The Partial-Quotients
Division Algorithm, Part 1
Objectives To introduce and provide practice with a
“low-stress” division algorithm for 1-digit divisors.
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
Teaching the Lesson
EM Facts
Workshop
Game™
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Reviewing Place Value in Decimals
• Identify and use multiples of 10. Math Journal 1, p. 146
Students solve problems involving
various basic skills with decimals.
[Number and Numeration Goal 3]
• Add multiples of 10. [Operations and Computation Goal 1]
• Subtract multidigit numbers. [Operations and Computation Goal 2]
• Apply extended multiplication facts to
long-division situations. [Operations and Computation Goal 3]
• Solve equal-grouping division
number stories. [Operations and Computation Goal 4]
Math Boxes 6 3
Math Journal 1, p. 147
Students practice and maintain skills
through Math Box problems.
Study Link 6 3
Math Masters, p. 180
Students practice and maintain skills
through Study Link activities.
Key Activities
Students learn and practice a paper-andpencil algorithm for division that permits
them to build up the quotient by working
with “easy” numbers.
Ongoing Assessment:
Informing Instruction
See pages 413 and 416.
Ongoing Assessment:
Recognizing Student Achievement
Use journal pages 144 and 145. [Operations and Computation Goal 4]
Key Vocabulary
dividend divisor partial quotient
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Playing Beat the Calculator
Student Reference Book, p. 233
Math Masters, p. 461
per group: 4 each of number cards 1–10
(from the Everything Math Deck, if available),
calculator
Students practice extended multiplication facts.
ENRICHMENT
Determining the Cost of Pens
Math Masters, p. 181
10 nickels (optional)
Students use clues to solve a division
number story.
EXTRA PRACTICE
Playing Division Dash
Student Reference Book, p. 241
Math Masters, p. 471
per partnership: 4 each of number cards 1–9
(from the Everything Math Deck, if available)
Students practice dividing 2- or 3-digit
dividends by 1-digit divisors.
ELL SUPPORT
Building Vocabulary
chart paper markers
Students display and label parts of division
number models.
Materials
Math Journal 1, pp. 144 and 145
Study Link 62
Math Masters, p. 403; pp. 436 and 438
(optional)
slate
Advance Preparation
Make 1 or 2 copies of Math Masters, page 403 for each student. Have extra copies available throughout this unit.
Teacher’s Reference Manual, Grades 4–6 pp. 132–140
412
Unit 6
Division; Map Reference Frames; Measures of Angles
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
4.OA.3, 4.NBT.6, 4.MD.2
Mental Math and Reflexes
Pose true or false problems involving equal groups. Suggestions:
There are at least 2 [5s] in 11. true
There are at least 3 [2s] in 5. false
There are at least 4 [3s] in 14. true
There are at least 7 [6s] in 45. true
There are at least 9 [8s] in 67. false
There are at least 8 [9s] in 75. true
There are at least 10 [8s] in 92. true
There are at least 30 [6s] in 157. false
There are at least 40 [9s] in 391. true
Math Message
Study Link 6 2 Follow-Up
How many days are there in 30 weeks?
Partners compare answers. Have volunteers explain
different strategies that could be used to solve Problem 3.
How many weeks are there in 98 days?
1 Teaching the Lesson
Ongoing Assessment:
Informing Instruction
Math Message
WHOLE-CLASS
ACTIVITY
Follow-Up
SOLVING
Algebraic Thinking In the first problem, the number of weeks is
known; students find the number of days by multiplying by 7. In
the second problem, the total number of days is given; students
divide by 7 to find the number of weeks.
weeks
days per week
total days
30
7
t 210
Watch for students who do not understand
where the numeral 7 came from because
they did not see a 7 in the problem and did
not equate a week with 7 days. Remind
students that using another name for an
equal quantity is often necessary to solve
problems.
Teaching Aid Master
weeks days per week
w 14
Name
total days
7
Date
Easy Multiples
98
100 ⴱ
Introducing the
Partial-Quotients Algorithm
Time
WHOLE-CLASS
ACTIVITY
ELL
(Math Masters, pp. 403 and 438)
17
⫽
100 ⴱ
⫽
50 ⴱ
⫽
50 ⴱ
⫽
20 ⴱ
⫽
20 ⴱ
⫽
10 ⴱ
⫽
10 ⴱ
⫽
5ⴱ
⫽
5ⴱ
⫽
2ⴱ
⫽
2ⴱ
⫽
1ⴱ
⫽
1ⴱ
⫽
100 ⴱ
⫽
100 ⴱ
⫽
50 ⴱ
⫽
50 ⴱ
⫽
⫽
This lesson formally introduces the partial-quotients algorithm.
Begin with an equal-grouping division problem like the following:
●
Amy is 127 days older than Bob. How many weeks older
is Amy?
Briefly work through the steps mentioned in the previous lessons:
Step 1: Decide what you need to find out. How many weeks older
is Amy? Write a variable in the “weeks” column.
20 ⴱ
⫽
20 ⴱ
10 ⴱ
⫽
10 ⴱ
⫽
5ⴱ
⫽
5ⴱ
⫽
2ⴱ
⫽
2ⴱ
⫽
1ⴱ
⫽
1ⴱ
⫽
Math Masters, p. 438
Lesson 6 3
413
weeks days per week
w
total days
7
127
Step 2: Identify the data you need to solve the problem. The
number of days in all and the number of days per week. Write
the data in the appropriate spaces in the diagram. (See margin.)
Step 3: Decide what to do to find the answer. 127 days must be
equally grouped (divided) so that there are 7 days in each week.
Write a number model to represent the problem using a variable
for the unknown. Number model with unknown: 127 / 7 = w
Step 4: Do the computation. Model the following algorithm while
students follow along with paper and pencil.
1. Write the problem in the traditional form: 7 127 . Point out
that the dividend—the number that is being divided—is 127.
The divisor—the number that the dividend is being divided
by—is 7. To support English language learners, label the
dividend and the divisor on the board.
2. Draw a vertical line so that the problem looks like the problem
below. The vertical line will separate subtractions from partial
quotients.
7 127
Adjusting the Activity
Suggest that students first make
a list of “easy” multiples of the divisor. For
example, if the divisor is 7, students might
make the following list:
100 ∗ 7 = 700
50 ∗ 7 = 350
20 ∗ 7 = 140
10 ∗ 7 = 70
5 ∗ 7 = 35
2 ∗ 7 = 14
1∗7 =7
Depending on the students and the divisor,
the list can be extended or reduced.
3. Suggest that one way to proceed is to use a series of “at
least/not more than” multiples of the divisor. A good strategy
is to start with easy numbers, such as 100 times the divisor or
10 times the divisor.
●
Are there at least 100 [7s] in 127? No, because 100 ∗ 7 =
700, which is more than 127.
●
Are there at least 10 [7s] in 127? Yes, because 10 ∗ 7 = 70,
which is less than 127.
●
Are there at least 20 [7s]? No, because 20 ∗ 7 = 140, which
is more than 127.
●
So there are at least 10 [7s], but not more than 20 [7s].
Try 10.
Write 10 ∗ 7, or 70, under 127. Write 10 at the right. 10 is
the first partial quotient. Partial quotients will be used to
build up the final quotient.
Students can use their list of easy multiples
to take the guesswork out of successive
multiples and focus on solving the division
problem.
Students will realize that they can work
more quickly if they begin with a more
extensive list of multiples. Math Masters,
page 438 provides an optional form for
writing multiples.
AUDITORY
414
KINESTHETIC
TACTILE
7 127
70 10 (The first partial quotient) 10 * 7 = 70
4. The next step is to find out how much is left to divide.
Subtract 70 from 127.
VISUAL
Unit 6 Division; Map Reference Frames; Measures of Angles
7 127
- 70
57
10 (The first partial quotient) 10 * 7 = 70
Subtract. 57 is left to divide.
5. Now find the number of 7s in 57. Following are two ways
to do this:
Use a fact family: 8 ∗ 7 = 56, so there are at least 8 [7s]
in 57. Record as follows:
7 127
- 70
57
56
(The first partial quotient) 10 ∗ 7 = 70
Subtract. 57 is left to divide.
8 (The second partial quotient) 8 ∗ 7 = 56
10
Use “at least / not more than” multiples with easy
numbers. For example, ask:
●
Are there at least 10 [7s] in 57? No, because 10 ∗ 7 = 70.
●
Are there at least 5 [7s]? Yes, because 5 ∗ 7 = 35.
Next, subtract 35 from 57 and continue by asking:
●
How many 7s are in 22? 3
7 127
- 70
57
- 35
22
21
(The first partial quotient) 10 ∗ 7 = 70
Subtract. 57 is left to divide.
5 (The second partial quotient) 5 ∗ 7 = 35
Subtract. 22 is left to divide.
3 (The third partial quotient) 3 ∗ 7 = 21
10
6. For both ways, the division is complete when the subtraction
leaves a number less than the divisor (7 in this example). The
final step is to add the partial quotients—the numbers of 7s
that were subtracted. 18 is the quotient. There is 1 left over.
So, 1 is the remainder.
7 127
- 70 10
57
- 56 8
1 18
7 127
- 70 10
57
- 35
5
22
- 21
3
1 18
7. Have students record the final answer in the traditional
position above the dividend. To support English language
learners, label the quotient and the remainder.
18 R1
127
7 8. Conclude by interpreting the answer: Amy is 18 weeks and
1 day older than Bob.
Step 5: Record the answer and write a number model to summarize
the number story.
Answer: 18 weeks and 1 day older than Bob
Summary number model: 127 / 7 → 18 R1
Lesson 6 3
415
Student Page
Date
Time
LESSON
Partial-Quotients Division Algorithm
63
22 23
Sample number models are given.
1.
There are 6 pencils in each pack. How
many packs can be made from 96 pencils?
Number model with unknown:
2.
Number model with unknown:
96 ÷ 6 = p
16 packs
Answer:
79 ÷ 7 = b
11 books
Answer:
0
How many pencils are left over?
pencils
2
How many dollars are left over?
Summary number model:
79 ÷ 7 ∑ 11 R2
There are 184 plants to be put into pots.
Each pot can hold 8 plants. How many pots
are needed?
4.
Number model with unknown:
A waiter distributed 1,325 drinking straws
evenly among 9 dispensers. How many
straws went in each dispenser?
Number model with unknown:
184 ÷ 8 = p
23 pots
Answer:
1325 ÷ 9 = s
Answer: 147 straws
0
How many plants are left over?
2
How many straws were left over?
plants
Summary number model:
How many [4s] are there in 92? 23
●
How many [3s] are there in 87? 29
●
How many [7s] are there in 301? 43
●
How many [8s] are there in 925? 115 R5
●
How many [12s] are there in 588? 49
●
How many [15s] are there in 556? 37 R1
straws
Summary number model:
184 ÷ 8 = 23
●
dollars
Summary number model:
96 ÷ 6 = 16
3.
Phil has $79 to purchase books. Books cost
$7 each. How many books can Phil buy?
Lead students through several more problems on the board. Ask:
How many [ns] are there in m? Each n should be a 1-digit number;
each m should be a 2- or 3-digit number. Some students may be
ready for a 2-digit divisor. Suggestions:
1325 ÷ 9 ∑ 147 R2
Have the class pose a division problem, and ask partnerships to try
to find the answer. Have volunteers share their work with the
class. Look for students who got the same results in different ways.
Ongoing Assessment: Informing Instruction
Math Journal 1, p. 144
EM3MJ1_G4_U06_137-169.indd 144
1/13/11 3:27 PM
Watch for students using multiples that are not too large and that are easy to
work with. Using such multiples may require more steps, but it will make the
work go faster. Also, students should not be concerned if they pick a multiple
that is too large. If that happens, they will quickly realize that they have a
subtraction problem involving a larger number below.
Using the Partial-Quotients
INDEPENDENT
ACTIVITY
Algorithm
(Math Journal 1, pp. 144 and 145)
Students use the partial-quotients algorithm to solve number
stories and computation problems. Have copies of Math Masters,
pages 436 and 438 readily available for students who choose to
use them. Encourage students to use the relationship between
multiplication and division to check their answers.
Student Page
Date
LESSON
63
Time
Partial-Quotients Division Algorithm
Divide.
___
5. 3 87
Answer:
Try This
7.
6.
29
1,081 ÷ 7
Answer:
Sample number
models are given.
A factory has 372 boxes of shirts to
distribute evenly among 12 stores. Each box
holds 15 shirts. How many shirts will each
store receive?
cont.
154 R3
Sample answers:
8.
There are 250 players in the league.
(Write a number greater than 100.)
There are 6 players on each team.
(Write a number between 3 and 9.)
Number model(s) with unknown:
(372 12) ∗ 15 = s
How many teams can be made?
Number model with unknown:
Answer:
465
shirts
How may shirts are left over?
0
shirts
Summary number model(s):
(372 12) ∗ 15 = 465
250 ÷ 6 = t
41 teams
Answer:
How many players are left over?
4
players
Summary number model:
Ongoing Assessment:
Recognizing Student Achievement
Use journal pages 144 and 145, Problems 1, 2, and 5 to assess students’
ability to solve division problems and number stories with 1-digit divisors and
2-digit dividends. Students are making adequate progress if they are able to
compute the answers using the partial-quotients division algorithm. Some
students may be able to solve Problems 3, 4, and 6–8, which involve
2-digit divisors or 3- or 4-digit dividends.
250 ÷ 6 ∑ 41 R4
[Operations and Computation Goal 4]
Math Journal 1, p. 145
137-169_EMCS_S_MJ1_G4_U06_576361.indd 145
416
Journal pages
144 and 145
Problems 1, 2, 5
2/15/11 5:52 PM
Unit 6 Division; Map Reference Frames; Measures of Angles
Student Page
2 Ongoing Learning & Practice
Date
Time
LESSON
Place Value in Decimals
63
Write these numbers in order from smallest
to largest.
1.
1.26
Reviewing Place Value
INDEPENDENT
ACTIVITY
in Decimals
(Math Journal 1, p. 146)
Students solve problems involving ordering of decimals, naming
place value in decimals, and reading and writing decimals.
Math Boxes 6 3
INDEPENDENT
ACTIVITY
1.09
1.091
3
6
in
in
in
in
tenths place,
ones place,
hundredths place, and
tens place.
.
6 5
largest
4
7
4.
2
6.
What is the value of the digit 4 in the
numerals below?
a.
37.48
b.
49.08
c.
0.942
d.
1.664
4 tenths
4 tens
4 hundredths
4 thousandths
I am a four-digit number less than 10.
The digit in the tenths place is the result
four and seventy-two hundredths
of dividing 36 by 4.
4.72
(Math Journal 1, p. 147)
the
the
the
the
9 4
Write each number using digits.
a.
31
Write the number.
23,467 (whole number)
and .23467 (decimal)
5.
30
A number has
6
4
5
9
0.35
smallest
Write the smallest number you can make
with the following digits:
3.
0.58
0.35
0.58
1.09
1.091
1.26
2.
The digit in the hundredths place is the
b.
nine hundred thirty-five thousandths
result of dividing 42 by 7.
0.935
The digit in the ones place is the result
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 6-1. The skill in Problem 5
previews Unit 7 content.
of dividing 72 by 8.
The digit in the thousandths place is the
result of dividing 35 by 5.
What number am I?
9
.
9 6 7
146
Writing/Reasoning Have students write a response to the
following: Explain how the exponent in Problem 4a changes
the value of 10. Sample answer: The exponent tells how many
times the base 10 is used as a factor. For example, 104 =
10 ∗ 10 ∗ 10 ∗ 10, or 10,000.
Study Link 6 3
Math Journal 1, p. 146
EM3MJ1_G4_U06_137-169.indd 146
1/13/11 2:24 PM
INDEPENDENT
ACTIVITY
(Math Masters, p. 180)
Home Connection Students practice using the partialquotients division algorithm.
Student Page
Study Link Master
Name
Date
Date
Time
6 3 Division
Sample number models are given.
22
1.
Bernardo divided a bag of 83 marbles
evenly among five friends and himself.
How many marbles did each get?
2.
Time
LESSON
STUDY LINK
Math Boxes
63
23
The carnival committee has 360 small prizes
to share equally with 5 carnival booths.
How many prizes will each booth get?
1.
Measure each line segment to the nearest millimeter.
a.
P
Number model with unknown:
Number model with unknown:
83 ÷ 6 = m
Answer:
13
marbles
Answer:
How many marbles are left over?
5
72
0
marbles
prizes
360 ÷ 5 = 72
22 R3
A
About
2.
83 ÷ 6 ∑ 13 R5
Answer:
cm
3
mm
6
cm
5
mm
B
128
How many prizes are left over?
Summary number model:
4 91
S
9
b.
prizes
Summary number model:
___
3.
About
360 ÷ 5 = p
4.
427 8
Answer:
53 R3
Round 5,906,245 to the nearest
3.
Multiply. Use a paper-and-pencil method.
3,016
a.
million.
b.
ten thousand.
c.
thousand.
d.
hundred.
6,000,000
= 58 ∗ 52
5,910,000
5,906,000
5,906,200
182 183
4.
Practice
5.
36.54
7.
43.27 - 12.67 =
= 34.96 + 1.58
30.60
6.
8.
302.578 = 300.2 + 2.378
43.545
Complete.
a.
104 =
b.
10
5
18
5.
19
1
Circle _
2 of the squares.
10,000
= 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10
2
c.
100 = 10
d.
10 to the seventh power =
10,000,000
74.6 - 31.055 =
5
44
147
Math Journal 1, p. 147
Math Masters, p. 180
EM3MJ1_G4_U06_137-169.indd 147
EM3MM_G4_U06_177-202.indd 180
1/13/11 2:24 PM
1/13/11 3:07 PM
Lesson 6 3
417
Name
Date
3 Differentiation Options
Time
A Pen Riddle
LESSON
63
䉬
Mrs. Swenson bought 2 pens for each of her 3 daughters.
◆
She gave the clerk a $10 bill.
◆
Each pen cost the same amount.
◆
Her change was all in nickels.
◆
No sales tax was charged.
◆
Her change was less than 50 cents.
READINESS
Playing Beat the Calculator
$1.60 or $1.65
1.
What was the cost of each pen?
2.
Show or explain how you got your answer.
Sample answer:
I needed an amount between $9.55 and $9.95 that can
be evenly divided by 6. Both $9.60 and $9.90 are evenly
divided by 6. $9.60 ⫼ 6 ⫽ $1.60; $9.90 ⫼ 6 ⫽ $1.65.
SMALL-GROUP
ACTIVITY
5–15 Min
(Student Reference Book, p. 233;
Math Masters, p. 461)
To provide experience with extended multiplication facts, students
play a version of Beat the Calculator in which the caller attaches
a 0 to one or both of the factors shown on the cards.
Math Masters, page 181
INDEPENDENT
ACTIVITY
ENRICHMENT
Determining the Cost of Pens
5–15 Min
(Math Masters, p. 181)
To apply students’ understanding of division, have them use clues
to find the cost of several pens. Encourage students to use money
to model the problem if necessary.
PARTNER
ACTIVITY
EXTRA PRACTICE
Playing Division Dash
5–15 Min
(Student Reference Book, p. 241; Math Masters, p. 471)
To practice dividing 2- or 3-digit dividends by 1-digit divisors,
have students play Division Dash. See Lesson 6-4 for additional
information.
Student Page
SMALL-GROUP
ACTIVITY
ELL SUPPORT
Games
Division Dash
Building Vocabulary
5–15 Min
Materials 䊐 number cards 1–9 (4 of each)
To provide language support for division, have students write
division number models on chart paper in the following ways:
䊐 1 score sheet
Players
1 or 2
Skill
Division of 2-digit by 1-digit numbers
Object of the game To reach 100 in the fewest
divisions possible.
Directions
1. Prepare a score sheet like the one shown at the right.
42 / 5 → 8 R2
2. Shuffle the cards and place the deck number-side down
on the table.
3. Each player follows the instructions below:
to right. Use the 3 cards to generate a division problem.
The 2 cards on the left form a 2-digit number. This is the
dividend. The number on the card at the right is the divisor.
the result. This result is your quotient. Remainders are
ignored. Calculate mentally or on paper.
♦ Add your quotient to your previous score and record your new
8 R2
5 42
Label and underline the divisor (the number the dividend is
being divided by) in blue.
score. (If this is your first turn, your previous score was 0.)
4. Players repeat Step 3 until one player’s score is 100 or more.
The first player to reach at least 100 wins. If there is only
one player, the object of the game is to reach 100 in as few
turns as possible.
Label and circle the quotient in a third color.
Turn 1: Bob draws 6, 4, and 5.
He divides 64 by 5. Quotient ⫽ 12.
Remainder is ignored. The score is 12 ⫹ 0 ⫽ 12.
64 is the dividend.
5 is the divisor.
Quotient
Score
12
82
7
12
94
101
Label and circle the remainder in a fourth color.
Point out that both the quotient and the remainder are part of the
answer. Display this chart throughout the division lessons.
Student Reference Book, p. 241
418
42 ÷ 5 → 8 R2
Label and underline the dividend (number being divided) in
red.
♦ Divide the 2-digit number by the 1-digit number and record
Turn 3: Bob then draws 5, 7, and 8.
He divides 57 by 8. Quotient ⫽ 7.
Remainder is ignored. The score is 7 ⫹ 94 ⫽ 101.
Bob has reached 100 in 3 turns and the game ends.
→ 8 R2
Have students do the following for each number model:
♦ Turn over 3 cards and lay them down in a row, from left
Turn 2: Bob then draws 8, 2, and 1.
He divides 82 by 1. Quotient ⫽ 82.
The score is 82 ⫹ 12 ⫽ 94.
42
_
5
Unit 6 Division; Map Reference Frames; Measures of Angles
Name
STUDY LINK
63
1.
Date
Division
Bernardo divided a bag of 83 marbles
evenly among five friends and himself.
How many marbles did each get?
22
2.
23
The carnival committee has 360 small prizes
to share equally with 5 carnival booths.
How many prizes will each booth get?
Number model with unknown:
Number model with unknown:
Answer:
Answer:
marbles
How many marbles are left over?
prizes
Summary number model:
___
4 91
Answer:
prizes
How many prizes are left over?
marbles
3.
Time
Summary number model:
4.
427 8
Answer:
Copyright © Wright Group/McGraw-Hill
Practice
= 34.96 + 1.58
5.
7.
43.27 - 12.67 =
180
= 300.2 + 2.378
6.
8.
74.6 - 31.055 =