Download Let`s do division with many different numbers.

Let's do division with many different numbers.
We are going to calculate the
number of sheets of colored paper
each student will get when we start
with different numbers of sheets.
Calculate the answers when you
are sharing among 4 people, and
6 people.
In
, we write 4 because
we must calculate 24 ÷ 6.
Divisor
Number of sheets of
colored paper
Dividend
Number of students sharing colored paper
6 people
Red:
24 sheets
4
Yellow:
32 sheets
5 remainder 2
Blue:
40 sheets
Green:
48 sheets
I think
should
be 48 ÷ 4.
4 people
If the number of
sheets keeps
increasing ...
38
© Koyo Publishing 2016
For review purposes only.
3
Division Algorithm (1) — 1-digit Divisors
Let's Think about How to
Calculate Division
We will switch
the card to
.
sheets of colored paper will be shared equally
among 4 people. How many sheets of paper will each
person get?
32 ÷ 4 = 8
Answer: 8 sheets
If there are 80
sheets, the math
sentence will be ...
1 Dividing Multiples of 10 and 100
80 sheets of colored paper will be shared equally among 4 people.
How many sheets of paper will each person get?
80 sheets
Write a math sentence.
sheets
0
1
2
Total number
of sheets
÷
3
Number of
people sharing
Math
Sentence
4 (people)
=
Number of
sheets for each
Explain the reason for your math sentence.
Eliza
Since we are finding the amount
for 1 person, the total number of
sheets must be ...
Sam
The number of sheets for 1 person is
1 of 4 equal parts of 80 sheets, so ...
80 sheets
sheets
Total number
÷ Number of
= Number of
of sheets
people sharing
sheets for each
0
Let’s think about how to calculate.
1
2
3
4 (people)
If we think in
terms of bundles
of 10 sheets ...
© Koyo Publishing 2016
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39
Since we are sharing 8 bundles
of 10 sheets among 4 people ...
Govind
8 ÷4= 2
80 ÷ 4 =
40 ÷ 2
90 ÷ 3
60 ÷ 2
70 ÷ 7
120 ÷ 4
450 ÷ 5
320 ÷ 8
630 ÷ 7
300 ÷ 6
200 ÷ 4
400 ÷ 8
100 ÷ 5
C
Additional
problems
Page 260
600 sheets of colored paper will be shared equally among 3 people.
How many sheets of paper will each person get?
Since we are sharing 6 bundles of
sheets among 3 people ...
Jayla
6
÷3= 2
600 ÷ 3 =
400 ÷ 2
800 ÷ 4
900 ÷ 3
500 ÷ 5
1600 ÷ 4
1500 ÷ 5
4200 ÷ 7
2700 ÷ 3
2000 ÷ 4
3000 ÷ 5
4000 ÷ 5
1000 ÷ 2
D
40
Additional
problems
Page 260
© Koyo Publishing 2016
For review purposes only.
2 Division Algorithm (1)
We will switch the
card to
.
60
sheets of colored paper will be
shared equally among 3 people.
How many sheets of paper will each
person get?
60 ÷ 3 = 20
Answer: 20 sheets
The number
of sheets of
paper is 72,
but the math
sentence is ...
72 sheets of colored paper will be shared equally among 3 people.
How many sheets of paper will each person get?
72 sheets
Math
Sentence
sheets
0
1
2
3 (people)
Yoko
First, estimate about
how many sheets
each person will get.
Since 60 ÷ 3 = 20, each
person should get more
than 20 sheets.
Let’s think about how to calculate.
Think about what is different
from the calculations we have
learned so far.
41
© Koyo Publishing 2016
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Eliza
Govind
I thought about the bundles
of 10 sheets and the loose
ones separately.
First, share the bundles
of 10 sheets.
I used the division we have
learned already.
Since 20 × 3 = 60, we
can give each person
20 sheets first.
Left over
60 ÷ 3 = 20
20 sheets
7 ÷ 3 = 2 R 1
One bundle left and 2 loose
sheets make 12 sheets to be
shared by 3 people.
72 − 60 = 12, so there
are 12 sheets to be shared
among 3 people.
12 ÷ 3 = 4
4 sheets
Since each person gets 20
sheets and 4 sheets, they will
each get 24 sheets.
12 ÷ 3 = 4
Each person gets 24 sheets.
72 ÷ 3 =
sheets
Answer:
Explain these 2 students’ thinking.
Both of them started
with bundles of 10
sheets, didn't they?
You can calculate 72 ÷ 3 by using the
division algorithm as shown next.
42
Jayla
372
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How are Eliza's and Govind's ideas being
used in the division algorithm?
4 24
20
3 72
6 0 20 × 3
12
12 4 × 3
0
Calculation of the ones place
Calculation of the tens place
How to calculate 72 ÷ 3
2
3 7 2
2
3 7 2
6
Divide 7 in the tens
place by 3 and write
the quotient, 2, in the
tens place.
Multiply 3 by 2.
2
3 7 2
6
1
Subtract 6 from 7.
2
3 7 2
6
1 2
Bring down the 2
from the ones place.
2 4
3 7 2
6
1 2
Divide 12 by 3 and
write the quotient,
4, in the ones place.
2
3 7
6
1
1
4
2
2
3 7
6
1
1
0
4
2
The division
algorithm starts
from the largest
place, doesn't it?
Check the answer.
72 ÷ 3 = 24
3 × 24 =
452
Multiply 3 by 4.
2
2
Subtract 12 from 12.
2
2
0
E
43
3 81
5 85
4 96
7 91
3 75
6 90
Additional
problems
Page 260
© Koyo Publishing 2016
For review purposes only.
76 sheets of colored paper will be
shared equally among 3 people.
How many sheets of paper will
each person get, and how many
will be left?
About how
many sheets will
each person get?
Math
Sentence
Victor
2
37
6
1
1
Let’s think about how to calculate using the
division algorithm.
Is the remainder less
than the divisor?
76 ÷ 3 = 25
R1
Answer: Each person will get
sheets and
5
6
6
5
1
sheet will be left.
What is the quotient in the above division problem?
What is the remainder?
Check the calculation.
3
Divisor
×
25
+
1
=
× Quotient + Remainder =
The calculations shown
on the right are incorrect.
Explain what is incorrect, and
calculate them correctly.
23
3 74
6
14
9
5
44
76
Dividend
19
4 94
4
54
36
18
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Calculate the following, and then check your answers.
97 ÷ 2
275
86 ÷ 7
83 ÷ 3
79 ÷ 5
6 93
5 89
4 97
8 98
3 71
8 95
7 82
4 91
6 70
Check your
answers, too.
F
Additional
problems
Page 260
There are 50 pieces of candy.
If 3 people share the pieces of
candy equally, how many will
each person get, and how many
will be left?
If we give 4 pieces of candy
to each person, how many
people can get candy, and
how many pieces of candy
will be left?
A 90-cm ribbon is cut into 8-cm pieces.
How many 8-cm pieces can we make,
and how many cm of ribbon will be left?
45
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21
486
8
6
4
2
Explain how each calculation is
done on the right.
The differences
from what we have
learned so far are...
20
362
6
2
0
2
Yoko
The answer
might be 4 with
a remainder of 1.
Sam
The calculation shown on the
right is incorrect.
Explain what is incorrect, and
calculate it correctly.
4
2 81
8
1
3 65
2 87
5 57
6 68
4 84
3 69
2 46
3 93
G
2 61
6 64
7 76
8 85
Additional
problems
3 91
2 80
Page 260
4 83
5 50
There are 52 cards. If we give each person 5 cards, how many
people can get cards, and how many cards will be left?
Calculate using the division algorithm.
Pay attention to where to write the quotient.
Eliza
We studied this kind
of division in third
grade, didn't we?
5 28
9 42
Victor
Since the quotient
is a single digit
number...
4 26
6 50
3 22
7 52
46
7 3 4
8 45
4 30
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734 sheets of colored paper will be shared equally among
5 people. How many sheets of paper will each person get,
and how many will be left?
About how
many sheets will
each person get?
Math
Sentence
Jayla
Let’s think about how to calculate using the division algorithm.
If we start with
the bundles of
100 sheets...
Govind
How to calculate 734 ÷ 5
1
5 7 34
5
2
This 23 shows
that there are
23 tens,
doesn't it?
1
5 7
5
2
2
0
4
3 4
1
5 7
5
2
2
0
0
4 6
3 4
3
0
3
3
0
3 4
3 0
4
Calculation of the
hundreds place
Calculate 7 ÷ 5, and
write the quotient, 1,
in the hundreds place.
7÷5=1R2
Calculation of the
tens place
Bring down the 3.
Calculate 23 ÷ 5, and
write the quotient, 4,
in the tens place.
23 ÷ 5 = 4 R 3
Calculation of the
ones place
Bring down the 4.
Calculate 34 ÷ 5, and
write the quotient, 6,
in the ones place.
34 ÷ 5 = 6 R 4
6
4 0 146
100
5 734
5 0 0 100 × 5
234
200
40 × 5
34
6×5
30
4
Check the answer.
5 × 146 + 4 =
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47
6 809
8 991
3 822
5 630
3 715
4 908
7 819
6 704
H
Explain how each calculation on
the right is done.
The differences
from what we
have learned
so far are...
Eliza
Additional
problems
2 1 0
4 8 4 3
8
4
4
3
0
3
Page 260
2 0 6
3 6 1 9
6
1
0
1 9
1 8
1
4 851
6 684
7 918
3 726
3 695
2 847
3 963
2 486
I
Additional
problems
Page 260
4 683
2 960
3 962
4 480
8 815
3 621
2 807
5 547
J
Additional
problems
Page 261
Calculate 625 ÷ 3 using the division algorithm.
3 6 2 5
Victor
What
modification
did Victor
make?
48
208
3 625
6
25
24
1
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3 Division Algorithm (2)
We will switch the
card to
.
183
4 732
4
33
32
12
Answer: 183 sheets
12
0
sheets of colored paper will be
shared equally among 4 people. How
many sheets of paper will each person
get, and how many will be left?
732 ÷ 4 = 183
Even if the number
of sheets is 256, the
math sentence is...
256 sheets of colored paper will be shared equally among 4 people.
How many sheets of paper will each person get, and how many will
be left?
0
0
256 (sheets)
1
2
3
Math
Sentence
4 (people)
Let’s think about how to calculate using the
division algorithm.
100
10
10
About how
many sheets will
each person get?
10
Sam
100
Govind
10
10
What if we split the
bundles of 100 into
bundles of 10 ...
The calculation in the
hundreds place is 2 ÷ 4,
but we can't do that ...
Yoko
Try using Yoko’s idea.
49
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Split the bundles of 100 into bundles of 10
100
10
10
100
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Share the bundles of 10 among 4 people
Left over
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
25 ÷ 4 =
Share 16 sheets
made from the
left over 10 and
16 ÷ 4 = 4
50
10
R1
among 4 people
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Each person gets 64 sheets
© Koyo Publishing 2016
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256 ÷ 4 may be calculated using the division algorithm as follows.
How to calculate 256 ÷ 4
4 256
0 6
4 256
24
01
064
4 256
24
016
016
0 0
4
6 0 64
4 256
2 4 0 60 × 4
16
16
4×4
0
Calculation of the
hundreds place
There is no digit in the
hundreds place of the
quotient since it is 2 ÷ 4.
Calculation of the
tens place
Calculate 25 ÷ 4, and
write the quotient, 6, in
the tens place.
25 ÷ 4 = 6 R 1
Calculation of the
ones place
Calculate 16 ÷ 4, and
write the quotient, 4, in
the ones place.
16 ÷ 4 = 4
Even though 25 ÷ 4 =6 R 1
is in the middle of the
algorithm, the remainder
is still less than the
divisor, isn't it?
Check the answer.
4 × 64 =
S U
M M
A R
Y
When the left-most digit in the dividend is less than the divisor, look
at the two-digit number formed by using the digit in the next place
to start the calculation.
Of the division problems below, which ones will have the quotients
that start in the tens place?
Explain your answers.
5 625
5 325
5 525
5 425
5 427
3 268
6 282
4 308
3 218
2 128
8 244
9 630
51
K
Additional
Page 261
problems © Koyo Publishing 2016
For review purposes only.
4 Calculation
with
Times
as Much
Calculation
with
Times
as Much
An adult whale is 15 m long, and its calf is 3 m long.
How many times as long is the adult whale as the
calf?
Think about it using the diagram below.
15 m
Adult
Whale
3m
Calf
0
1
 times
Write a math sentence, and then find the answer.
15 ÷ 3 =
Answer:
times
3×=5
5 times means that if we consider 3 m as 1, 15 m corresponds to 5.
If we say 1 piece is 3 m long, 15 m is
the same as 5 pieces together, isn’t it?
Paige’s class has pet hamsters. At the beginning
of the school year, there were 4 hamsters, but
now there are 24 of them. Answer the following
questions:
How many times as many hamsters are
there now as there were at the beginning
of the school year?
If we consider 4 hamsters as 1, what
number will 24 hamsters correspond to?
52
© Koyo Publishing 2016
For review purposes only.
A young giraffe is 180 cm tall, and
the adult giraffe is 3 times as tall.
How tall is the adult giraffe in cm?
Think about it using the diagram below.
cm
Adult Giraffe
180 cm
Young Giraffe
0
1
3 times
Write a math sentence, and then find the answer.
180 × 3 =
Answer:
cm
Check your answer by using division.
180 × 3 = 540
540 ÷ 3 =
A child airfare is $370, and an
adult fare is twice as much. How
much is an adult airfare?
Check your answer by
using division, too.
53
© Koyo Publishing 2016
For review purposes only.
An adult jaguar weighs 6 times as much
as a jaguar cub.
If the adult jaguar weighs 72 kg, how
much does the cub weigh in kg?
Let’s think about how to calculate the amount for 1.
Write a multiplication sentence by considering the weight of
the jaguar cub as  kg.
What calculation is necessary to find the number in the ?
Think about it using the diagram below.
72 kg
Adult
kg
Cub
0
1
 × 6 = 72
 = 72 ÷ 6
= 12
6 times
Answer: 12 kg
A laptop costs $920. The price of the
laptop is 4 times as much as the price
of a tablet.
Write a multiplication sentence by
considering the price of the tablet as
 dollars.
Then, find the number that goes in the .
54
© Koyo Publishing 2016
For review purposes only.
Problems with times as much
We are going to buy 12 pear and cherry trees.
A nursery sells a set of three
pear trees for $240. How much
will it cost to buy 12 pear trees?
1 set
$240
Explain the two students’ thinking.
Victor
Eliza
I thought about the price
of each pear tree.
I thought about how many
times 12 is compared to 3.
240 ÷ 3 = 80
12 ÷ 3 = 4
80 × 12 = 960
240 × 4 = 960
Answer: $960
Answer: $960
The nursery also sells cherry trees in sets of three, and each set costs
$200. How much will it cost to buy 12 cherry trees?
Victor
If we calculate
the price of each
cherry tree ...
Calculate the cost of the following items.
The price of 5 flower pots is $120.
How much will it cost to buy 15
flower pots?
The price of 3 sets of lawn lanterns
(4 lanterns in each set) is $100.
How much will it cost to buy 21
sets of lanterns?
55
© Koyo Publishing 2016
For review purposes only.
5 Mental
Calculation
Calculation
with Times as Much
74 sheets of colored paper will be shared equally between 2 people.
How many sheets of paper will each person get?
Math
Sentence
Jayla
Since 80 ÷ 2 = 40, it
must be fewer than
40 sheets ...
Since 60 ÷ 2 = 30,
it must be more
than 30 sheets ...
Think about how to calculate
74 ÷ 2 mentally.
Sam
74 ÷ 2 = 37
60 14
60 ÷ 2 = 30
14 ÷ 2 = 7
together 37
36 ÷ 3
28 ÷ 2
66 ÷ 3
64 ÷ 4
58 ÷ 2
75 ÷ 5
60 ÷ 4
90 ÷ 5
74 ÷ 2 = 37
740 ÷ 2 = 370
Think about how to calculate
740 ÷ 2 mentally.
240 ÷ 2
680 ÷ 2
690 ÷ 3
840 ÷ 4
960 ÷ 2
750 ÷ 3
900 ÷ 6
1,000 ÷ 4
56
© Koyo Publishing 2016
For review purposes only.
Power Builder
Calculate the following.
60 ÷ 3
800 ÷ 2
84 ÷ 7
92 ÷ 3
590 ÷ 7
100 ÷ 4
628 ÷ 6
849 ÷ 4
542 ÷ 3
808 ÷ 4
458 ÷ 5
109 ÷ 9
803 ÷ 8
314 ÷ 7
420 ÷ 6
870 ÷ 3
Copy the chart and color the answers.
12
12 R 1
20 25 30 R 2
44 R 6
70 84 R 2
91 R 3
100 R 3 104 R 4
180 R 2 202 212 R 1 290 400
We will put 640 beads into 5
containers equally.
How many beads do we need
to put in each container?
Cassandra is reading a book that is 96 pages long.
If she reads 7 pages each day, how many days will it take for her
to finish the book?
A building is 6 times as high as
an electrical pole.
If the building is 42 m tall, how
high is the electrical pole?
Students from 4 classrooms are going on a field trip by bus.
There are 36 students in each classroom.
If there are 3 buses, how many students will be on each bus if the
same number of students go on each bus?
57
© Koyo Publishing 2016
For review purposes only.
3
1
Mastery
Problems
Division Algorithm (1) — 1-digit Divisors
The calculations shown below are incorrect.
Explain what is incorrect and calculate them
correctly.
12
6 79
6
19
12
7
2
3
290
3 627
6
27
27
0
With each of the division
problems shown on the
right, what digits should be
in each so that there will
be a digit in the hundreds
place in the quotient?
801
4 324
32
4
4
0
6
20
543
Below is a math sentence to check the answer
of a division problem. Write the original division
sentence.
Answer the questions about the
division shown on the right.
14 means 14 of what number?
25 means 25 of what number?
Write a math sentence to check
the answer, and then calculate.
To P
a
Page 48
Page 49
Do you
understand in
which place the
quotient will start
within the division
algorithm?
Page 49
4 × 28 + 3 = 115
4
Can you use
the division
algorithm
correctly?
Page 44
ge 2
73
58
248
3 745
6
14
12
25
24
1
Can you figure
out the original
division sentence
from the math
sentence for
checking the
answer?
Page 49
Do you
understand how
the division
algorithm works?
Page 47
© Koyo Publishing 2016
For review purposes only.
Do You Remember?
Write the following numbers with numerals.
Eight hundred six trillion, three hundred seventy-five billion
563 one hundred millions, and 7,428 ten thousands combined
The number 1 less than 10 billion
The number 10 times as much as 400 billion
Write the numbers indicated by A, B, and C on the number line below.
0
10 trillion
A
20 trillion
30 trillion
40 trillion
B
C
Estimate the size of angles
and
, then measure them.
ulation
Calc
C ha
ge
What Does the Sentence Say?
ll e n
Calculate the following and write the letters in the
correct boxes.
A 0.6 + 0.2
Z 0.3 + 0.9
N 0.8 + 0.8
M 0.7 − 0.2
A 1 − 0.6
I 1.8 − 0.4
M 25 + 15
!
H 37 + 47
G 34 − 24
T 67 − 27
A 1 −
0.4
0.5
0.8
1.2
1.4
1.6
1
4
3
5
2
6
4
7
3
8
7
7
(1)
2
+ 8
4
6
5
8
© Koyo Publishing 2016
For review purposes only.
59