Download Lesson 3.5b - Estimation, Mental Math 4.4e.1 and

3.5b Estimation
Objectives
• Estimatetheanswerstodivisionproblems.
• Solvewordproblems.
Materials
• Place-valuediscs
• Appendix3.5b(RenamedfromAppendix3.5c)
Common Core State Standards
3.OA.3
3.OA.4
3.OA.5
3.OA.7
3.OA.8
3.MD.4
Mathematical Practices
MP1
MP4
MP6
MP7
digital
Divide tens, hundreds, or thousands mentally
• Studentshavealreadybeendividingbyones,tens,and
hundredsindivisionproblems.Onthispage,focuson
thenumberofzerosinthedividendandinthequotient
andtheplacevalueofthedigits.Forallofthese
problems,ifyoutakeoffsomeofthezeros,thedigits
thatremaincanberelatedtothedivisionfactsstudents
shouldalreadyknow.Thequotientwillhaveonlyone
nonzerodigit.
• InTask4,therearethesamenumberofzerosinthe
quotientasinthenumberwearedividingby3.InTask
5,thereisoneless.Askstudentstoexplainwhy.They
cannotdivide4onesby5,buttheycandivide40ones
by5.Sotodivide400by5,theyaredividing40tens.
Thequotientwillbetens,andsowillhave1zero.
Answers:
5. (a) 2
2
(b) 20
2
(c) 200
2
(d) 2,000
2
6. 8
80
800
7. (a) 3
(d) 20
(g) 80
(a) 6 ÷ 3 =
1
1
1
1
1
6 ones ÷ 3 =
1
ones
(b) 60 ÷ 3 =
10
10
10
10
10
6 tens ÷ 3 =
10
tens
(c) 600 ÷ 3 =
100
100
100
100
100
6 hundreds ÷ 3 =
100
hundreds
(d) 6,000 ÷ 3 =
1,000 1,000
1,000 1,000
1,000 1,000
6 thousands ÷ 3 =
thousands
6. Complete the equations.
4,0 00 ÷ 5 = 8 00
40 ÷ 5 =
400 ÷ 5 =
4,000 ÷ 5 =
(b) 90 ÷ 3
(e) 360 ÷ 4
(h) 2,400 ÷ 4
(c) 900 ÷ 3
(f ) 400 ÷ 5
(i) 1,000 ÷ 5
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(b) 30
(e) 90
(h)600
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5. Complete the equations.
7. Divide.
(a) 9 ÷ 3
(d) 40 ÷ 2
(g) 320 ÷ 4
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(c) 300
(f) 80
(i) 200
• Writeproblemssuchasthoseontherightandhave
studentsfindtheanswermentallyortellyouthatthey
willneedtodothedivisionalgorithm.Theyneedto
recognizewhenthefirstnumberisamultipleof10,100,
or1,000thatcanbeevenlydividedbythesecond,and
whenitcannot.
226
Textbook, p. 115
720÷8
(90)
6,300÷7
(900)
3,500÷6 (divisionalgorithm)
4,200÷9 (divisionalgorithm)
4,200÷7
(600)
10,000÷3 (divisionalgorithm)
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Estimation
Textbook, p. 116
• Remindstudentsthattheycanestimatetheanswerto
addition,subtraction,andmultiplicationproblemsby
roundingthenumberstotensorhundreds,soitiseasy
tomentallycalculateinordertogetanapproximate
answer.
• Askstudentstoestimatetheanswerto317×5.
317×5
• HavestudentslookattheprobleminTask8.
• Tellthemthattheycanalsoroundtoestimatethe
answerfordivisionproblems,andtheystillwantto
roundtosomethingthatmakesthecalculationseasy.
Becauseofremaindersindivision,roundingtothe
nearesthundredswillnothelp.Theycannoteasily
divide200by3.Sowithdivision,theyusuallyroundto
anumberthathas1or2nonzerodigitsthattheycan
easilydividementally.15and18canbedividedby3.
• Askstudentswhether150or180iscloserto167.180is
closer,butnotbymuch.
• Tellstudentsthatoften,itwillbeeasiertopickthe
closestnumbertoroundto.Ifbothnumbersareclose,
theyshouldnotspendtoomuchtimedecidingwhichis
closer;estimationshouldbesomethingthattheycando
quickly.Anestimateusing150÷3=50willstillgivean
estimatedanswercloseenoughtotellwhethertheactual
answerto167÷3isreasonable.
• Askstudentstofindtheactualanswerto167÷3(55R2).
• HavestudentsdoTask8.
300×5=1,500
8. Estimate the value of 167 ÷ 3.
200 ÷ 3 = ?
I can’t round to the
nearest hundred.
There will be a remainder.
150 ÷ 3
167
180 ÷ 3
180 ÷ 3 =
.
The value of 167 ÷ 3 is about
9. Ashley made 276 muffins.
She put them into boxes of 4 muffins each.
How many boxes of muffins were there?
How many were left over?
(a) Estimate the number of boxes.
276 ÷ 4 = ?
4 × 6 = 24
4 × 7 = 28
I will use 280.
280 ÷ 4 =
The answer will be around 70.
(b) Find the answer.
There were
boxes of muffins.
muffins were left over.
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Answers:
8. 60
60
9. (a) 70
(b) 69
0
Assessment
• Youcanalsohavestudentsfindtheactualanswersfor
Task9.
• Estimatedanswerscanvary.Astudentgoodatmental
mathcouldestimate(a),using840÷2=420.
Answers:
10. (a) 840÷4=210or800÷4=200
(b) 380÷2=190or400÷2=200
(c) 100÷5=20
(d) 600÷3=200
(e) 500÷2=250
(f) 1,600÷4=400
exact answers:
(a) 210R2
(b) 189
(c) 19R3
(d) 189R1
(e) 256
(f) 423R1
11. $150
12. 105
2
Textbook, p. 117
10. Estimate the value of
(a) 842 ÷ 4
(b) 378 ÷ 2
(c) 98 ÷ 5
(d) 568 ÷ 3
(e) 512 ÷ 2
(f ) 1,693 ÷ 4
11. 5 bicycles cost $750.
How much does one bicycle cost?
?
750
1 bicycle costs
12. Craig has 317 oranges.
He puts 3 oranges in each bag.
How many bags of oranges can he make?
How many oranges will be left over?
He can make
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bags of oranges.
oranges will be left over.
Exercise 16, pages 114–115
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Other word problems
Appendix 3.5b
• ProvidestudentswithcopiesofAppendix3.5bandlet
themworkontheproblems.
• Problem4ischallenging.Studentsmaynotbeableto
solveitindependently.
1. AmyandBradhave$240.
• Studentswillprobablydrawacomparisonmodel,as
Bradhas$60morethan
theyaretoldhowmuchmoreBradhasthanAmy.Ifthey
Amy.Howmuchmoney
maketheamountBradhasthesameaswhatAmyhas,
doesAmyhave?
thentheywillhavetwoequalunits.Todothat,they
needtosubtract$60fromtheamountBradhas,and
therefore$60fromthetotal.
?
Amy
Brad
$60
$240
2units=240–6=180
1units=180÷2=90
Amyhas$90.
• Ifstudentshavedifficultywiththis,drawapart-whole
modelwiththreeparts.Twopartsareequal(theamount
AmyhasandthepartofwhatBradhasthatisthesame
aswhatAmyhas),andthethirdpartistheamountBrad
hasmorethanAmy.Althoughcomparisonmodelsare
quiteusefulformultistepproblems,thisisessentiallya
3-part-wholeproblem.Withtwoofthepartsequal,and
withapart-wholemodel,itcanbeeasiertovisualize
thattheyaresubtracting$60fromthetotaltogettwo
equalunits.
• Groupingproblemsaredifficulttomodel,asthenumber 2. Abakeryispacking2
ofgroupsisnotknown.Somestudentsmaybeableto
banana-nutmuffinsand
easilysolvethiswithoutamodel;forothers,sometype
3orange-cranberrymuffins
ofdiagrammaybeusefulinhelpingthemvisualizethe
ineachpackage.Thereare
situation.
430muffins.Howmanyof
• Asthereare5muffinsineachpackage,theyfirstdivide
themareorange-cranberry
by5togetthenumberofpackages.Thentheycan
muffins?
multiplythatbythenumberoforange-cranberrymuffins
ineach.
430÷5=86
86×3=258
Thereare258orange-cranberrymuffins.
$153
153÷3=51
51×5=255
Hespends$255.
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3. T
omwantstobuythe
samegameforeachofhis
niecesandnephews.He
hasenoughmoneytobuy
8games,buthebuysonly
5games.Hehas$153left.
Howmuchmoneydoeshe
spend?
© 2014 Singapore Math Inc.®
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• Studentsneedtorememberfromapreviouslessonthat 4. CharlieandDrewcollected
tohavethesamenumberofcards,thepersonwithmore
650gamecards.Charlie
cardshastogivehalfthedifference.
has4timesasmanygame
cardsasDrew.Howmany
?
wouldhegivetoDrewso
theybothhavethesame
C
numberofcards?
650
D
?
650÷5=130;130×3=390
OR:130×4=520,520−130=390
390÷2=195
CharlieneedstogiveDrew195cards.
Practice
Workbook Exercise 16,
pp. 114–115
• HavestudentscompleteWorkbookExercise16
onpages114–115.
EXERCISE 16
4. David saved $900 in 4 months.
He saved the same amount of money each month.
How much did he save each month?
1. Write the missing numbers.
(a) 8 tens ÷ 2 = 4 tens
80 ÷ 2 = 40
(b) 40 tens ÷ 4 = 10 tens
400 ÷ 4 = 100
(c) 10 tens ÷ 5 = 2 tens
100 ÷ 5 = 20
(d) 21 tens ÷ 3 = 7 tens
210 ÷ 3 = 70
$900 ÷ 4 = $225
He saved $225 each month.
5. Ms. Holt had 186 stickers.
She gave 5 stickers to each student in her class.
How many students were there in her class?
How many stickers were left over?
2. Divide.
6÷2=3
60 ÷ 2 = 30
600 ÷ 2 = 300
186 ÷ 5 = 37 R 1
24 ÷ 3 = 8
240 ÷ 3 = 80
2,400 ÷ 3 = 800
There were 37 students in her class.
32 ÷ 4 = 8
320 ÷ 4 = 80
3,200 ÷ 4 = 800
1 sticker was left over.
3. Estimate, and then divide.
(a)
222 ÷ 4 is about
200
(b)
÷ 4 = 50
285 ÷ 3 is about
270
÷ 3 = 90
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6. 3 students sold 243 concert tickets altogether.
Each student sold the same number of tickets.
How many tickets did each student sell?
4)222
55 R 2
243 ÷ 3 = 81
Each student sold 81 tickets.
3)285
95
Unit 3: Multiplication and Division
Unit 3: Multiplication and Division
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Primary Mathematics Teacher’s Guide 3A
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Copying is permitted.
Mental Math
4.4e.1
Mental Math
4.4e.2
7,200 ÷ 9 =
435 + 98 =
120 ÷ 6 =
81 ÷ 9 =
6,300 ÷ 9 =
6×8=
560 ÷ 8 =
328 + 671 =
3,000 ÷ 5 =
3,420 − 700 =
4,200 ÷ 7 =
64 ÷ 8 =
810 ÷ 9 =
52 × 4 =
4,800 ÷ 8 =
438 + 90 =
2,400 ÷ 6 =
800 × 7 =
450 ÷ 9 =
326 − 97 =
3,600 ÷ 6 =
387 + 8 =
2,800 ÷ 4 =
56 ÷ 7 =
9,000 ÷ 3 =
7 × 60 =
2,400 ÷ 8 =
48 + 85 =
2,500 ÷ 5 =
4,000 ÷ 8 =
10,000 ÷ 5 =
83 − 58 =
3,000 ÷ 6 =
357 ÷ 7 =
210 ÷ 7 =
12 × 7 =
150 ÷ 5 =
200 ÷ 5 =
2,000 ÷ 4 =
480 + 50 =
378
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