Volume–BETA VERSIONSummer2013

Grade5

MathematicsFormativeAssessmentLesson

DesignedbyKentuckyDepartmentofEducationMathematicsSpecialiststobeField-testedbyKentuckyMathematicsLeadershipNetworkTeachers

Ifyouencountererrorsorotherissues,please contacttheKDEteamat:

Createdfor thesolepurposeofassistingteachersastheydevelopstudentunderstandingofKentucky’sCoreAcademicStandardthroughtheuseofhighlyeffectiveteachingandlearning.

Notintendedforsale.

VolumeFifthGrade

Mathematicalgoals

This lessonis intendedtohelpyouassess howwellstudentsareable tomodel three dimensionalfigures andfindtheir volume.Inparticular,this unit aims toidentifyandhelpstudents whohavedifficultieswith:

  • Recognizing volume as anattribute ofthree-dimensional space.
  • Measuring volume byfindingthe totalnumber ofsame-sizeunits ofvolume requiredtofill the spacewithout gaps or overlaps.
  • Measuringnecessaryattributes ofshapes,inparticular thebase area,inorder todetermine volumes tosolve realworldandmathematical problems.

CommonCoreStateStandards

Thislessoninvolvesmathematical contentinthe standards fromacross the grade,withemphasis on:

Measurement andData5.MD

  • Geometricmeasurement:understandconceptsofvolumeandrelatevolumeto multiplicationandtoaddition.

Thislessoninvolves arangeofStandards forMathematicalPracticewithemphasis on:

1.Makesenseofproblemsandperseverein solvingthem.

4. Model withmathematics.

7. Lookforandmakeuseofstructure.

Introduction

This lessonunit is structuredinthe followingway:

•Before the lesson,students workindividuallyonanassessment taskthat is designedtorevealtheir currentunderstanding anddifficulties.You thenreviewtheirworkandformulate questionsforstudents toanswertohelpthemimprove their solutions.

•Duringthe lesson,students workinpairstomatch thewordproblemandmodels ofthe3-dimensional figures.

•Ina whole-class discussion,explaintheiranswers.

•Finally,students returntotheir original assessment task,andtry toimprove their ownresponses.

Materialsrequired

•Eachindividual studentwill needtwocopies ofthe worksheetHowManyCubes?.

•Eachsmall groupofstudentswill needapacket ofCardSetA andB.

•Eachsmall groupofstudentswill alsoneedatleast 30cubes or blocks sothat theycanmodelthewordproblems ifneeded.

Timeneeded

Approximatelyfifteenminutesfortheassessmenttask, aone-hourlesson, and 15 minutesforthestudentstoreviewtheirworkforchanges.Alltimingsareapproximate. Exacttimingswilldependonthe needsofthe class.

Beforethelesson

Assessmenttask:

Have the students dothis taskinclassa dayor more before the FormativeAssessmentLesson.This will give youanopportunitytoassess the workandtofindout the kinds ofdifficultiesstudentshave withit.Thenyouwill be able totargetyourhelpmore effectivelyinthefollow-uplesson.

Give eachstudent acopyofHowManyCubes?.Introduce thetaskbrieflyandhelptheclass tounderstandthe problemanditscontext.

Spend fifteenminutes onyourown,answeringthisquestions.

Don’tworry if youcan’tfigure it out.There will bealessononthis material [tomorrow] that will helpyouimprove yourwork.Your goal is tobe able toanswerthese questions withconfidence by the endofthatlesson.

It is important that studentscomplete thetaskwithout assistance,as far as possible.

Ifstudents are strugglingtoget started,ask themquestions thathelpthemunderstandwhat isrequired,but donotdothe taskforthem.

Assessingstudents’responses

Collect students’ responses tothetask.Makesome notes onwhat theirworkreveals about theircurrent levels ofunderstandingas they figureout the volume oftheboxes.The purpose ofthisistoforewarnyouoftheissues thatwillarise duringthe lesson,sothatyoumayprepare carefully.

We suggest thatyoudonot score students’work.The researchshows that thisiscounterproductive,as itencourages students tocompare scores,anddistracts their attentionfromhowtheymayimprove theirmathematics.

Instead,helpstudents tomake further progressbyaskingquestions that focus attentiononaspectsoftheirwork.Some suggestions for these aregivenonthe next page.These have beendrawnfromcommondifficultiesanticipated.

We suggest thatyouwriteyour ownlistsofquestions,basedonyour ownstudents’work,usingthe ideas below.Youmaychoose towrite questions oneachstudent’s work.Ifyoudonot havetime todothis,select afewquestions thatwillbe ofhelptothemajority ofstudents.These canbewrittenon the boardat the beginningofthe lesson.

Common issues:Suggestedquestionsand prompts:

CommonIssues / Suggestedquestionsandprompts
Studentwhohas trouble gettingstarted. / •Whatinformationdoyou know?
•Howcanyouusewhatyouknowto begintheproblem?
Student confuses areaandvolumebecause theydonotunderstandwhat eachdescribes. / •Howmanycubeswillfitintoaprismthissize?
•Comparethisproblem/modeltotheareaproblems/modelsyouhaveseen.Howisthissimilar?Howisthisdifferent?
Student does not connect the 3dimensional model cards tothewordproblem. /
  • Howcanyouusethecubesprovidedtobuild amodel?WhichcardfromsetBmatchesyourmodel?

Student does notseehowthe baseareacanbe usedtofindvolume. /
  • Howmanycubeswill fitinthebottomlayer?Whatdoesthisrepresent?
  • Whatifyouknowhowmanylayersare inthemodel?Canyouuse thistohelpyoufindthevolume?

Suggestedlessonoutline

Improveindividualsolutionstotheassessmenttask(10minutes)

Returnyourstudents’workontheHowManyCubes?problem.Askstudentstore-read boththeHowManyCubes?questionsandtheirsolutions.Ifyouhavenotaddedquestionstostudents’work, writeashortlistofyourmostcommonquestionsontheboard. Studentscanthenselectafewquestionsappropriatetotheirownworkand beginansweringthem.

Recallwhatwewereworkingonpreviously.Whatwasthetask?

Drawstudents’ attentiontothe questions youhave written.

IhavereadyoursolutionsandIhavesomequestionsaboutyourwork.

Iwouldlikeyoutoworkonyourowntoanswermyquestionsfor tenminutes.

Collaborative activity1–MatchingCard SetATaskCardsand CardSetB3-DModels(15minutes)

Organize thestudents intosmall groups oftwoorthree.Intrials,teachers foundkeepingsmallhomogenousgroupshelpedmore students playanactive role.

Introducethelesson carefully:

Iwantyoutoworkasateam.BeginwithaTaskcardfromCardSet A.Model this problemwiththe blocks first.

ThenfindacardfromCardSet Bthatmatches the model youbuilt.

Continue this withall taskcards.Eachtime youdothis; explainyourthinking clearlyand carefully.

If your partner disagrees withthemodel youchose,thenchallenge him/her.It isimportant that youbothunderstandthemathfor all themodels.

There is alot of worktodotoday,andit doesn't matter if youdon't all finish.Theimportant thingis tolearnsomethingnew,sotake your time.

You have twotasks duringsmall-groupwork,tonote different student approaches tothe task,andtosupport student problemsolving.You can thenusethis informationtofocus awhole-classdiscussiontowards theendofthe lesson.Inparticular,notice anycommon mistakes.

Notedifferentstudentapproachestothetask

Listenandwatchstudents carefully.Inparticular,listentoseewhether they areaddressingthedifficultiesoutlinedintheCommonIssuestable.You can use this informationtofocus awhole-class discussiontowards theendofthelesson.

Supportstudentproblem solving

Try nottomakesuggestionsthatmovestudentstowardsaparticularapproachto thistask. Instead, askquestions tohelpstudentsclarifytheirthinking.Ifseveralstudentsintheclassarestrugglingwiththesameissue, youcouldwritearelevantquestionontheboard.Youmightalsoaskastudentwhohasperformedwellononepartofthe tasktohelpastudentstrugglingwiththatpartofthe task.

Thefollowingquestionsandpromptswould behelpful:

What informationhave youbeengiven?What doyouneedtofindout?

Howcanyoumodel the 3-dimensional figurethatwasgiveninthe problem?

Ifone studenthas modeledwithasetofbar model cards,challenge theirpartner toprovide anexplanation.

Mariamodeledthe problemwiththese cards.Martin,why does Mariamodelit thisway?

Ifyoufindstudentshave difficultyarticulatingtheir decisions,thenyoumaywant touse thequestions fromtheCommonIssuestable tosupport your questioning.

Collaborative Activity2: PlacingCardSet C:Base AreaCards (15minutes)

As students finishmatchingthe wordproblemandmodelcards handout CardSet C:Base Area.(Donot collectCardSet AandB.)Thissetof cardsprovides studentswithanopportunitytofocus on the base ofthe prism.An important part ofthis cluster of5thgrade standards is that

students discover that the volume ofa prismis the basearea xheight. Thiswill leadtotheunderstandingthat the volume ofother 3dimensional shapes is base area xheight.

CollaborativeActivity3:PlacingCARD SETD –V=Lx Wx H formulacards (15minutes)

As students finishplacingtheBaseAreacardshandout CardSetD: FormulaCards.Theseprovide studentswitha different wayofmodelingthe situationwithanumerical equation.Donotcollectanyofthe previouscards.

Plenarywhole-class discussioncomparingdifferent approaches (20minutes)

Organizeawhole-classdiscussionto allowstudentstoexplaintheirmodels.Theintentionisforyoutofocusongettingstudentstounderstandtherepresentationsofthetasktobuildtheirconceptualunderstandingofvolumeratherthanshowingthem theformula. Focusyourdiscussiononpartsofthe small-grouptasksstudentsfounddifficult.

Improveindividualsolutionstotheassessmenttask(10minutes)

Returnto thestudentstheiroriginalassessment,HowManyCubes?, aswellasasecondblankcopyofthetask.

Lookatyouroriginalresponsesandthinkaboutwhatyouhavelearnedthislesson.Usingwhatyouhavelearned, trytoimproveyourwork.

Ifyouhavenotaddedquestions toindividualpieces ofworkthenwriteyourlistofquestionsontheboard.

Studentsshouldselectfromthislistonlythequestionsappropriateto theirownwork.

Ifyoufindyouare runningoutoftime,thenyoucouldset thistaskinthenextlesson, orforhomework.

ThisFormativeAssessment Lessonwas createdaroundtaskstakenfromInsideMathematics.

HowManyCubes?AnswerKEY

(Itisagoodideatohave woodencubesorsquaredpaperforkidstomodel thisproblem.)

1.BoxAcanhold30 cubes.Asampleexplanationcould be thatthebottom layerwouldconsistof6cubes and5layersof6cubes would be30cubes. (Other explanationsshouldbeaccepted.)

2.BoxBcanhold24cubes. Asamplecalculation–The basecanbefoundby2x2 is4cubes.Sincethereare6layers,4x6is24 cubes.

3.BoxAcanholdmorecubes.

4.Thisboxcanhold36cubes.

5.Acceptanycombinationofnumberswhoseproductis36, i.e.,4,3,3.

AssessmentTaskHowManyCubes?

HowManyCubes?

StevefillsBox A andBox Bwithonecentimetercubes.

1.HowmanycubescanSteve fitintoBoxA?

Explainhow you figured itout.

2.HowmanycubescanSteve fitintoBoxB?

Showyourcalculations.

Sample Solutions for CardSort

Arectangularaquariumwillhold24cubicfeet ofwater whenfilledtothe top.Ifitis4ftlong and 3fttall,howwideisthetank? / V=4x2x
Whatisthevolumeofarectangularprismwithaheightof2ft.andabaseareaof6ft2? / / / V=x2x2
Cube-shapedboxesofcandyareshippedinlargerboxes.Thelargerboxesaresixfeetlong,onefootwide,andtwofeethigh.Howmanyonecubicfootboxesofcandywillthelargeboxhold? / / 12=6xx
Arectangularjuiceboxcontains 24millilitersofapplejuice.Theboxis2cmhighand 3 cmwide.What isthelengthofthejuicebox?
(1milliliter=1cubiccentimeter) / / 24=x3x
Awadingpool willbe2mlong and5m wide.Ifit willhold30cubicmetersofwater,howdeepshouldthiswadingpoolbe? / / V=5xx3
Amovingcompanyistryingtostore1cubicmeterboxesinastorageroomwithalength of5m,widthof 3mandheightof2m.Howmanyboxescanfitinthisspace? / / 30=x3x
Whatisthevolumeofa cubewithanedge thatmeasures3cm? / / / V=3x3x
Atoy companyisplanningtomarketasetofwoodenalphabetblocks.
Eachblock isacubewith1-inchedges.How manyblockswillfitintoacontainerthatis3cm long,2 cmwide,and3cmtall? / / 18=x2x

CARDSETA:TASKCARDS

Arectangularaquariumwillhold24cubicfeetofwaterwhenfilledtothetop.Ifitis 4ft longand 3 ft tall,howwideisthetank? / Whatisthevolumeofarectangularprismwithaheightof2ft.andabase areaof6ft2?
Cube-shapedboxesofcandyareshippedinlarger boxes.Thelargerboxes are sixfeetlong,onefootwide,and twofeethigh.
Howmanyonecubicfootboxesofcandywillthelarge boxhold? / Arectangularjuiceboxcontains24millilitersofapplejuice.Theboxis2cmhighand3cmwide.Whatisthelengthofthejuicebox?
(1milliliter=1 cubiccentimeter)
Awadingpoolwillbe 2m longand5m wide.Ifitwillhold30cubicmetersofwater,howdeepshouldthiswadingpool be? / Amovingcompany istryingtostore1cubicmeter boxesin astorage roomwithalengthof5m,width of3m andheight of2
m.Howmany boxescanfitinthisspace?
Whatisthevolumeofacubewithanedgethatmeasures3cm? / Atoycompanyis planningtomarketasetofwoodenalphabetblocks.Eachblockis acube with1-cmedges.Howmanyblockswillfit intoacontainerthatis 3cmlong,2cmwide,and 3cmtall?

CARDSETB–3-DimensionalFigures

CARDSET C–BaseArea

CARDSETD–V=LxWxHformulacards

24=___x3x___ / V=5x___ x 3
30=__ x 3x___ / V=4x2x___
18=__ x 2x___ / V=___x2x2
12=6x ___x___ / V=3x3x___