifferentline.Haveacloselookatthenextpicturetobesureyoucanseehowitshowshowthenewlinecreatesdifferentsquares.

This area represents thesum of the squares of thevertical distances of thepoints to the line we draw

20

10

050454035302520

Mean Annual Rainfall (in inches)

Average annual runoff per unit area (in inches)

Water Runoff Data

=

++ ++++Area=

This area represents thesum of the squares of thevertical distances of thepoints to the line we draw

20

10

050454035302520

Mean Annual Rainfall (in inches)

Average annual runoff per unit area (in inches)

Water Runoff Data

=

++ ++++Area=

143

Thissecondlineisa‘better’linebecausethesumofthesquares(thefinalgraysquare)issmaller.Nowimaginethatyoursoftwaredoesthisprocesswithallpossiblelinesandthentellsyouwhichoneofthemgavetheverysmallestfinalsum.Thatlineiscalledtheleastsquaresregressionline(seehowthenamefitstheprocess?)orthelineofbestfit.Spendsometimestudyingourexplanationsothatyouunderstanditwellenoughtoexplainittosomeone.Thewayyourcomputerwillreturnthisinformationtoyouisprobablyintheformofaslopeanday-intercept–inotherwordsyou’llgettheequationoftheline.Youmayalsogetanumber(r)calledthecorrelationcoefficientthattellsyouhowgoodthefitreallyis–butthatisatopicforthenextsection.Forthewaterrunoffdata,thelineofbestfitisgivenbytheequation:y=0.619x-9.82whereyisannualwaterrunoffperunitarea(ininches)andxisannualrainfall(ininches).The0.619istheslopeoftheline,andsoittellsusthatifwearestandingontheline,foreveryunitwemovehorizontallyinthepositivedirection,weneedtomoveup0.619unitstostayontheline.Orinthecaseofthesedata,ittellsusthatforevery1inchincreaseintheannualrainfall,wegetanincreaseof0.619inchesperunitareaofrunoff.Makesurethatyouunderstandthatlastsentence–itisimportantbecauseitshowshowtomakesenseofthenumbersintheequationintermsofthephysicalsituation.Inthiscontextthey-interceptof-9.82meansthatiftherewasnorainfallatall(x=0),we’dgetnegativerunoff(whichsuggeststhatthedatalooksdifferentforverysmallamountsofrain).Itisimportantthatyouthinkaboutthemeaningofyourlineofbestfit.Wesaidthataregression(orbestfit)lineisfoundsothatwecanmakepredictionsaboutvaluesoftheresponsevariable(inthiscasewaterrunoffinstreams)givenvaluesoftheexplanatoryvariable(inthiscasetheannualrainfall).Usetheequationofthebestfitlinetopredicttherunoffinastreamthatgets60inchesofannualrainfall.Didyoujustinterpolateordidyouextrapolate?Why?Howconfidentareyouinyourprediction?Explain.

144

Homework

Theonlythingthatcomestouswithouteffortisoldage. GloriaPitzer

1) DoalltheitalicizedthingsintheReadandStudysection.

2) DecidewhethereachofthefollowingisTrueorFalseandexplainyourthinking.

a) Outliersshouldberemovedfromadatasetbeforefindingalineofbestfit.b) Thecomputerwillfindabestlineevenifthedataisnotlinear.c) Apositiveslopeforabestfitlinethroughdatameanstheexplanatoryand

responsevariablehaveapositiveassociation.d) Apositiveassociationbetweentheexplanatoryandresponsevariablesmeans

therewillbeapositiveslopeforthelineofbestfit.e) Thebetterthelinefitsthedata,thebetteritspredictivepower.f) Thebetterthelinefitsthedata,themorelikelytheexplanatoryvariable

causesvariationsintheresponsevariable.

3) Alinearrelationshipwasfoundbetweenthenumberofhoursstudentsstudiedforanexam(t)andthescoreoftheexam(s).Theleastsquaresregressionlineforthedataisgivenbythefollowingequation:s=6.5t+52.

a) Whatexactlydoesthe6.5tellyouintermsofthissituation?Whatareitsunits?

b) Whatdoesthe52tellyouandwhatareitsunits?c) Ifastudentstudiedfor4hours,whatisherpredictedscoreontheexam?d) Ifastudentscoreda90ontheexam,howlongwouldyoupredictshestudied?e) Giveatleasttwodifferentpossibleexplanationsfortheassociationbetween

hoursstudyingandexamscore.

4) Thedataforthegoldmedalperformancesforeachyearsince1900isprovidedininchesinthetableonthenextpage.Forthisproblemyoushoulduseacalculatororanonlineappletforcomputingregressionline(askyourteachertorecommendone).

a) Foreachdataset,enterthedataintotheAppletwithtimeasthex-variableandrecordperformanceasthey-variable.

b) Beforeyoudotheregressiondescribetheshapeofthedata.Istherelationshiplinear?

c) Ifthedataislinear,dotheregressionandusethelineofbestfittopredicttheOlympicperformancefortheyear2012.Howconfidentareyouinthatprediction?Explain.

145

YEAR LongJump(in) HighJump(in) Discus(in)1900 282.9 74.8 1418.91904 289.0 71.0 1546.51908 294.5 75.0 1610.01912 299.2 76.0 1780.01920 281.5 76.3 1759.31924 293.1 78.0 1817.11928 304.8 76.4 1863.01932 300.8 77.7 1948.91936 317.3 79.9 1987.41948 308.0 78.0 2078.01952 298.0 80.3 2166.91956 308.0 83.3 2218.51960 319.8 85.0 2330.01964 317.8 85.8 2401.51968 350.5 88.3 2550.51972 324.5 87.8 2535.01976 328.5 88.5 2657.41980 336.3 92.8 2624.01984 336.3 92.5 2622.01988 343.3 93.5 2709.31992 342.5 92 2563.8

5) NowaddanextrapointvariousplacesonthegraphoftheabovedataandusetheApplettoseehowthepositionofanoutlieraffectsthepositionoftheline.Printouttwoofyourpicturestobringtoclass.

6) Useanonlineappletoryourcalculatortofindthelineofbestfitforthemanatee

problem.Useyourlinetopredictthenumberofmanateedeathswhenthereare700,000boatregistrations.Howconfidentareyouintheprediction?Whatdoestheslopeofthelinetellyouintermsofthissituation?Whatdoesthey-intercepttellyou?Explain.

146

ClassActivity23:Correlation

Themostmercifulthingintheworld,Ithink,istheinabilityofthehumanmindtocorrelateallitscontents.

H.P.LovecraftConsiderthescatterplotsbelow.Ineachcasedescribehowthevariablesonthexandy-axesappeartoberelated.Whatexactlydoyouthink“r”ismeasuring?

Correlation(r)=0.05 Correlation(r)=0.99

Correlation(r)=0.70 Correlation(r)=-0.70

(Thisactivityiscontinuedonthenextpage.)

0

5

10

15

20

25

0 10 20 300

5

10

15

20

25

0 10 20 30

0

5

10

15

20

25

0 10 20 300

5

10

15

20

25

0 10 20 30

147

Thecorrelationcoefficientrisastatisticthatmeasuresthestrengthofthelinearrelationshipbetweentwovariables.Itisdefinedbythefollowingequationwherenisthenumberofvalues,mxisthemeanofthexvariable,myisthemeanoftheyvariable,sxissomethingcalledthestandarddeviationofthexvariable,andsyisthestandarddeviationoftheyvariable.(Don’tworry-youwon’tneedtocomputestandarddeviationsorcorrelationcoefficients.Justhavealookattheequationtoseewhatcluesitgivesyou):

1) Doesitmatterwhichvariableistheexplanatoryvariableandwhichistheresponse

whencomputingcorrelation?Whyorwhynot?

2) Supposexistime(inmonths)andyisheightofaplant(ininches).Whatunitsdoesrhave?Explain.

3) Doesthevalueofrchangewhenwechangetheunitsofmeasurementforxory?Whyorwhynot?

4) Isrresistanttooutliers?Whyorwhynot?

5) TrueorFalse?Ifr=0,thatmeansthereisnorelationshipbetweenthevariablesxandy.Explainyourthinking.

+++×=y

yn

x

xn

y

y

x

x

y

y

x

x

smy

smx

smy

smx

smy

smx

nr ...

11 2211

148

ReadandStudy

[Inart]there’snocorrelationbetweencreativityandequipmentownership.None.Nada.

HughMacleodPeopleusetheterm‘correlated’whentheyshouldsay‘associated.’Inmathematics,twonumericalvariablearehighlycorrelatedifhaveastronglinearrelationship.Ifthevariablesarenotmeasuredinsomenumericalmanner,thentheycannotbecorrelated.Ifthevariablesshowsomeotherrelationship(besideslinear)theymightbeverystronglyassociated,butnotveryhighlycorrelated.ReadthequotesthatheadtheClassActivity,ReadandStudy,andHomeworksectionsaboutcorrelation.Dotheauthorsmeantousetheterm?Orwould‘association’bebetter?Explain.Asmathematicsteachersyouwillneedtopracticecarefulmathematicallanguage.Correlationiscapturedbyastatisticcalledthecorrelationcoefficient,r,thatcanvarybetweenr=1,meaningthatthedatahasaperfectlinearrelationshipwithapositiveslopetor=-1,meaningthatthedatashowsaperfectlinearrelationshipwithanegativeslope.Whenr=0,thereisnolinearrelationshipatall.Wewillnotcomputeacorrelationourselves–ourcalculatororcomputerwilldothat–butwhatwewilldoisinterpretit.Belowyouwillseelotsof‘clouds’ofdata(eachlittlepictureisadataset)alongwiththeircorrelationcoefficients(fromWikipedia.com).Studythem.Notethatthosewithapositiveslopehaveapositivevalueofrandthosewithnegativeslopehaveanegativevalueofr.Noticethatlotsofcloudsthatshowarelationshipbetweenthevariableshaveanrofzero(checkoutthatbottomrow).

(usedbypermission)

149

Wehavetoldyouthatthecorrelationcoefficientmeasuresthestrengthofalinearrelationshipbetweentwovariables–butwehavetoaddawordofcaution.Thenumberrshouldbecomputedonlywhenitmakessensetodoso.Wewillusethefourdatasetsbelowtoshowyouwhatwemean.Allofthesedatasetshaveacorrelationcoefficientof0.8,yetinonlythefirstcase(DataSet1)doesthevaluemakesense.ItmakesnosensetofitalinetoDataSet2whenthatdatahasanonlinearshape.DataSets3and4bothhaveoutliersthatprobablyshouldhavebeendiscardedbeforefittingalinetothedata–afterthat,thelinearrelationshipislikelytobemuchstrongerthan0.8.

Inthisclass,anysetofnicelinear-typedata(likethatinCase1above)datawithacorrelationcoefficientbiggerthan0.75willbeconsideredlineardata,butdifferentpeoplewhousestatisticswillhavedifferentcriteriaforwhatconstitutesagoodline.Onemorepoint.Evenastrongcorrelationdoesnotimplyacausalrelationshipbetweenthevariables.Here’sanexample:thereisastrongcorrelationbetweenteachers’raisesandalcoholsalesinthestateofWisconsin.Thebiggertheraise,themostalcoholissold.Isthisacausalrelationship?Maybe.Perhapsteacherscelebratetheirraisesbyhavingparties.Butmorelikelyitisthecasethatinyearswhenthereisagoodeconomyteachersgethigherraisesandalcoholsalesarehigher.Soitmaybethattheeconomyisalurkingvariablehere–ahiddenvariablethatcausesbotheffects.

0

5

10

15

0 5 10 15

DataSet1

024681012

0 5 10 15

DataSet2

0

5

10

15

0 5 10 15

DataSet3

0

5

10

15

0 5 10 15 20

DataSet4

150

Sohowdowedetermineacausalrelationship?Weperformaclinicalstudy.We’lltalkmoreaboutthatinthenextsection.

Homework

IfIhadtoselectonequality,onepersonalcharacteristicthatIregardasbeingmosthighlycorrelatedwithsuccess,whateverthefield,Iwouldpickthetraitofpersistence.

RichardM.DeVos

1) DoalltheitalicizedthingsintheReadandStudysection.2) DecidewhethereachofthefollowingisTrueorFalse.Explainyourchoice.

a) Ifthecorrelationcoefficientis1,thenthatmeansthatonevariablecausesthe

other.b) Ifthedatahasacorrelationcoefficientofzero,thenthatmeansthevariables

arenotrelated.c) Wecouldmeasurethecorrelationbetweenfavoritecolorandpersonality.

3) Hereisadatasetthatshowsdatafor12perchcaughtinalakeinFinland:

Length(cms) 8.819.222.523.524.025.528.730.139.041.442.546.6

Weight(grams) 61001101201501453003006856508201000

a) Makeacarefulscatterplotoflengthversusweightbyhandforthesedataanddescribethepattern.Arethereanyoutliers?Explain.

b) UseacalculatororonlineApplettoperformalinearregressionandtofindthecorrelationcoefficient.Whatdoesrtellyou?Usethislinetopredicttheweightofafishthatis12cmlong.30cmlong.60cmlong.Wouldyoutrustthesepredictions?Explain.

c) Doesitmakesensethattheweightoffishwouldbealinearfunctionoflengthorwouldanothercurvebetterdescribetherelationship?Explainyourthinking.

High Teachers Salaries High Alcohol Sales

Good Economy

151

4) Ineachcase,giveapossiblelurkingvariablethatmayexplaintheassociation

described:

a) AsthenumberofboatingpermitsincreasesinFLsodothenumberofmanateedeaths.Aresearcherclaimsthatboatersarekillingthemanatees.Canyouarguewithher?

b) Thereisastrongpositiveassociationbetweenheightandreadingabilityamongelementarychildren.Why?

c) Peoplewhouseartificialsweetenerstendtobeheavierthanthosewhousesugar.Aresearcherclaimsthateatingartificialsweetenerscausespeopletogainweight.Canyouarguewithhim?

d) Thebiggerthehospital,thehigherthepercentageofdeaths/peradmission.Doesthismeanlargerhospitalsgivepoorercare?

e) AresearcherfindsthatpeoplewholistentoclassicalmusichaveahigherIQthanthosewhodonot.Basedonthisfinding,isitreasonabletoassumethatlisteningtoclassicalmusicboostsone’sIQ?Explainyourthinking.

152

ClassActivity24:CaffeineandHeartRate

Realistsdonotfeartheresultsoftheirstudy. FyodorDostoevsky

Todayinclasssomeofyouwillbeparticipatinginaclinicalstudytodeterminewhetherconsuming8ouncesofcaffeineraisestheheartrateofadults.Thoseofyouwhochoosenottoparticipatewillcomposetheresearchteam.Yourinstructorhasbroughttwobottlesofsodathatareexactlyalikeexceptthatonecontainscaffeineandtheotherdoesnot.Neitheryounortheresearchteamwillknowwhichbottleiswhichuntiltheexperimentiscomplete.Inordertoparticipate,youmustbewillingtodrink8ouncesofsoda(thatmayormaynotcontaincaffeine)duringthenexttenminutes.Thosewhodonotparticipatewillpourandservesoda,andrunandrecordheartratesessions.Decidenowifyouwillparticipateintheclinicalstudy.

I) Participantsshouldbeassignedrandomlytotwogroups.(EventuallyonegroupwillconsumesodafromBottleIandtheotherwillconsumesodafromBottleII).Theresearchteamshoulddecidehowtodothis.

II) Allparticipantsinbothgroupswilltakeaone-minuterestingheartrate.

III) Researchersshouldrecordheartratesforeachparticipantandrecordwhichgroup

heorsheisin(IorII).Theparticipantswilldrink8ouncesofsodafromtheirdesignatedbottle.

IV) Afterparticipantsarefinisheddrinkingand15minuteshaveelapsed(duringwhich

youwilldiscussingroupsthebelowquestions),everyoneshouldonceagaintakea1-minuteheartrate.

V) Finallytheresearchteamshouldmeasurechangesinheartrateandreportresults

oftherawdata.Questionsforsmall-groupdiscussion:

1) Doesdrinking8ouncesofcaffeineraiseheart-rate?Whatstatisticsmightyoucomputeinordertoanswerthisquestion?

2) Inthecasewhereonegroupwasdifferentfromtheother,howmightyoudecideif

thatdifferenceisjustduetorandomchance?3) Whatfactorsmighthaveaffectedtheresults?Howmightyouimprovethedesignof

thisstudy?

153

ReadandStudy

Alllifeisanexperiment. RalphWaldoEmerson

Inthissectionwe’lldiscussthelanguageinvolvedindoingaclinicalstudy.First,thepurposeofaclinicalstudyistotrytodetermineacauseandeffectrelationshipbetweentwovariables.Thecaffeineandheart-ratestudyyoudidintheclassactivitywasanexampleofsuchastudy.Thegoaltherewastoseewhetherdrinkingsomecaffeineraisedheart-rate.Intheworldofclinicalstudies,goodonesarerandomized-controlled–meaningthatthereisatreatmentgroupandacontrolgroup(towhichparticipantswererandomlyassigned)andtheonlydifferencebetweenthosetwogroupsisthetreatmentvariable.Evenbetteraretheblindrandomizedcontrolledstudies-meaningthattheparticipantsdonotknowwhethertheyaregettingthetreatment.Inthatcase,usuallya‘faketreatment’isgivencalledaplacebo.Theverybestclinicalstudydesignisonethatisadouble-blindrandomizedcontrolledstudy–inthatcaseneithertheparticipantsnortheresearchersknowwhoisgettingthetreatment.Thestudyyoudidinclasswasofthislasttype.Thinkthatthroughandmakesureyouagree.Whatwastheplacebointhatstudy?Justbecauseastudyisarandomized-controlled,double-blindclinicalstudydoesn’tnecessarilymeanitwillyieldgoodresults.Asyoumayhavefoundintheclassactivity,theconditionsoftheclassroommaynothavebeenidealformeasuringheartrate,theremaynothavebeensufficienttimetometabolizethecaffeine,orperhapsthesampleofparticipantswasflawed(toosmallortoohomogeneous–allthedifficultieswithsamplingcancomeupindesigningclinicalstudiestoo).Finallythereistheproblemofdecidingwhethertherewasreallyan‘effect’orwhetherthedifference(ifany)foundbetweenthetwogroupswassimplyduetorandomchance.Let’sconsideranexampleofarealclinicalstudy:TheSalkPolioVaccineTrialof1954.Itisanamazingclinicalstudybecauseitisunprecedentedinscopeandscale.Here’sthestory(assummarizedfromapaperbyDawson,L.(2004).TheSalkpoliovaccinetrialof1954:risks,randomizationandpublicinvolvement,ClinicalTrials,1,pp.122-130.)In1954,aterrifyingdiseasecalledpoliowasaleadingcripplerofchildrenacrosstheglobe.Justtwoyearsbefore,abreakthroughhadoccurred:JonasSalkandhisteamhaddevelopedavaccineusingakilled-formofthepoliovirus.TheNationalFoundationforInfantileParalysis(NFIP)sponsoredthestudyinwhich1.8millionchildrenparticipated.Thestudywastobedouble-blindandrandomized-controlled(althoughintheend,someofthechildrenincertaincitiesdidnotreceivetheplacebo).Thestudyneededtobehuge–onlyabout5childrenper10,000typicallygotpolioinanygivenyear–anditneededtoberandomized-controlledbecausethenumberofcasesvariedalotfromyeartoyear(sojustcomparinghowmanychildrenhadpolioinyearsbeforetohowmanygotin1954wouldnotgiveenoughevidenceforeffectivenessofthevaccine).

154

Thinkoftherisk.Whatifthevaccinewasn’tsafe?Whatifsomelivevirusremainedinthevaccines?Whatiftherewereunpredictableside-effects?Still,parentsweresoafraidofthedisease,andtheNFIPsoeffectiveatpromotingthetrial,thatover60%ofparentssigneduptheirchildrentoparticipate.ChildrenwereinoculatedwitheitheraplacebooravaccinedesignedtoprotectagainstthreeformsofpoliobetweenApril26andJune15,1954.AboutoneyearlateronApril12,1955,morethanonehundredandfortyreportersgatheredattheUniversityofMichigantoheartheresultsofthetrial:protectionwas68%againsttypeI,100%againsttypeII,and92%againsttypeIIIinfection.ThevaccinewaslicensedbytheUSPublicHealthServicethatsameday.TodaypolioisallbuteradicatedintheUS.

Homework

Youknowhowitiswhenyougotobethesubjectofapsychologyexperiment,andnobodyelseshowsup,andyouthinkmaybethat’spartoftheexperiment?I’mlikethatallthetime.

StevenWright

1) DoanyitalicizedthingsintheReadandStudysection.2) Thefollowing(fictitious)studywasconductedtodeterminewhetheranew

readingprogram“ReadOn”wasmoreeffectiveatteachingchildrenreadingthanatraditionalapproachusingphonics.ThreeelementaryschoolsinSanDiegothathadbeenusingthephonicsapproachforseveralyearsagreedtoparticipateinthestudy.The“ReadOn”approachwasusedbyallfirstgradeteachersin2002andreadingscoresfromthe120firstgradersattheendof2002werecomparedwiththedatafromthe112firstgraderswhohadusedthephonicsapproachin2001.

b) Whatwasthepopulationforthisstudy?c) Whatwasthesample?d) TrueorFalse?Thisstudyisanexampleofarandomizedcontrolledstudy.

Explain.e) TrueorFalse?Thisstudywasblind.Explain.f) Givetwogoodexamplesofpossiblesourcesofselectionbiasforthisstudy.g) Ifthisstudyshowedthatthe“ReadOn”studentsscoredsignificantlyhigheron

testsofreadingthandidthephonicsstudents,wouldyoubuytheargumentthat“ReadOn”isthemoreeffectiveprogram?Explain.

h) Aconfoundingvariableinaclinicalstudyisavariablethatwasnotcontrolledintheexperimentandmayhavecausetheobservedeffect.Giveanexampleofsuchavariableforthisstudy.

155

3) Aresearcherbelievesthatanewdietsupplementwillimprovemarathon

performance.Inordertotesthertheory,shecontactedbyemail500peoplewhohadregisteredearlyfortheGreenBayMarathon.42peopleagreedtoparticipateinherstudy.Half(21runners)wereselectedatrandomtotakehersupplementsdailyfortwomonthspriortotheevent.Theremaininghalftrainedasusual.

Inthetreatmentgroup,16runnerscompletedtherace.Herearetheirfinishingtimes(inminutes):

302,288,276,265,256,230,220,212,210,210,201,200,170,150,148,146

Inthecontrolgroup,15runnersfinishedtherace.Herearetherefinishingtimes(inminutes):

476,310,278,266,264,230,210,210,208,202,200,180,176,146,142

b) Whatisthepopulationforthisstudy?c) Findthemeanofeachsetoffinishingtimes.d) Findthe5-number-summaryofeachsetandmakeboxplotstohelpyou

comparethesedata.e) Arethereanyoutliersineitherset?Useourcriteriontodecide.f) Theresearcherclaimsthathersupplementloweredfinishingtimesforrunners

inthetreatmentgroup.Doyouagree?Makeagoodargument.g) Theresearcherclaimsthathersupplementlowersfinishingtimesinmarathon

runners.Doyouagree?Makeanothergoodargument.h) Nowputbothsetsofdatatogetherandmakeaside-by-sidebargraph.What

doesthispresentationofthedatashow?

156

ClassActivity25:MusicandMath

Wherewordsfail,musicspeaks. HansChristianAnderson

Aresearchteambelievesthattakingpianolessonswillimproveelementaryschoolchildren’sspatialreasoningabilities.Inordertotestthisconjecture,theyselectedatrandom20elementaryschoolsinWinnebagoCounty,WIandthenselected20firstandsecondgradersatrandomfromeachschool.Ofthese400childrenwhowereidentified,360agreedtoparticipate.Halfofthechildren(againchosenatrandomfromthe360)weregiven1yearofpianolessonsforonehoureachweek.Theotherhalfweregiven1yearofcomputerlessonsforonehoureachweek.Allchildrenweretestedbothbeforeandaftertheyearoflessonsusingastandardizedtestofspatialvisualization.Thepianogroupmadesignificantlybettergainsonthetestafter1yearthandidthecomputergroup.DecidewhethereachofthefollowingisTrueorFalseandbrieflyjustifyyourgroup’schoice.

1) Thestudydescribedusesclustersampling.2) Thepopulationiselementaryschoolchildrenin20WinnebagoCountyschools.

3) Thesamplingrateis90%.

4) Thestudydescribedisarandomizedcontrolledstudy.

5) Thestudydescribedisblind.

6) Thestudydescribedisdoubleblind.

7) Thisstudyusesaplacebo.

8) ThisstudymaybebiasedbecausechildreninWisconsinmaynotberepresentativeof

allchildren.

9) Basedonthisstudy,yourgroupwouldconcludethattakingpianolessonsdoescauseimprovementinthespatial-visualizationofelementaryschoolchildren.

Now,giveexamplesofsomepossibleconfoundingvariablesforthisstudy.

157

ClassActivity26:ClassSurveyRevisited

Statistics:Theonlysciencethatenablesdifferentexpertsusingthesamefigurestodrawdifferentconclusions.

EvanEsarInanearlieractivity,youdesignedaclasssurveyinordertoexamineteacherpreparationatyourschool.Today,youwilltakeallthosesurveysasaclass,andthenyourgroupwillbegiventherawdatageneratedbyyourquestions.

1) YourgroupshoulddecidehowtodisplayeachdatasetforyourreporttotheAmericanFederationofTeachersusinggraphicalrepresentationswehavediscussedinthischapter.Youwillneedtodecidehowtohandleanydifficultiesthatarise.Remembertolabelyourgraphs.Hereisareminderofsomepossibletypesofdisplays:

PictogramsStemandLeafplots(Stemplots)LineplotsBargraphsHistogramsBoxplots

2) Whatcautionsregardingthedatadoyouwanttohigh-lightinyourreporttotheAFT?

3) Whatdidyoulearnaboutwritingsurveyitemsfromthisactivity?Bespecific.

4) Selecttwoofyouritemsthatmightberelated(e.g.,Numberofhoursofstudiedper

weekandCollegeGPA)andmakeadisplaythatwouldhelpyouanalyzetherelationship.

158

ChapterSix

ProbabilityRevisited

159

ClassActivity27:CasinoRoyale

AlltheevidenceshowsthatGodwasactuallyquiteagambler,andtheuniverseisagreatcasino,wheredicearethrown,androulettewheelsspinoneveryoccasion.

StephenHawkingTheCasinoRoyalegameattheCountyFairallowsyoutospinawheelcontaining16spacesnumbered1through16.Ifyouspinanyoddnumber,youloseandgetnothing.Ifyouspinamultipleof4(thenumbers4,8,12,or16),youwin$3.Ifyouspinanythingelseyouwintheamountofmoneyspecifiedbythenumber(e.g.,spina2,get$2;spina6,get$6,etc.).Ifyouplaythisgamemanytimes,whatcanyouexpecttowinonaverage?Afairgameisoneinwhichyourexpectednetwinningswouldbe$0.Whatshouldaticketcosttomakethisafairgame?Supposetheticketcost$3andtherulesofthegamedonotchange.Howcanyouchangethepayoutstomakeitafairgame?Istheremorethanonesetofpayoffsthatwouldwork?Explain.

160

ReadandStudyYourbestchancetogetaRoyalFlushinacasinoisinthebathroom.

V.P.PappyIntheCasinoRoyaleclassactivityyouwereaskedtodetermineyourexpectedwinnings.Aswithallmathematicalterms,“expected”doesnotmeanexactlywhatyoumightexpectittomean.Ifyouareaveryoptimisticperson,youmayexpecttowinthelotteryeverytimeyouplay(afterall,someonehastowin–whynotyou?).Whenthestarofyourschool’sbasketballteamstepstothefreethrowline,youmayexpecthimorhertomaketheshot.However,whenamathematiciantalksabouttheexpectedvalueofaneventshemeanstheaveragevalueoftheeventoveraverylargenumberofrepeatedtrials.Forexample,supposewerollafairdie.Whatistheexpectedvalueoftheoutcome?Thinkaboutthisquestionforaminute–whatvaluemakessenseastheexpectedvalueoftherollofadie?Noticethatexpectedvaluecannotmean“themostprobable”outcome,sinceeachofthepossiblesixoutcomes(1,2,3,4,5,6)isequallylikely.Let’ssetupanexperimenttodeterminetheexpectedvalueasanaverage.Supposewerollthedie10,002timesandrecordtheoutcomeofeachroll.Nowweaddupalltheoutcomesanddivideby10,002tofindtheaveragevalue.(Ifyouareinclinedtocarryouttheexperiment,dosonowandbringyourresultstoclass.)Instead,supposeweprefertoreasontheoretically(whichwedo).Inimaginingthe10,002rollswewouldexpecta1tooccur1/6thofthetime,a2tooccur1/6thofthetime,etc.,sinceeachoutcomeisequallylikely.Sowewouldexpectabout16671’s,16672’setc.Sothetheoreticalaveragevaluewouldbe

1667×1 + 1667×2 + 1667×3 + 1667×4 + 1667×5 + 1667×610002 = 3.5

Wecanalsocomputetheexpectedvalueasthesumoftheproductsofeachoutcomewiththeprobabilityofthatoutcome:

1× !*+ 2× !

*+ 3× !

*+ 4× !

*+ 5× !

*+ 6× !

*= 3.5.

161

Explainhowthetwocalculationsarerelatedmathematically.Aretheyreallytwodifferentwaysoffindingthetheoreticalaveragevalue?Whyorwhynot?Sowecandetermineanexpectedvalueintwoways:1)theaveragevalueoftheoutcomeovermany,manytrials(experimental),or2)theprobability-weightedsumoftheoutcomes(theoretical).Practicallyspeaking,sinceweneedavery,verylargenumberoftrialstofindexpectedvalueexperimentally,weusethetheoreticalmethodinalmostallcases.Wecangeneralizethecomputationofexpectedvaluetogivethefollowingdefinition.HereeachxisanoutcomeandP(x)istheprobabilityofthatoutcome.Studythisformulatoseehowitfitswithoursecondcomputationabove.

Expectedvalueofx=x1×P(x1)+x2×P(x2)+…+xn×P(xn)Noticethattheexpectedvalue(sinceitisanaverage)canbeavaluethatisnotoneofpossibleoutcomes.ConnectionstotheElementaryGrades

Therewasnorationalreasonwhymonkeysmightpreferoneoftheseoptionsovertheotherbecause,accordingtothetheoryofexpectedvalue,they’reidentical.

MichaelPlattIntheMiddleGradesMathematicsProjectunitonProbability,theauthorssuggestthatstudentsuseareamodelstoanalyzeeventsinthecourseofdeterminingtheaveragepayoffoveraverylongrunoftrials(theexpectedvalue).Forexample,havealookatthebelowspinner.Theredsectorcontains1/4oftheareaofthecircle,thebluesectorcontains1/3ofthearea,andgreensectorcontainsthebalance.Sotheprobabilityoflandinginredis1/4,theprobabilityoflandinginblueis1/3,andtheprobabilityoflandingingreenis5/12.Colorthespinnerandthencheckthatlastprobabilityforyourself.Noticethatitdoesn’tmakesensetocomputetheexpectedvalue(color?)ofthespinner.Butifyouwon$5forlandingonRed,$1forlandingonBlueandnothingforlandingGreen,thenyoucouldcomputeyourexpectedwinningslikethis:

($5)+ ($1)+ ($0)≈$1.58.Whatdoesthe$1.58reallymean?Explainitasyouwouldtoyourupperelementarystudents.

41

31

125

162

Nowconsideraproblemaboutabasketballgame.SupposeTerryisa60%free-throwshooterandisinaone-and-onesituation.(Recallthatateamisinaone-and-onesituationwhentheopposingteamhascollectedatotalofsevenfouls;eachtimetheteamgoesupforafreethrow,theywillgetasecondfreethrowonlyiftheymakethefirstone.)Thechildrenareaskedtoinvestigatethefollowingthreequestions:

1) Onanygiventriptothefreethrowline(trial)isTerrymostlikelytoget0points,1point,or2points?

2) Overmanytrials,say100,abouthowmanytotalpointswouldTerryexpecttomake?

3) Overthelongrunwhatistheaveragenumberofpointspertrip(expectedvalue)?

ChildrenarefirstaskedtopredictanswerstoeachquestionTaketimenowtomakeyourpredictions.Nextchildrenareaskedtodeterminethetheoreticalprobabilitiesforquestion1.Taketimetodothis(aratmazemighthelp).Whatarethetheoreticalprobabilitiesof0,1,or2points?Finally,thechildrenareaskedtodesignasimulationusingaspinnertoexperimentallydeterminetheprobabilityofTerrygetting0,1,or2pointsonanyonegiventriptothefreethrowline.Whywouldthespinneraboveproveusefulinthisregard?Howcouldyouimprovetheaccuracyofthesimulation?

Homework

Oneofthehealthiestwaystogambleiswithaspadeandapackageofgardenseeds.

DanBennett

1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.

163

2) AnswerthethreequestionsintheConnectionssectionfora50%free-throwshooter,an80%shooter,anda90%shooter.Foreachcaseseeifyoucancomeupwithmorethanonesimulationmodelandmorethanonetheoreticalmethodtodeterminetheanswers.

3) Explainhowthelawoflargenumbersisrelatedtotheconceptofexpectedvalue.

4) Whatareyourexpectedwinningsinafairgameofchance?Explain.

5) Everydayatyourjobyouareofferedthefollowingchoicesforanhourlyrate:$7anhour,ortheamountofmoneyyouselectfromabagcontainingone$20bill,two$5billsanda$1bill.Whichplanwouldyouchooseandwhy?Useexpectedvaluetohelpyoudecide.

6) Whatistheexpectedvalueofthesumofarolloftwofairdice?Whatistheexpectedvalueofthedifferenceofthetwodice?

7) Supposethepremiumofahomeinsurancepolicyis$300andincaseofafirethe

insurancecompanywillpayout$200,000.Supposetheprobabilityofafireis0.0002.Iftheinsurancecompanyissues10,000suchpolicies,whatisitsexpectedprofit?

8) Hereisafungame!Yourolladietwotimes.Ifyourollexactlyonesix,youwin$10.Ifyourollexactlytwosixes,youwin$50.Otherwiseyouwinnothing.Whatareyourexpectedwinningsifyouplaythisgameonce?Explainhowtointerpretthatnumber.

164

ClassActivity28:BadBulbBinomial

Yourlifecanchangeontheflipofacoin.TraceyGold

1) Ijustreceivedashipmentof4lightbulbs.Ifeachbulbhasa15%chanceofbeingdefective.Whatistheprobabilitythatnoneofthebulbsaredefective?Exactlyone?Exactlytwo?Exactly3?Exactly4?Makeamaze,anduseittosee.

2) Ijustreceivedashipmentof30lightbulbs.Whatistheprobabilitythatexactly4aredefective?

3) IfIflipafaircoin12times,whatistheprobabilitythatIgetfewerthanthreeHeads?

165

ReadandStudy

Musicisthepleasurethehumanmindexperiencesfromcountingwithoutbeingawarethatitiscounting.

GottfriedLeibnizIntheClassActivityyouexploredthebehaviorofawholefamilyofprobabilityproblems:thosethathaveabinomialdistribution.Inorderforaproblemtofitintothisfamily,youmusthaveabinomialsettingandthismeansthatfourcriteriamustbemet:

1) Youareperformingafixednumberofindependentrandomtrials(likeselectingthirtybulbs).Let’scallitthatnumbern.

2) Foreachtrialthereareonlytwopossibleoutcomes(likegoodandbad).Let’scallone

ofthoseoutcomes“success”andtheother“failure.”(Thisistheconditionthatputsthe“bi”inbinomial.)

3) Theprobabilityofsuccessisthesameforeverytrial.Ifwecallthatprobabilityp,thentheprobabilityoffailurewillbe(1-p).(Why?)

4) Youareinterestedincomputingtheprobabilityofanumberofsuccesses(orfailures).(Forexample,tryingtoanswer,“Whatistheprobabilitythatyougetexactly2badbulbs.)

ExplainwhythecoinproblemintheClassActivitymeetseachcriterion.Whatisnforthatproblem?Whatisp?Thegoodnewsisthatifyoucanidentifyaprobabilityproblemasabinomialsetting,thenyoucansolveit.IntheClassActivityyoubuiltthetoolforsolvingthesetypesofproblems.Let’slookmorecloselyataminiversionofthatbulbproblemtomakesureyouseewhatwemean.Here’sthatproblem:

Ijustreceivedashipmentof3lightbulbs.Ifeachbulbhasa15%chanceofbeingdefective.Whatistheprobabilitythatnoneofthebulbsaredefective?Exactlyone?Exactlytwo?Exactlythree?

Thefirstthingwedowhenfacedwithaprobabilityproblemistobeginatreediagram.Notethatthefirstbulbhasa0.15probabilityofbeingdefectiveanda0.85probabilityofbeinggood.Sodoesthesecond.Andthethird.Soourtreelookslikethis:

166

Usingthetree,theprobabilityofnobadbulbsis(0.85)×(0.85)×(0.85)=(0.85)3=0.614125

We’llwriteitlikethis:P(#bad=0)=0.614125Nowtogettheprobabilitythatexactlyonebulbisbad,weneedtoaddupthefinalprobabilitiesofeachbranchwherethereisexactlyonebadbulb.Therearethreepathslikethat: BGG GBG GGB

P(#bad=1)=3×(0.15)×(0.85)×(0.85)=3×(0.15)1×(0.85)2=0.325125

Likewisetheprobabilityofexactlytwobadbulbsis P(#bad=2)=3×(0.15)×(0.15)×(0.85)=3×(0.15)2×(0.85)1=0.057375Whydoesthethreeappearintheaboveequation?Explain.Finally,thereisonlyonewaytogetallbadbulbs.Computethatprobabilitynow.Whenyouaredone,addupP(#bad=0)+P(#bad=1)+P(#bad=2)+P(#bad=3)andseewhatyouget.Thenexplainwhythatanswermakessense.

Bulb 3 bad .15

Bulb 3 bad .15

Bulb 3 bad .15

Bulb 2 bad .15

Bulb 3 bad .15

Bulb 1 bad .15

Bulb 3good .85

Bulb 3good .85

Bulb 3good .85

Bulb 2good .85

Bulb 2 bad .15

Bulb 3good .85

Bulb 2good .85

Bulb 1good .85

167

Soyoumighthavenoticedthatallthesebinomialprobabilitiesaregoingtobeproductsinvolvingtheprobabilityofsuccessptoapowerandtheprobabilityoffailure(1–p)toapoweralongwitha“coefficient”representingthenumberofpathsthatcontainthatmanybadbulbs.Nowwhatifthenumberoftrialsisbigger?Whatifwehaveashipmentof40bulbsandwanttoknowthatprobabilitythatexactly5arebad?Wellweknowthatweareinterestedinpathsthatcontain5badbulbsand35goodones.Eachpathlikethatwillhaveprobability(0.15)5×(0.85)35,buthowmanypathsaretherelikethat?Toanswerthisquestion,weneedtothinkbacktothecountingsectionsofthistext.Whatwereallyneedtocounthereisthenumberofstringsthatcontain35goodbulbsand5badbulbs.Hereisonesuchstring:

BBGGGGGGGBGGGGGGBGGGGGGGBGGGGGGGGGGGGGGGSooutof40blankswemustchoosewheretoplacetheBadbulbs(thentherestmustbeGood).Whatistheanswertothatquestionandwhy?Ifyouknow,thencarefullyexplainit.Ifyoudon’t,thenspendtenminutescheckingsmallerexamplesinordertofigureitoutandbringyourworktoclass.

Homework

Mathematicsisasmuchasaspectofcultureasitisacollectionofalgorithms.

CarlBoyer

1) DoalloftheitalicizedthingsintheReadandStudysection.

2) DidyoureallyworkonthatlastReadandStudyquestionabouttheG’sandB’s?Ifnot,doitnow;it’simportant.

3) Decidewhethereachofthefollowingisabinomialsituation.Ineachcaseexplainyourthinking.

168

a) Irolladie30times.WhatistheprobabilitythatIgetexactly5twos?b) IflipacoinuntilIgetahead.WhatistheprobabilitythatIneedtoflipit4times?c) Ihaveadishcontaining12redcandiesand6greencandies.Iselect5candiesone

byoneatrandomwithoutreplacingthem.WhatistheprobabilitythatIgetexactly1greencandy?

d) Irolladie20times.WhatistheprobabilitythatIgetatleast7twos?

4) Useatreediagramtosolvethisproblem:Aspinnerhasa25%chanceoflanding“red”anda75%chanceoflanding“blue.”IfIspinit4times,whatistheprobabilityofgettingexactly2“reds”?

5) Aspinnerhasa25%chanceoflanding“red”anda75%chanceoflanding“blue.”IfIspin30times,whatistheprobabilitythatIgetexactlyseven“reds”?Lessthanseven“reds”?(Youcandothatsecondquestion–stopandthinkaboutitforawhile.)

6) IfIchooseasampleof100peopleatrandomfromalargepopulationcontaining37%

menand63%women,whatistheprobabilitythatmysamplecontainsexactly37men?(Notethatthissituationisn’tquitebinomial–explainwhynot.Butwestillcangetareallygoodestimateofthedesiredprobability–explainwhythatisso.)

169

ClassActivity29:TheProblemofPointsBuildupyourweaknessesuntiltheybecomeyourstrongpoints.

KnuteRochne

AnnandBob(whoarebothequally-skilledplayers)areplayingagameofchanceforhighstakeswhensuddenlythegameisindefinitelyinterrupted.Eachroundofthegameisworthapoint.Bobstillneeds3pointstowin.Annneedsonlytwopointstowin.Ifeachplayerisjustaslikelytowinanygivenpointastheother,howshouldthe$10,000prizemoneybedividedbetweenthem?Arguethatyouarecorrect.

170

ReadandStudyGodnotonlyplaysdice.Healsosometimesthrowsthedicewheretheycannotbeseen.

StephenHawkingSometimesnewmathematicsiscreatedwhichseemstohaveno“realworld”applicationatthetime,butwhichlaterplaysanimportantroleinadevelopingarea,suchascomputergraphics.Atothertimes,newareasofmathematicsarecreatedspecificallytosolverealworldproblems.Suchisthecasewiththestudyofprobability,whicharosetoanswerquestionsthatsurfacedintheworldsofgamingandgambling.Thesystematicstudyofthetheoryofprobabilityisthoughttohaveoriginatedinthe17thcenturywiththesolutionofa“problemofpoints”bytwoFrenchmathematicians:PascalandFermat.In1654theywrotelettersbackandforthinwhichtheysolvedthegeneralcase(fortwoplayers)oftheexactsameproblemyousolvedintheClassActivity.Notonlydidtheysolvethisproblem,buttheyalsolaidthegroundworkforthestudyofprobabilityasamathematicaldiscipline.TheirmainideaswerepopularizedbyHuygens,inhisDeratiociniisinludoaleae,publishedin1657.Duringthecenturythatfollowedtheirwork,othermathematicians,includingJamesandNicholasBernoulliandDeMoivre,developedmorepowerfulmathematicaltoolsinordertocalculateoddsinmorecomplicatedgames.DeMoivre,andothersalsousedthetheorytocalculatefairpricesforannuitiesandinsurancepolicies.Thoughprobabilitywasamathematicaltheoryby1750,theapplicationsofthetheorywerestillonlytoquestionsoffair-distribution.Noonehadconsideredhowtouseprobabilityindataanalysis.Thiswastrueevenintheworkonannuitiesandlifeinsurance.Itwasworkoncombiningobservationsinastronomyandgeodesythatfinallybroughtdataanalysisandprobabilitytogether.ThisworkinspiredLaplace'sinventionofthemethodofinverseprobability(whichweknowasBaysianmethodofstatisticalinference),andculminatedinLegendre'spublicationofthemethodofleastsquaresin1805.TheseideaswerebroughttogetherinLaplace'sgreattreatiseonprobability,Théorieanalytiquedesprobabilités,publishedin1812.(Shafer,1993)Youmaywonderwhyprobabilityissuchayoungmathematicaldiscipline.Wedid.Afterall,geometryis2500yearsold–sowhyisprobabilityonlyaround350yearsold?Thereareseveraltheoriesaboutthat,butthemainoneisthat“chance”itselfmaybeafairlynewhumanidea;untilrecently,culturestendedtoascribealloutcomestothewillofGodorgods.Yourolledadieandwon?ThatisbecauseGodwilledit.Youwerestruckbylightning?Youmusthavedonesomethingreallybad.Youcanimaginethatinsuchaculture,itwouldmakenosensetotalkabout“random”events,andsoitwouldmakenosensetostudyprobability.

171

Todayprobabilityhasevolvedarichbranchofpuremathematics,anditsroleasafoundationformathematicalandappliedstatisticsisonlyoneofitsmanyrolesinthesciences.Forthebibliography:Theearlydevelopmentofmathematicalprobability.p.1293-1302ofCompanionEncyclopediaoftheHistoryandPhilosophyoftheMathematicalSciences,editedbyI.Grattan-Guinness.Routledge,London,1993.

Homework

You'llalwaysmiss100%oftheshotsyoudon'ttake.WayneGretzky

1) SupposeBobhadbeenwinning19to17whentheyheandAnnwereinterrupted.Howshouldthe$10,000bedivided?Whatifhewerewinning19to16?18to16?

2) Makeaconjecture:Doesthefairdivisionofthemoneydependonthenumberofpointstheleadingplayerisleadingby?Doesitdependonthenumberofgamestheleadingplayerhasalreadywon?Doesitdependonthenumberofgamestheleadingplayerstillneedstowin?Isthenumberofgamesthelosingplayerhaswon/needstowinalsomatter?Collectenoughdatatomakeaconjectureandthenarguethatyourconjectureistrue.

3) Supposeyouhadcomeuponthemearlierandnoticedthattheyweretied15to15.Youleaveandcomeback5flipsofthecoinlater.What'stheprobabilityofBobbeingahead19to16?

172

Appendices

173

ProbabilitySimulationPosterProject

Project:Forthisgroupprojectyouwillanalyzeagameofchanceusingbothsimulation(tocomputeanexperimentalprobability)andthestructureoftheproblem(tocomputeatheoreticalprobability).Eachgroupwillbeassignedtostudyadifferentgamebelowandthencreateaposterdescribingthework.Yourgroup’spostershouldbevisuallyinterestingandincludefoursections:

1) IntroductiontotheGame.Provideadescriptionofthegameandexplanationsofthequestionsyouwilladdress(Convinceusthatyouunderstandtheproblem(s)anddefineambiguoustermsorideas.)

2) SimulationInformation.Whenyouplayedthegame,whatdidyoulookfor?What

didyoukeeptrackof?Appendthetablesofdatayoucollected.

3) Solution.Whataretheanswerstoyourquestion(s)?

4) Justification.Whydoesyoursolutionmakesensemathematically?Arguethatitiscompleteandcorrect.

TheGames:

1) PassthePigs2) Roulette3) DiceGame4) Snake5) SumitUp6) DoubleUp7) RandomWalk

Rules:Thisisaprojectforyourgroup,soyouwillworktogetheronthinkingaboutthequestionsandcreatingtheposter.Youmaynotlookupanything(ontheweb,inanytextotherthanourtextbook,orfromsomeoneelse’soldwork…)aboutyourspecificgameorprojectquestions.

174

PassthePigs(Seethegameforinstructionsonhowtoplayandscore.)

1) Isthescoringforthisgameappropriatebasedonexperimentalprobabilities?Ifnot,

whatdoyousuggestforchangingthescoringandwhy?2) Basedontherulesasgiven,howlongshouldyouroll(towhatpointaccumulation)

eachroundtomaximizeyourchancesofwinning?Arguethatyouarecorrect.

RouletteARoulettewheelcontainsthenumbers1through36,0and00.Thusithas38placeswheretheballcanstop.Youchooseanumberorcombinationofnumbers,thewheelspins,andiftheballstopsonyournumber,youwin.Ifyoupickasinglenumberandiftheballstopsonyournumber,youget$36backforevery$1thatyoubet.Ifyoupickoddorevenandyouwin,yougetdoubleyourmoneyback.Ifyoupickonethirdofthenumbers(1,2,...36),youwintripleyourmoneybackifoneofyournumberscomesup.

Thequestion:Ihaveastrategyforwinningatroulette!Youfirstbet$1ontheoddnumbers.Ifyouwin,youhavemadeadollar.Ifyoulosethefirsttime,youbet$2ontheoddnumbersthesecondtime.Ifyouwinthistime,youreceive$4backanditcostyou$3,soyouhaveagainmadeadollar.Ifyoulostthesecondtime,youbet$4ontheoddnumbers.Ifyouwinthistime,youreceive$8andthecosthasbeen$7,soyouhavestillmadeadollar!Continuethisprocess...Sincetheballcannotgoonavoidingtheoddnumbersforever,soonerorlateryougetaheadbyadollarandthenyoustartthewholeprocessoveragain!Whatdoyouthink?Willthisstrategywork?Whyorwhynot?Canyouproveit?

WinningTicket

Aboxcontains12tickets.Eachtickethasadifferentnumberwrittenonit.Otherthanthat,weknownothingaboutthem.Thenumberscanbepositiveornegative,integersordecimals,largeorsmall–anythinggoes.Amongalltheticketsintheboxthereis,ofcourse,onethatbearsthebiggestnumberofall.That’sthewinningticket.IfweturnthatticketI,wewinaprize.Ifweturnanyotherticketin,wegetnothing.Thegroundrulesarethatwecandrawaticketoutofthebox,lookatit,andifwethinkit’sthewinningticket,turnitin.Ifwedon’t,wegettodrawagain,butfirstwemustteartheotherticketup(oncewepassonaticket,wecannotuseitagain).Wecancontinuedrawingticketsthiswayuntilwefindonewelikeorwerunoutoftickets…Whatstrategygivesustheverybestprobabilityofwinning?Canyouproveit?

175

DiceGameInthisgame,theplayer’sfirstrollofapairofdiceisveryimportant.Ifthefirstrollisa7oran11,theplayerwins.Ifthefirstrollisa2,3,or12theplayerloses.Ifthefirstrollisanyothernumber(4,5,6,8,9,10),thisnumberiscalledtheplayer’s“point.”Theplayercontinuestorolluntileitherthepointreappears,inwhichcasetheplayerwins,oruntila7showsupbeforethepointinwhichcasetheplayerloses.Whatistheprobabilityofwinningthisgame?

SnakeTheobjectofthegameistoaccumulateasmanypointsaspossiblein5rounds(theroundsarenamesbythelettersintheword“Snake”).Ineachroundapairofdiceisrolled.Ifneitherofthediceshowsa1,thenthetotalvalueoftherollisrecorded,andadecisionismadewhethertorollthepairofdiceagain.Ifeitherdieshowsa1,thentheperson’sturnisoverandallpointsaccumulatedbythatplayerinthatroundarelost.Ifsnakeeyes(doubleones)isrolledthenallpointsaccumulatedbytheplayer(inallpreviousrounds)islostaswell.

1) DescribeasafestrategyandthenariskystrategyforplayingSnakeandseewhatkindsofdistributionsofscoreseachproduces.

2) WhatistheoptimalstrategyforwinningatSnake?Canyouproveit?

SumitUp

Toplaythisgame,yourollapairofdiceandalwaystaketheirsum.Eachtimeyourollasumof7,youwin$2andmaycontinuerolling.Ifyourollanyotheroddsum,thegameisover.Ifyourolldoubles,youget$1andcancontinuerolling.Ifyourollanyotherevensum,thegameisover.Howmuchshouldthegamecosttoplaytomakeitafairgame?Canyouproveit?(Agameisfairiftheexpectedreturnoneach$1is$1.)

DoubleUpThisisagameplayedbyrollingapairofdiceandtakingtheirsum.Ifyourollanoddsum,thegameisover.Ifyourolldoublesofanynumber,youwinthatvalue(forexample,ifyourolldoublefives,youwin$5)andthegameisover.Ifyourollanevensum(otherthandoubles)youmaycontinuerolling.Acasinoneedstosetapriceforplayingthisgamesothattheymakeaprofitinthelongrun.Howmuchshouldthegamecosttoplay?

176

RandomWalk

Randomwalkisagameinvolvingtheboardbelowandasingledie.AplayermaystartfrompositionAorpositionB.Theplayerrollsthedie,andifa1or2appears,theplayermovesonespacehorizontally(towardtheprizediagonal).Ifa3,4,5,or6appears,theplayermovesonepositionvertically(towardtheprizediagonal).Theplayercontinuestorollandmoveuntilheorshelandsontheprizediagonalandwinstheprizeindicated.

1) IsitbettertostartatpositionAorB?Givemeyourevidence.

2) Whatshoulditcosttoplaythisgameinorderforthegametobe“fair?”(Agameisfairiftheexpectedreturnona$1betis$1.Theexpectedreturnisdefinedtobetheproductoftheprobabilityoftheoutcomeandtheamountofmoneyyouwillreceiveifyouwin.)Proveyouarecorrect.

177

References

Givecreditwherecreditisdue. Authorunknown

• CommonCoreStateStandardsforMathematics.AsfoundMarch2015at

http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.

• Dawson,L.(2004).TheSalkpoliovaccinetrialof1954:risks,randomizationandpublicinvolvement,ClinicalTrials,1,pp.122-130.

• Fuson,K.(2006).MathExpressions.Boston:HoughtonMifflin.

• MathematicalQuotationsServer(MQS)atmath.furman.edu.

• NationalCouncilofTeachersofMathematics.(2000).PrinciplesandStandardsforSchoolMathematics.Reston,VA:NCTM.

• Shulman,L.S.(1985).Onteachingproblemsolvingandsolvingtheproblemsof

teaching.InE.A.Silver(Ed.),TeachingandLearningMathematicalProblemSolving:multipleresearchperspectives(pp.439-450).Hillsdale,NJ:Erlbaum.

• Steen,L.A.andD.J.Albers(eds.).(1981).TeachingTeachers,TeachingStudents,

Boston:Birkhïser.

• TERC.(2008).Investigations.Glenview,IL:Pearson/ScottForesmanpublishing.

• TheUniversityofChicagoSchoolMathematicsProject.(2007).EverydayMathematics.Chicago,IL:McGrawHill,WrightGroup.

• Theearlydevelopmentofmathematicalprobability.p.1293-1302ofCompanionEncyclopediaoftheHistoryandPhilosophyoftheMathematicalSciences,editedbyI.Grattan-Guinness.Routledge,London,1993.

178

Glossary

AvailabilityFallacy:Thetendencytomakedecisionsregardingchancebasedonthehoweasilycertaineventscanbecalledtomind.Bargraph:adatadisplayinwhichtheverticalaxisrepresentsacount(frequency)orapercent(relativefrequency)andthehorizontalaxisrepresentsvaluesofacategoricalvariable(likecolors),valuesofanumericdiscretevariable(likeshoesize),orvaluesofacontinuousvariable(likeheights),groupedintointervals.BaseRateFallacy:Thetendencytomakedecisionsregardingchancethatneglecttheratesatwhichrelevantcharacteristicsappearinthepopulation.BinomialSetting:Arandomexperimentinwhichthereareafixednumberofindependentrandomtrials,eachwithtwopossibleoutcomes(successandfailure).Theprobabilityofsuccessmustbethesameforeverytrial,andthecomputationofinterestistheprobabilityofanumberofsuccesses(orfailures).Blind:Aclinicalstudyisblindiftheparticipantsdonotknowwhetherornottheyhavereceivedthetreatment.CategoricalData:Dataforwhichitmakessensetoplaceanindividualobservationintooneofseveralgroups(orcategories).ChanceError:Errorinsamplingduesimplytothefactthatasampleisnotexactlythesameasapopulationduetochancealone.Census:Anydatacollectionmethodinwhichdataiscollectedfromeachandeverymemberofthepopulation.Clinicalstudy:Anystudydesignedtodetermineacauseandeffectrelationshipbetweentwovariables.Clustersampling:Randomsamplinginstages.Confoundingvariable:Anyvariablethatwasnotcontrolledintheexperimentandmayhavecausedtheobservedeffect.ConjunctionFallacy:ThetendencytomakedecisionsregardingchancewithoutregardtothefactthatAandBbothoccurringislesslikely(orequallylikely)thanAoccurringalone.Controlgroup:Inacontrolledstudy,thecontrolgroupisagroupofparticipantswhodoesnotreceivethetreatmentunderstudyandisusedforcomparisonwiththetreatmentgroup.

179

Correlated:twonumericalvariablesaresaidtobecorrelatedifhaveastronglinearrelationship.Correlationcoefficient(r)isastatisticthatmeasuresthestrengthofthelinearrelationshipbetweentwonumericalvariables.Data:Anyinformationcollectedfromasample.Distributionofdata:adisplaythatshowsthevaluesthedatatakesalongwiththefrequency(orrelativefrequency)ofthosevalues.Disjoint(events):Twoeventsaredisjointiftheyhavenooutcomesincommon.Double-blind:Aclinicalstudyisdouble-blindifneithertheparticipantsnortheresearchersknowwhohasreceivedthetreatment(untilafterthestudyisover).Equallylikelyoutcomes:Alltheoutcomesofarandomexperimentaresaidtobeequallylikelyif,inthelongrun,theyalloccurwiththesamefrequency(theyallhavethesameprobabilityofoccurring).Event:anysubsetofthesamplespaceofarandomexperiment.Expectedvalue(ofanumericalevent):theaveragevalueoftheeventoveraverylargenumberoftrials.Experimentalprobabilitiesareprobabilitiesthatarebasedondatacollectedbyconductingexperiments,playinggames,orresearchingstatistics.Explanatoryvariableisonethatisusedtoexplainvariationsinanothervariable,calledtheresponsevariable.Extrapolation:Makingapredictionoutsidetheboundsofthedata.Firstquartile(Q1):themedianofthedatapositionedstrictlybeforethemedianwhenthedataisplacedinnumericalorder.FiveNumberSummary:ofnumericaldataconsistsofthefollowingfivestatistics:Minimumvalue,Q1,Median,Q3,Maximumvalue.Gambler’sFallacy:Thetendencytomakedecisionsregardingchancebasedonthe(mistaken)beliefthatafteralongstringoflosses,awinbecomesmorelikely.(Thisisatypeofrepresentativenessfallacy.)

180

Histogram:Adatadisplayinwhichtheverticalaxisrepresentsarateandthehorizontalaxisisactuallythex-axis-inotherwords,itisacontinuouspieceoftherealnumberlineandassuchitcontainsallrealnumbersinaninterval.Valuesofthevariablearebrokenintointerval‘classes.’(Note:manytextsusetermshistogramandbargraphinterchangeably.Wewillnotdosointhistext.)Independent(events):Twoeventsaresaidtobeindependentifknowingthatoneoccurs(orknowingitdoesnotoccur)doesnotchangetheprobabilityoftheotheroccurring.Interpolation:Makingapredictionwithintheboundsofthedata.Inter-QuartileRange(IQR)=Q3–Q1.Itisthespreadofthemiddlehalfofthedata.Lawoflargenumbers:Inrepeated,independenttrialsofarandomexperiment,asnumberoftrialsincreases,theexperimentalprobabilitiesobservedwillconvergetothetheoreticalprobability. Inotherwords,theaverageoftheresultsobtainedfromalargenumberoftrialsshouldbeclosetotheexpectedvalue,andwilltendtobecomecloserasmoretrialsareperformed.Mean:Themean(oraverage)valueofasetofnumericdataisthesumofallthevaluesdividedbythenumberofvalues.

MeanAbsoluteDeviation(d)= [|(x1–m)|+|(x2–m)|+|(x3–m)|+…+|(xn–m)|].Itistheaveragedistanceofthendatapointstotheirmean.Median:Themedianofasetofnumericvaluesisthemiddlevaluewhenthedataisplacedinnumericalorder.Inthecasewheretherearetwo“middle”values,themedianistheaverageofthetwo.Mode:thevaluethatoccursmostfrequently.Multiplicationprinciple:IfyouhaveAdifferentwaysofdoingonethingandBdifferentwaysofdoinganother(andonechoicedoesnotaffecttheother)thenthetotalnumberofwaystodobothAandBisA×B.Negativeassociation:Twovariablesmeasuredonthesameindividualshaveanegativeassociationifincreasesinonevariabletendtocorrespondtodecreasesintheothervariable.Nonresponsebias:Biasedinformationthatresultsfromthefactthatsomegroupsinthesampleweremorelikelytorespondthanothers.NumericalData:numberdataforwhichitmakessensetoperformarithmeticoperations(suchasaveraging).

n1

181

Outcome:Anyonethingthatcouldhappeninarandomexperiment.Outlier:aspecificvaluethatlieswelloutsidetheoverallpatternofthedata.Parameter:Somenumericalinformationthatisdesiredfromanentirepopulation.Usuallyaparameterisanunknownvaluethatisestimatedbyastatistic.Percentile:Ascoreinthenthpercentilemeansthat“npercent”ofthepeoplewhotookthetestscoredatorbelowthatscore.Placebo:afaketreatmentgiventothecontrolgroupinaclinicalstudy.Population:thelargestgroupaboutwhichtheresearcherorsurveyorwouldlikeinformation.PositiveAssociation:Twovariablesmeasuredonthesameindividualshaveapositiveassociationifincreasesinonevariabletendtocorrespondtoincreasesintheothervariable.Probability:referstotheproportionoftimesaneventwouldoccuriftherandomexperimentwasperformedaverylargenumberoftimes.Proportion:Aproportionisastatementthattworatiosareequivalent.Range=maximumvalue–minimumvalue.RandomExperiment:anyexperimentwheretheoutcomedependsonchanceandcannotbeknownbeforehand.Whileinarandomexperiment,thespecificoutcomecannotbeknown,thereisnonethelessaregulardistributionofoutcomesafteraverylargenumberoftrials.Randomized-controlled:aclinicalstudyissaidtoberandomized-controlledifparticipantsareassignedtotreatmentandcontrolgroupsatrandom.Ratio:acomparisonoftwocountsormeasuresthathavethesameunit.RepresentativenessFallacy:Thetendencytomakedecisionsregardingchancebasedonthe(mistaken)beliefthatevensmallsamplesshouldberepresentativeofpopulation.Regressionline:thelinethatminimizesthesumofthesquaresoftheverticaldistancesofthepointstothelineonascatterplot.ResponseRate:Theratioofthenumberofrespondents(peoplewhoactuallytookpartinthestudy)tothenumberwhowereinvitedtoparticipate.Responsevariable:avariablewhosevariationisexplainedbyanothervariable(calledanexplanatoryvariable).

182

Sample:asubsetofapopulationfromwhichdataiscollectedSamplebias:Biased(inaccurate)informationthatresultsfromapoorlychosensample.Samplingerror:thedifferencebetweenthevalueofaparameterandthevalueofthestatisticthatrepresentsit.Samplingerrorcanincludechanceerror,samplingbias,andnonresponsebias.Samplingrate:[(#inthesample)÷(#inthepopulation)]Samplespace:Thesamplespaceofarandomexperimentisthesetofallpossibleoutcomes.Scalefactor,avaluebywhichwemultiplyeachoriginaldimensiontofindthenewlengths.Scatterplot:Adisplaythatshowstherelationshipbetweentwonumericalvariablesmeasuredonthesamesampleofindividuals.Thevaluesofonevariableareshownonthehorizontalaxis,andvaluesoftheothervariableareshownontheverticalaxis.Eachindividualisplottedasapointrepresentinganorderedpair(variable1,variable2).Self-selectedsample:asampleinwhichrespondentsvolunteeredtoparticipate.Similar:Twogeometricfiguresaresimilarifthereissomescalefactorwecouldusetomakethemcoincide.(Similarfigurehavethesameshape,butarenotnecessarilythesamesize.)SimpleRandomSample(SRS):Asampleinwhicheverysubsetofthepopulationhasthesamechanceofbeinginthesampleasanyothersubsetofthesamesize.Skewedleft:Adistributionofdataisskewedleftifithasanasymmetrictailtotheleft.Skewedright:Adistributionofdataisskewedrightifithasanasymmetrictailtotheright.Slope(m):theslopeofalineisthechangeinyforaoneunitchangeinx

StandardDeviation(s)= whereμisthemeanofthevalues,nisthenumberofvalues,andeachxiisanindividualvalue.Standarddeviationisastatisticthatmeasuresspreadaroundthemean.Statistic:Anynumericalinformationcomputedfromasample.(Forexample,ameancalculatedfromasampleisastatistic,whichcanbeusedasanestimateforthepopulationmean,whichwouldbeaparameter.

[ ]222

21

1 )(...)()( +++ nn xxx

183

Stemandleafplot(orsimplystemplot):isalistingofallthedatatypicallyarrangedsothetensplacemakesthestemandtheonesarethe‘leaves.’Survey:Anydatacollectionmethodinwhichdataiscollectedfromjustasubsetofthepopulation.SymmetricDistribution:Adistributionthatissymmetricaboutthemedian.TheoreticalProbabilities:probabilitiesassignedbasedonassumptionsaboutthephysicaluniformityandsymmetryofanobject(suchasadieorcoinorbagofcandies).Thirdquartile(Q3):themedianofthedatapositionedstrictlyafterthemedianwhenthedataisplacedinnumericalorder.Treatmentgroup:inacontrolledstudy,thetreatmentgroupreferstotheparticipantswhoaregiventhetreatmentthatisbeingtestedtoseeifitcausesacertaineffect.