Math g3-m1-topic-a-lesson-2

Lesson 2NYS COMMON CORE MATHEMATICS CURRICULUM 3•1
Lesson 2: Relate multiplication to thearraymodel.
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Lesson 2
Objective: Relate multiplication to the array model.
Suggested Lesson Structure
Fluency Practice (15 minutes)
Application Problem (5 minutes)
Concept Development (30 minutes)
Student Debrief (10 minutes)
Total Time (60 minutes)
Fluency Practice (15 minutes)
 Sprint: Addor SubtractUsing 2 3.OA.1 (9 minutes)
 Group Counting 3.OA.1 (3 minutes)
 AddEqual Groups 3.OA.1 (3 minutes)
Sprint: Add or Subtract Using 2 (9 minutes)
Materials: (S) Add or SubtractUsing 2 Sprint
Note: ThisSprintsupportsgroupcountingskillsthatare foundational to interpretingmultiplicationas
repeatedaddition.
DirectionsforAdministrationofSprints
A Sprinthastwo parts,A and B, withcloselyrelatedproblemsoneach. Each part isorganizedinto four
quadrantsthat move fromsimple tocomplex. Thisbuildsachallenge intoeach Sprintforeverylearner.
Before the lesson,print SprintA andSprintB on twoseparate sheetsof paper. Studentscomplete the two
parts of the Sprintinquicksuccessionwiththe goal of improving forthe secondpart,evenif onlybyone
more. With practice, the followingroutinetakesabout 9minutes.
SprintA
Place SprintA face downon studentdesks,andinstructstudents nottolookat the problemsuntil asignal is
given.
T: You will have 60 secondstodo as many problemsasyoucan. I do not expectyoutofinishall of
them,justas manyas you can, tryingforyour personal best.
T: Take your mark! Get set! THINK!

31
Studentsturnpapersoverand workfuriouslytofinishasmanyproblemsastheycanin 60 seconds. Time
precisely.
T: Stop! Circle the lastproblemyou completed. Iwill readjustthe answers. If you gotthe answer
right,call out “Yes!” If you made a mistake,circle it. Ready?
Repeatto the endof SprintA or until no studenthasa correct answer.
T: Now,at the top of the page,write the number of problems yougotcorrect. This isyour personal
goal for SprintB.
T: How manyof yougot one right? (All handsshouldgoup.)
T: Keepyourhandup until Isay a numberthat is one more thanthe numberyougotright. So,if you
got 14 right,whenI say15, your handgoesdown. Ready?
T: (Continue quickly.) Howmanygot two right? Three? Four? Five? (Continue until all handsare
down.)
If the classneedsmore practice with SprintA,continue withthe optional routine presentedbelow.
T: Take one minute todomore problemsonthishalf of the Sprint.
As studentswork,the studentwhoscoredhigheston SprintA mightpassout SprintB.
T: Stop! I will readjust the answers. If yougot itright,call out “Yes!” If youmade a mistake,circle it.
Ready?
Readthe answerstothe firsthalf again as studentsstand.
Movement: To keepthe energyandfungoing,doa stretchor a movementgame inbetween Sprints.
SprintB
Place SprintB face downon studentdesks,andinstructstudents nottolookat the problemsuntil asignal is
given. Repeatthe procedure for SprintA upthroughthe show of handsfor how many correct answers.
T: Standup if you got more correct onthe secondSprintthanon the first.
S: (Stand.)
T: Keepstandinguntil Isaythe numberthattellshow manymore you got righton SprintB. If you got
three more righton SprintB than on SprintA, whenIsay three, yousitdown. Ready?
Call out numbers,startingwith one. Studentssitasthe numberby whichtheyimprovediscalled. Students
may take Sprintshome.
Group Counting (3 minutes)
Note: Basic skip-countingskillsfromGrade 2 shiftfocusinthisGrade 3 activity. Group countinglaysa
foundationforinterpretingmultiplicationasrepeatedaddition. Whenstudentscountgroupsinthisactivity,
theyadd andsubtract groupsof three whencountingupanddown.
T: Let’scount to 18 forwardand backward. I wantyou to whisper,whisper,andthenspeaknumbers.
T: Watch my fingerstoknowwhethertocountup or down. A closedhandmeansstop. (Show signals
while explaining.)

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T: (Rhythmicallypoint upuntil achange isdesired. Show aclosedhandthenpointdown.)
S: (Whisper) 1, (whisper) 2,(speak) 3,etc.
T: Let’scount to 18 forwardand backwardagain. This time,thinkeverynumberinsteadof whispering.
S: (Think),(think),3,(think),(think),6,(think),(think),9,etc.
T: What didwe justcount by? Turn and talkto yourpartner.
S: Threes.
T: Let’scount by threes. (Directstudentstocountforwardand backwardto 18, periodicallychanging
directions. Emphasizethe 9to 12 transition.)
Add Equal Groups (3 minutes)
Materials: (S) Personal white board
Note: ThisactivityreviewsLesson1. Studentsdirectlyrelate repeatedadditiontomultiplication. They
interpretproductsas the numberof equal groupstimes the numberof objectsin eachgroup.
T: (Projectapicture array with3 groupsof 2 circled.) How many groupsare circled?
S: 3.
T: How manyare ineach group?
S: 2.
T: Write thisas an additionsentence.
S: (Write 2 + 2 + 2 = 6.)
T: Write a multiplicationsentence for3twosequals6.
S: (Write 3 × 2 = 6.)
Continue with thispossiblesequence: 3 groupsof 5, 5 groupsof 10, and 3 groupsof 4.
Application Problem (5 minutes)
Jordanuses3 lemonstomake 1 pitcherof lemonade. He makes4 pitchers. How many lemonsdoeshe use
altogether?Use the RDW processto showyoursolution.
Note: Presentthe image of 4 groupsof 3 lemonswiththe wordproblemasascaffold. Thisproblemreviews
multiplyingequal groupsfromLesson1. It alsoleadsintotoday’s ConceptDevelopmentinwhichstudents
relate multiplicationtothe array model.

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NOTES ON
MULTIPLE MEANS
OF REPRESENTATION:
The words arrayand row were
introduced inGrade 2, Module 6 but
are treatedas new vocabularyinthis
lesson.
When reviewing the concept, have
students trace a rowon the arraywith
a finger while sayingthe word row.
Provide a real-worldexample byhaving
students count the rows onvarious
cupcake pans (miniature andregular
size) before usingthe template.
NOTES ON
MULTIPLE MEANS
OF REPRESENTATION:
When presentingthe concept of array,
it maybe beneficialto askstudents to
turn and talk, describingor defining an
arrayfor their partner.
Concept Development (30 minutes)
Materials: (S) Personal white boardwith threesarray
(Template) inserted(picturedbelow),lemonsimage
fromApplicationProblem,1sheetof blankpaper
Problem1: Relate equal groupstoarrays.
Note: Students’ templatesshouldbe vertical rather
than horizontal,asshownbelow.
T: Look back at Jordan’slemons. Compare the way his
lemonsare organizedwiththe groupsof 3 circleson
your template.
S: The lemonsare touchingeachother,but the circles
have space betweenthem.  Each line onthe
template showsthree,like eachgroupof lemons.
 The template isorganizedwitheverythingin
straightlines.
T: Many studentsare noticingstraightlinesonthe
template. Let’scall astraightline goingacrossa row.
Use yourblankpaperto cover all but the toprow.
S: (Coverall butthe top row.)
T: Uncover1 row at a time inthe picture. Asyouuncover
each row,write the newtotal numberof circlestothe
rightof it.
S: (Skip-countbythree usingthe threesarraytemplate.)
T: At the signal,saythe total numberof circlesyou
counted. (Signal.)
S: 30 circles!
T: Take 10 secondstofindhowmanyrows of 3 you
counted. Atthe signal sayhowmany. (Signal.)
S: 10 rows!
T: True or false: 10 rows of 3 circlesequals30 circles?
S: True!
T: (Write 10 × 3 = 30 on the board.) Use the picture on
your template totalkwithyourpartneraboutwhythis
equationistrue.
S: Yesterday, we learnedthatwe canmultiplyequal
groups.  We skip-counted10 rowsof 3 circleseach
and the total is 30.  It means10 groupsof 3. When
youadd 10 threes,youget30.  Yeah, butwriting10
× 3 is a loteasierthanwritingout3 + 3 + 3 + 3 +…
T: We call thistype of organized picture an array.
Threes array template (with student work)

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NOTES ON
MULTIPLE MEANS
OF ENGAGEMENT:
Provide a challenge in this part of the
lesson bygivinganequation
(e.g., 5 × 4 = _____) and no picture.
Have students drawboththe equal
groups and the arrayto represent the
equation. Then, theyskip-count to find
the total.
T: (Projectordraw the image on the right.) Take a lookat thisarray. At the signal,tell
howmany rectanglesare inthe top row. (Signal.)
S: 4 rectangles.
T: The size of 1 row is4 rectangles. Eachrow of 4 can alsobe calleda groupof 4. Atthe
signal,tell howmanygroupsof fourare inthe array. (Signal.)
S: 3 groupsof four.
T: To write thisas an equation,we firstwrite the numberofgroups. How many groups?
S: 3 groups!
T: (Write 3 × ____ = ____.) Next, we write the size ofthe group. How many rectanglesare ineach
group?
S: 4 rectangles!
T: (Fill inthe equation toread3 × 4 = ____.) Skip-counttofindthe total numberof rectanglesinthe
array.
S: 4, 8, 12.
T: (Fill inthe equationtoread3 × 4 = 12.) We justfoundthe answertothe multiplication equation
that representsthe array. Inmultiplication, the answer,ortotal,iscalledthe product.
Showan array of 2 rowsof 6 and repeatthe process.
Problem2: Redrawequal groupsasarrays.
T: (Projectordraw the image on the right.) The drawing
shows3 equal groupsof 5. On your personal white
board,re-drawthe picture as an array with3 rowsof 5.
S: (Draw3 rows of 5.)
T: Write a multiplication expression todescribeyour
array. Remember,anexpression isdifferentfroman
equationbecause it doesn’thave anequal sign.
S: (Write 3 × 5.)
T: Skip-counttofindthe product.
S: 5, 10, 15.
T: Withyour partner,compare my drawingwithyour
array. Which iseasiertocount? Why?
S: (Discuss.)
Show6 groupsof 2 and repeatthe process.

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Problem Set (10 minutes)
Studentsshoulddotheirpersonal besttocompletethe
ProblemSetwithinthe allotted10minutes. Forsome
classes,itmaybe appropriate tomodifythe assignmentby
specifyingwhichproblems they workonfirst. Some
problemsdonotspecifyamethodforsolving. Students
should solve these problemsusingthe RDWapproach
usedforApplicationProblems.
DirectionsonthisProblemSetincludethe words
expression andequation. Remindstudents thatwhile an
answerisnot requiredwithan expression,itshouldbe
includedwithan equation.
Student Debrief (10 minutes)
LessonObjective: Relate multiplicationtothe array
model.
The StudentDebrief isintendedtoinvitereflectionand
active processingof the total lessonexperience.
Invite studentstoreviewtheirsolutionsforthe Problem
Set. Theyshouldcheckworkby comparinganswerswitha
partnerbefore goingoveranswersasa class. Lookfor
misconceptionsormisunderstandingsthatcanbe
addressedinthe Debrief. Guide studentsina
conversationtodebrief the ProblemSetandprocessthe
lesson.
Anycombinationof the questionsbelow maybe usedto
leadthe discussion.
 In Problems5and 6, how dothe arrays represent
equal groups?
 Compare Problems6and 7. (Arrayshave the
same numberineach groupbut a different
numberof groups.)
 Compare equal groupsinscattered
configurationsandarrays.
 Review new vocabulary: row, array, numberof
groups, size ofgroups, and product.
 Promptstudentstonotice arrays aroundthe
room andpossiblythinkof arraysin real-world
situations.

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Exit Ticket (3 minutes)
Afterthe StudentDebrief,instruct studentstocomplete the ExitTicket. A review of theirworkwillhelpwith
assessingstudents’understandingof the conceptsthatwere presentedintoday’slessonandplanningmore
effectivelyforfuture lessons. The questionsmaybe readaloudtothe students.

Lesson 2 SprintNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
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Addor Subtract Using 2
1. 0 + 2 = 23. 2 + 4 =
2. 2 + 2 = 24. 2 + 6 =
3. 4 + 2 = 25. 2 + 8 =
4. 6 + 2 = 26. 2 + 10 =
5. 8 + 2 = 27. 2 + 12 =
6. 10 + 2 = 28. 2 + 14 =
7. 12 + 2 = 29. 2 + 16 =
8. 14 + 2 = 30. 2 + 18 =
9. 16 + 2 = 31. 0 + 22 =
10. 18 + 2 = 32. 22 + 22 =
11. 20 – 2 = 33. 44 + 22 =
12. 18 – 2 = 34. 66 + 22 =
13. 16 – 2 = 35. 88 – 22 =
14. 14 – 2 = 36. 66 – 22 =
15. 12 – 2 = 37. 44 – 22 =
16. 10 – 2 = 38. 22 – 22 =
17. 8 – 2 = 39. 22 + 0 =
18. 6 – 2 = 40. 22 + 22 =
19. 4 – 2 = 41. 22 + 44 =
20. 2 – 2 = 42. 66 + 22 =
21. 2 + 0 = 43. 888 – 222 =
22. 2 + 2 = 44. 666 – 222 =
A
NumberCorrect: _______

Lesson 2 SprintNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
38
Addor Subtract Using 2
1. 2 + 0 = 23. 4 + 2 =
2. 2 + 2 = 24. 6 + 2 =
3. 2 + 4 = 25. 8 + 2 =
4. 2 + 6 = 26. 10 + 2 =
5. 2 + 8 = 27. 12 + 2 =
6. 2 + 10 = 28. 14 + 2 =
7. 2 + 12 = 29. 16 + 2 =
8. 2 + 14 = 30. 18 + 2 =
9. 2 + 16 = 31. 0 + 22 =
10. 2 + 18 = 32. 22 + 22 =
11. 20 − 2 = 33. 22 + 44 =
12. 18 − 2 = 34. 66 + 22 =
13. 16 − 2 = 35. 88 − 22 =
14. 14 − 2 = 36. 66 − 22 =
15. 12 − 2 = 37. 44 − 22 =
16. 10 − 2 = 38. 22 − 22 =
17. 8 − 2 = 39. 22 + 0 =
18. 6 − 2 = 40. 22 + 22 =
19. 4 − 2 = 41. 22 + 44 =
20. 2 − 2 = 42. 66 + 22 =
21. 0 + 2 = 43. 666 − 222 =
22. 2 + 2 = 44. 888 − 222 =
B
[KEY]
NumberCorrect: _______
Improvement: _______

Lesson 2 Problem SetNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
39
Name Date
Use the arrays belowtoanswereachsetof questions.
1. a. How manyrows of cars are there? __________
b. How manycars are there ineach row? __________
2. a. What is the numberof rows? __________
b. What isthe numberof objectsineachrow? __________
3. a. There are 4 spoonsineachrow. How manyspoonsare in2 rows? __________
b. Write a multiplicationexpressiontodescribethe array. ____________________
4. a. There are 5 rowsof triangles. How manytrianglesare ineachrow? _________
b. Write a multiplicationexpressiontodescribethe total numberof triangles.
______________________

Lesson 2 Problem SetNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
40
5. The dots belowshow2 groupsof 5.
a. Redrawthe dotsas an array that shows2 rowsof 5.
b. Compare the drawingto yourarray. Write at least1 reasonwhytheyare the same
and 1 reasonwhy theyare different.
6. Emma collectsrocks. She arrangesthemin 4 rows of 3. Draw Emma’sarray to show how manyrocks she
has altogether. Then,write amultiplication equation todescribethe array.
7. Joshuaorganizescansof foodinto an array. He thinks,“Mycans show 5 × 3!” Draw Joshua’sarrayto find
the total numberof cans he organizes.

Lesson 2 Exit TicketNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
41
Name Date
1. a. There are 4 rowsof stars. How many stars are in eachrow? __________
b. Write a multiplicationequationtodescribe the array. _______________
2. Judycollectsseashells. She arrangesthemin3 rows of 6. Draw Judy’sarray to show how many seashells
she has altogether. Then, write amultiplication equation todescribe the array.

Lesson 2 HomeworkNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
42
Name Date
Use the arrays belowtoanswereachsetof questions.
1. a. How manyrows of erasersare there? __________
b. How manyerasersare there ineachrow? __________
2. a. What is the numberof rows? __________
b. What isthe numberof objectsineachrow? __________
3. a. There are 3 squaresineachrow. How manysquaresare in5 rows? __________
b. Write a multiplicationexpressiontodescribethe array. __________
4. a. There are 6 rowsof stars. How many stars are in eachrow? __________
b. Write a multiplicationexpressiontodescribethe array. __________

Lesson 2 HomeworkNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
43
5. The trianglesbelowshow3groupsof four.
a. Redraw the trianglesasan array that shows3 rows of four.
b. Compare the drawingto yourarray. How are theythe same?
How are theydifferent?
6. Rogerhas a collectionof stamps. He arrangesthe stampsinto5 rowsof four. Draw an array to represent
Roger’sstamps. Then,write a multiplication equation todescribe the array.
7. Kimberly arranges her18 markers as an array. Draw an array that Kimberlymightmake. Then, write a
multiplication equation todescribeyourarray.

Lesson 2 TemplateNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
44
0F

threes array

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