Understanding Equal Groups as Multiplication

Lesson 1NYS COMMON CORE MATHEMATICS CURRICULUM 3•1
Lesson 1: Understandequal groups ofas multiplication.
19
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NOTES ON
FLUENCYPRACTICE:
Think offluencyas having three goals:
1. Maintenance (stayingsharpon
previouslylearnedskills).
2. Preparation (targetedpractice for
the current lesson).
3. Anticipation(skills that ensure that
students will be readyfor the in-
depth workof upcoming lessons).
Lesson 1
Objective: Understand equal groups of as multiplication.
Suggested Lesson Structure
Fluency Practice (5 minutes)
Application Problem (10 minutes)
Concept Development (35 minutes)
Student Debrief (10 minutes)
Total Time (60 minutes)
Fluency Practice (5 minutes)
 Group Counting 3.OA.1 (5 minutes)
Group Counting (5 minutes)
Note: Basic skip-countingskillsfromGrade 2 shiftfocusinthis
Grade 3 activity. Group countinglaysa foundationfor
interpretingmultiplicationasrepeatedaddition. When
studentscountgroupsinthisactivity,theyadd and subtract
groupsof 2 whencountingup anddown.
T: Let’scount to 20 forwardand backward. Watch my
fingerstoknowwhethertocountup or down. A
closedhandmeansstop. (Showsignals duringthe
explanation.)
T: (Rhythmicallypoint upuntil achange isdesired. Show aclosedhand;thenpointdown.)
S: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9,
8, 7, 6, 5, 4, 3, 2, 1, 0.
T: Let’scount to 20 forwardand backwardagain. This time whispereveryother number. Saythe other
numbersina regularvoice.
S: (Whisper) 1,(speak) 2,(whisper) 3,(speak) 4,(whisper) 5,(speak) 6,etc.
T: Let’scount to 20 forwardand backwardagain. This time,humeveryothernumberinsteadof
whispering. Asyouhum,thinkof the number.
S: (Hum),2, (hum),4, (hum),6, etc.
T: Let’scount to 20 forwardand backwardagain. This time,thinkevery othernumberinsteadof
humming.
S: (Think),2,(think),4,(think),6,etc.

20
T: What didwe justcount by? Turn and talkto yourpartner.
S: Twos.
T: Let’scount bytwos. (Directstudentstocountforward to and backwardfrom 20, changingdirections
at times.)
Application Problem (10 minutes)
There are 83 girls and 76 boys in the third grade. How many total students are in the third grade?
Note: Students maychoose touse a tape diagramor a numberbondto model the problem. They are also
likelytosolve today’s ApplicationProblem inlessthan10 minutes. Ten minuteshave beenallotted toallow
for review of the RDW(Read,Draw,Write) process forproblem solving.
Directions onthe Read,Draw,Write (RDW) process: Readthe problem, draw and label,write anequation,
and write a wordsentence. The more studentsparticipateinreasoningthroughproblemswitha systematic
approach,the more theyinternalize those behaviorsandthoughtprocesses.
(Excerpted from“Howto ImplementA Story of Units.” A more complete explanationcanalsobe foundinthe
Grade 3 Module 1 Overview.)
Concept Development (35 minutes)
Materials: (S) 12 counters,personal white board
Problem1: Skip-counttofindthe total numberofobjects.
T: (Select10 studentstocome to the front.) At the signal,sayhow many armsyou eachhave. (Signal.)
S: 2 arms!
T: Since we each representagroup of 2 arms, let’sskip-countourvolunteers bytwostofindhow many
arms theyhave altogether. Tokeeptrackof our count,students will raise uptheirarmswhenwe
count them.
S: (Count2, 4, 6, … 20.)

21
Sample Teacher Board
T: How many raisedarms dowe have in all?
S: 20.
T: Armsdown. How manytwos didwe countto findthe total? Turn andwhispertoyour partner.
S: 10 twos.
T: What didyou count to findthe numberof twos?
S: I countedthe numberof volunteers becauseeachperson hasagroup of two arms.
T: Skip-counttofindthe total numberof arms.
S: (Say 2, 4, 6, …)
T: (Asthey count,write 2 + 2 + 2 +…)
T: Look at our additionsentence. Showthumbs
up if yousee the correct numberof twos.
S: (Showthumbsup.)
T: (Underthe additionsentence, write 10twos.) Clap3 timesif youagree that 10 groups of two is 20.
S: (Clap3 times.)
T: (Write 10 groupsof two is 20 underthe othernumbersentences.)
Problem2: Understandthe relationshipbetween repeatedaddition,countinggroupsinunitform,and
multiplicationsentences.
Seatstudents at tableswith personal white boardsand 12 counterseach.
T: You have 12 counters. Use yourcountersto make equal groups of two. How manycounterswill
youput ineach group? Showwithyourfingers.
S: (Holdup 2 fingers andmake groupsof two.)
T: How many equal groupsof two didyou make? Tell at the signal. (Signal.)
S: 6 groups.
T: 6 equal groupsof howmany counters?
S: 6 equal groupsof 2 counters.
T: 6 equal groupsof 2 counters equal how many countersaltogether?
S: 12 counters.
T: Write an additionsentencetoshowyourgroups on your
personal white board.
S: (Write 2 + 2 + 2 + 2 + 2 + 2 = 12.)
T: (Recordthe additionsentenceonthe board.) Inunit
form,howmany twosdidwe add to make 12?
S: 6 twos.
T: (Record 6 twos= 12 underthe additionsentence.) 6× 2 is anotherwayto write 2 + 2 + 2 + 2 + 2 + 2
or 6 twos. (Record 6 × 2 = 12 under6 twos= 12 on the board.) These numbersentences are all
sayingthe same thing.
Sample Teacher Board

22
NOTES ON
MULTIPLE MEANS
OF ACTION AND
EXPRESSION:
Some students mayneedmore
scaffolding to realize that multiplication
cannot be used to find totals with
groups that are not equal. Use the
following questions to scaffold.
 Does the drawing show3 fours?
 Does 3 times 4 represent this
drawing?
 How might we redrawthe picture
to make it show 3 × 4?
NOTES ON
MULTIPLE MEANS
OF REPRESENTATION:
It maybe necessaryto explicitly
connect timesand the symbol ×. Have
students analyze the model. “How
manytimes doyou see a groupof
two?” Have themcount the groups,
write the number sentence, andsay
the words together.
 6 groups of two equal12.
 6 times 2 equals 12.
T: Turn and talkto your partner. Howdo you think 6 × 2 = 12 relatestothe othernumbersentences?
S: Theyall have twosin them,andthe answeris 12. 
I thinkthe 6 showshowmany twosthere are.  You
have to count two6 timesbecause there are 6 groups
of them. That’showyou get 6 times2.  6 × 2 might
be an easierway to write a longadditionsentence.
T: Ways that are easierandfasterare efficient. Whenwe
have equal groups,multiplicationisamore efficient
wayto find the total than repeatedaddition.
Repeatthe processwith4 threes,3 fours,and 2 sixestoget
studentscomfortable withthe relationshipbetweenrepeated
addition,countinggroupsinunitform,andmultiplication
sentences.
Problem3: Write multiplicationsentencesfromequal groups.
Draw or projectthe picture to the right.
T: These are equal groups. Turn and tell yourpartner
whytheyare equal.
S: There isthe same numberof grey circlesineach group.
 All of the grey circlesare the same size andshape,
and there are 4 ineach group.
T: Work withyourpartnerto write a repeatedaddition
and a multiplicationsentenceforthispicture.
S: (Write 4 + 4 = 8 and either2 × 4 = 8 or 4 × 2 = 8.)
T: (Projectordraw the following.) Lookatmy new
drawingandthe multiplicationsentence Iwrote to
representit. Checkmyworkby writinganaddition
sentence andcountingtofindthe total numberof
objects.
S: (Write 4 + 4 + 3 = 11.)
T: Use youradditionsentence asyoutalktoyour partnerabout whyyouagree or disagree withmy
work.
S: I disagree because myadditionsentence equals11,not 12.  It’sbecause that lastgroupdoesn’t
have 4 circles.  You can domultiplication whenthe groupsare equal.  Here, the groupsaren’t
equal,sothe drawingdoesn’tshow 3 × 4.
T: I hearmost studentsdisagreeingbecause mygroupsare notequal. True, to multiplyyoumust have
equal groups.
3 × 4 = 12
MP.3

23
Problem Set (10 minutes)
Studentsshoulddotheirpersonal besttocompletethe
ProblemSetwithinthe allotted10minutes. Some
problemsdonotspecifyamethodforsolving. Thisisan
intentionalreductionof scaffoldingthatinvokesMP.5,Use
Appropriate ToolsStrategically. Studentsshouldsolve
these problemsusingthe RDWapproachusedfor
ApplicationProblems.
For some classes,itmaybe appropriate tomodifythe
assignmentbyspecifyingwhichproblems studentsshould
workon first. Withthisoption,letthe purposeful
sequencingof the ProblemSetguidethe selectionssothat
problemscontinue tobe scaffolded. Balance word
problemswithotherproblemtypestoensure arange of
practice. Considerassigningincompleteproblemsfor
homeworkorat anothertime duringthe day.
Student Debrief (10 minutes)
LessonObjective: Understand equalgroupsof as
multiplication.
The StudentDebrief isintendedtoinvitereflectionand
active processingof the total lesson experience.
Invite studentstoreviewtheirsolutionsforthe Problem
Set. Theyshouldcheckworkby comparinganswerswitha
partnerbefore goingoveranswersasa class. Lookfor
misconceptionsormisunderstandingsthatcanbe
addressedinthe Debrief. Guide studentsinaconversation
to debrief the ProblemSetandprocessthe lesson.
Anycombinationof the questionsbelow maybe usedto
leadthe discussion.
 On the firstpage,whatdidyou notice aboutthe
answerstoyour problems?
 Discussthe relationshipbetween repeated
addition andthe unitform 2 groupsof threeor 3
groupsof two, dependingonthe drawing.
 Discussthe relationshipbetweenrepeated
addition,unitform,andthe multiplication
sentence 3× 2 = 6.
 Reviewthe newvocabularypresentedinthe
lesson:equal groups, multiplication,andmultiply.

24
Exit Ticket (3 minutes)
Afterthe StudentDebrief,instructstudentstocomplete the ExitTicket. A review of theirworkwillhelpwith
assessingstudents’understandingof the conceptsthatwere presentedintoday’slessonandplanningmore
effectivelyforfuture lessons. The questionsmaybe readaloudtothe students.

Lesson 1 Problem SetNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
25
Name Date
1. Fill inthe blankstomake true statements.
a. 3 groupsof five =_________
3 fives=_________
3 × 5 = _________
b. 3 + 3 + 3 + 3 + 3 = _________
5 groupsof three = _________
5 × 3 = _________
c. 6 + 6 + 6 + 6 = ___________
_______ groups of six = __________
4 × ______ = __________
d. 4 +____ + ____ + ____ + ____ + ____ = _________
6 groupsof ________ = ___________
6 × ______ = __________

Lesson 1 Problem SetNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
26
2. The picture belowshows2 groupsof apples. Doesthe picture show 2 × 3? Explainwhyorwhynot.
3. Draw a picture to show 2 × 3 = 6.
4. Caroline,Brian,andMarta share a box of chocolates. They eachgetthe same amount. Circle the
chocolatesbelowtoshow3 groupsof 4. Then,write a repeated addition sentence andamultiplication
sentence torepresentthe picture.

Lesson 1 Exit TicketNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
27
Name Date
1. The picture belowshows4 groupsof 2 slicesof watermelon. Fill inthe blankstomake true repeated
additionandmultiplication sentences thatrepresentthe picture.
2 + ____ + ____ + ____ = ___________
4 × ______ = __________
2. Draw a picture to show3 + 3 + 3 = 9. Then,write a multiplication sentence torepresentthe picture.

Lesson 1 HomeworkNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
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Name Date
1. Fill inthe blankstomake true statements.
a. 4 groupsof five =_________
4 fives=_________
4 × 5 = _________
b. 5 groups of four = _________
5 fours= _________
5 × 4 = _________
c. 6 + 6 + 6 = ___________ d. 3 + ____ + ____ + ____ + ____ + ____ = ______
_______ groupsof six = __________ 6 groupsof ________ = ___________
3 × ______ = __________ 6 × ______ = __________

Lesson 1 HomeworkNYS COMMON CORE MATHEMATICS CURRICULUM 3•1
29
2. The picture belowshows3 groupsof hot dogs. Does the picture show 3 × 3? Explainwhyorwhynot.
3. Draw a picture to show4 × 2 = 8.
4. Circle the pencils belowtoshow3 groupsof 6. Write a repeated additionand amultiplicationsentence
to representthe picture.

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