Download materials - Everyday Math

EM07TLG1_G5_U04_LOP02.qxd 2/2/06 1:40 PM Page 236
Objective
To review the partial-quotients division algorithm
with whole numbers.
1
materials
Teaching the Lesson
Key Activities
Students review and practice the use of a friendly number paper-and-pencil division algorithm
strategy. They play Division Dash to practice mental division with 1-digit divisors.
Math Journal 1, p. 101
Student Reference Book,
pp. 22, 23, and 303
Study Link 4 1
Key Concepts and Skills
Teaching Aid Master
(Math Masters, p. 415)
Class Data Pad
• Use the partial-quotients algorithm for problems.
[Operations and Computation Goal 3]
• Apply friendly numbers to identify partial quotients.
slates
Per partnership: 4 each of the
number cards 1–9, (from the
Everything Math Deck, if available)
[Operations and Computation Goal 3]
• Factor numbers to identify partial quotients.
[Operations and Computation Goal 3]
Key Vocabulary
See Advance Preparation
dividend • divisor • partial quotient • quotient • remainder
Ongoing Assessment: Recognizing Student Achievement Use journal page 101.
[Operations and Computation Goal 3]
Ongoing Assessment: Informing Instruction See page 240.
2
materials
Ongoing Learning & Practice
Students practice and maintain skills through Math Boxes and Study Link activities.
3
materials
Differentiation Options
READINESS
Students review divisibility
rules for 1-digit divisors.
ENRICHMENT
Students find numbers to
meet divisibility criteria.
Math Journal 1, p. 102
Study Link Master
(Math Masters, p. 104)
ELL SUPPORT
Students review vocabulary
for the parts of a division
problem.
Student Reference Book, p. 11
Teaching Master
(Math Masters, p. 105)
Class Data Pad
See Advance Preparation
Additional Information
Advance Preparation For Part 1, you will need 2 copies of the computation grid
(Math Masters, page 415) for each student.
236
Unit 4 Division
Technology
Assessment Management System
Journal page 101
See the iTLG.
EM07TLG1_G5_U04_L02.qxd 2/2/06 1:43 PM Page 237
Getting Started
Mental Math and
Reflexes
Pose multiplication and division problems
like the following.
How many 5s are in 45? 9
What number times 9 equals 27? 3
What is 3 times 120? 360
How many 4s are in 32? 8
What number times 8 equals 40? 5
Multiply 5 times 80. 400
What number times 7 equals 35? 5
Multiply 12 by 7. 84
Multiply 55 by 3. 165
Math Message
Amy is 127 days older than Bob. How many weeks is that?
Study Link 4 1 Follow-Up
Have partners compare answers. Explain that fact family relationships can
be used to check computations. Write 605 67 528 on the board or a
transparency. An addition problem from this fact family will check the subtraction. Write
528 67 605. Ask: Are there any problems with this approach? Most students will
recognize either the subtraction or the addition error. It is important to calculate the check
problem, not just rewrite the numbers. 528 67 595, not 605, so the subtraction
was incorrect in the initial number sentence. Change the equal sign to not equal, and
then write 605 67 538. Encourage students to use number relationships to check
their calculations.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Ask volunteers to share their solution strategies. Expect that some
students will suggest breaking 127 into friendly numbers.
Survey the class for clues that the Math Message was a division
problem. The problem gave the whole (127 days) and asked how
many groups (weeks); because there are 7 days in a week, the
problem was to figure out how many 7s are in 127. Ask volunteers
127
to write a number model for this problem. 127 / 7; 7; 127 7;
and 71
2
7
Reviewing the Partial-
WHOLE-CLASS
ACTIVITY
Quotients Algorithm
(Math Masters, p. 415)
Given a dividend and a divisor, the partial-quotients algorithm
is one pencil-and-paper strategy for division. Model the following
steps on the Class Data Pad:
1. Write the problem in traditional form: 71
2
7
.
2. Draw a vertical line to the right of the problem to separate the
subtraction part of the algorithm from the partial quotients.
2
7
71
Lesson 4 2
237
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Explain that with this notation, students will list their partial
quotients on the right of the vertical line and then subtract the
related multiples on the left of the vertical line, until the
remaining dividend is smaller than the divisor.
Links to the Future
Students will practice the partial-quotients algorithm in Lesson 4-4, using an
easy-multiples strategy to find partial quotients, and in Lesson 4-5 with decimal
dividends.
One strategy for finding partial quotients is to use friendly
numbers. Rename the dividend as an expression that contains
multiples of the divisor. Make a name-collection box for 127, and
add the expression 70 57. Use this expression to model the
algorithm.
3. Ask: How many 7s are in 70? 10, because 10 7 70. Write
70 under 127 and 10 next to it, to the right of the vertical line.
Subtract, saying: 127 minus 70 equals 57. Explain that 10 is
the first partial quotient and 57 is what remains to be divided.
71
2
7
– 70
57
127 70 57
10
57 is left to divide.
4. Ask: How many 7s are in 57? 8, because 8 * 7 56. Write 56
under 57 and 8 next to it, to the right of the vertical line.
Subtract, saying: 57 minus 56 equals 1. Explain that 8 is the
second partial quotient, and 1 is what remains to be divided.
71
2
7
– 70
57
– 56
1
NOTE When the result of division is
expressed as a quotient and a nonzero
remainder, Everyday Mathematics uses an
arrow rather than an equal sign, as in
246 12 → 20 R6. Everyday Mathematics
prefers this notation because 246 12 20
R6 is not a proper number sentence. The
arrow is read as is, yields, or results in.
Model this expression for students in your
examples of the partial-quotients algorithm.
Label the arrow on the Class Data Pad for
display throughout this unit.
10
8
1 is left to divide.
5. Explain that they can stop this process when the number left
to be divided is smaller than the divisor. This number can be
written in the quotient as a whole-number remainder.
6. Combine the partial quotients, saying: 10 8 equals 18. Write
18 above the dividend. Circle the 1 and write R1 next to 18.
There are 18 [7s] in 127, with a remainder of 1. So Amy is
how many weeks older than Bob? About 18 weeks older, or 18
weeks and 1 day older
18 R1
71
2
7
– 70
57
– 56
1
0
10
8
18
127 7 → 18 R1
238
Unit 4 Division
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Student Page
Date
ELL
Adjusting the Activity
K I N E S T H E T I C
4 2
䉬
The Partial-Quotients Division Algorithm
夹
Use the partial-quotients algorithm to solve these problems.
The arrow as a mathematical symbol is used to represent several
different concepts. To support English language learners, remind students of the
relationship between multiplication and division. Explain that when a quotient is
written to show a whole number remainder, the remainder is not part of the
multiplication expression for that fact family. So we need a different way, the
arrow, to show that the division results in the quotient and the whole-number
remainder.
A U D I T O R Y
Time
LESSON
T A C T I L E
82 R3
55 R7
3,518 / 32 ∑ 109 R30
5,360
ᎏᎏ ∑ 99 R14
54
1. 6冄4
苶9
苶5
苶
2. 832 ⫼ 15 ∑
3.
4.
5. Jerry was sorting 389 marbles into bags. He put a
dozen in each bag. How many bags does he need?
33 bags
V I S U A L
Ask students for other ways to rename 127 using multiples of 7.
Add these to the name-collection box. Sample answers: 105 22;
70 49 7 1; 35 35 35 21 1 Have students choose
one of these expressions to use with the partial-quotients
algorithm. Remind them to write the problem and draw the
vertical line, using the problem on the Class Data Pad as a model.
To help students remember place value as they write digits, have
them use a computation grid. Circulate and assist.
101
Math Journal 1, p. 101
Using the Partial-Quotients
INDEPENDENT
ACTIVITY
Algorithm
(Math Journal 1, p. 101; Student Reference Book, pp. 22 and 23)
Remind students that pages 22 and 23 in the Student Reference
Book and the samples on the Class Data Pad can be used to verify
correct usage of the steps in this algorithm. Have students
complete the page. Circulate and assist.
Links to the Future
Problem 5 on journal page 101 will provide some information about students’
ability to interpret remainders. Interpreting remainders will be covered in
Lesson 4-6.
Student Page
Whole Numbers
Division Algorithms
Different symbols may be used to indicate division. For example,
94
“94 divided by 6” may be written as 94 6, 69
4
, 94 / 6, or 6.
Four ways to show
“123 divided by 4”
♦ The number that is being divided is called the dividend.
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 101
Use journal page 101 to assess students’ understanding of the partial-quotients
algorithm. Students are making adequate progress if they demonstrate accurate
use of the notation for the algorithm.
[Operations and Computation Goal 3]
♦ The number that divides the dividend is called the divisor.
123 4
123 / 4
41
2
3
123
4
♦ The answer to a division problem is called the quotient.
♦ Some numbers cannot be divided evenly. When this happens,
the answer includes a quotient and a remainder.
123 is the dividend.
4 is the divisor.
Partial-Quotients Method
In the partial-quotients method, it takes several steps to find
the quotient. At each step, you find a partial answer (called a
partial quotient). These partial answers are then added to
find the quotient.
Study the example below. To find the number of 6s in 1,010 first
find partial quotients and then add them. Record the partial
quotients in a column to the right of the original problem.
Example
1,010 / 6 ?
Write partial quotients in this column.
61
,0
1
0
600
↓
100
50
The second partial quotient is 50. 50 ∗ 6 300
Subtract. At least 10 [6s] are left in 110.
110
60
The first partial quotient is 100. 100 ∗ 6 600
Subtract 600 from 1,010. At least 50 [6s] are left in 410.
410
300
Think: How many [6s] are in 1,010? At least 100.
10
The third partial quotient is 10. 10 ∗ 6 60
Subtract. At least 8 [6s] are left in 50.
50
48
8
2
168
↑
↑
The fourth partial quotient is 8. 8 ∗ 6 48
Subtract. Add the partial quotients.
Remainder Quotient
168 R2
The answer is 168 R2. Record the answer as 61
,0
1
0
or write 1,010 / 6 → 168 R2.
Student Reference Book, p. 22
Lesson 4 2
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Student Page
Games
Division Dash
Materials number cards 1–9 (4 of each)
Player 1
1 score sheet
Player 2
Quotient Score
Players
1 or 2
Skill
Division of 2-digit by 1-digit numbers
Quotient
(Student Reference Book, p. 303)
Division Dash uses randomly generated numbers to obtain values
for 1-digit divisors and 2-digit dividends. Encourage students to
calculate mentally, but do not restrict paper-and-pencil use.
Directions
1. Prepare a score sheet like the one shown at the right.
2. Shuffle the cards and place the deck number-side
down on the table.
3. Each player follows the instructions below:
Discuss the example on the Student Reference Book page. Then
have the class play a round of Division Dash together. The whole
class mentally calculates the division. Remind students that only
the whole-number part of the quotient is recorded. If the dividend
is less than the divisor, the quotient should be recorded as 0.
♦ Turn over 3 cards and lay them down in a row, from
left 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.
♦ Divide the 2-digit number by the 1-digit number and
record 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 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.
6
After students understand the rules, have partners play the game.
Circulate and assist.
4
5
4
5
Turn 1: Bob draws 6, 4, and 5. He divides
64 by 5. Quotient 12. Remainder is ignored.
The score is 12 0 12.
6
Turn 2: Bob then draws 8, 2, and 1. He
WHOLE-CLASS
ACTIVITY
Score
Object of the game To reach 100 in the fewest
divisions possible.
Example
Introducing Division Dash
64 is the dividend.
divides 82 by 1. Quotient 82. The score is
82 12 94.
5 is the divisor.
Quotient
Score
Turn 3: Bob then draws 5, 7, and 8. He divides
12
12
57 by 8. Quotient 7. Remainder is ignored.
The score is 7 94 101.
82
94
7
101
Bob has reached 100 in 3 turns and the game ends.
Ongoing Assessment: Informing Instruction
Watch for students who use paper-and-pencil, rather than mental strategies, to
calculate the division. To help them bridge into mental math, ask them to write
the division expression 44
9
but then use multiplication facts and friendly parts
to calculate mentally.
Student Reference Book, p. 303
2 Ongoing Learning & Practice
Math Boxes 4 2
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 102)
Mixed Review Math Boxes in this lesson are paired with
Math Boxes in Lesson 4-4 and 4-6. The skills in Problems
5 and 6 preview Unit 5 content.
Student Page
Date
Time
LESSON
Math Boxes
4 2
䉬
1. Write or .
2. Sasha earns $4.50 per day on her paper
34
180
0.89
4
0.54
5
1
0.35
3
7
0.9
8
a. 0.45
route. She delivers papers every day. How
much does she earn in two weeks?
b.
Open sentence:
c.
d.
e.
4.50 ⴱ 7 ⴱ 2 d
Solution:
d 63
Answer:
$63
9 83
89
3. Write the prime factorization of 80.
2 ⴱ 2 ⴱ 2 ⴱ 2 ⴱ 5,11
or 24 ⴱ 5
38–40
243
4. Without using a protractor, find the
90°
135.5
c.
4.339
6.671
11.01
b.
$21.98
d.
40%
Urban 60%
Rural
The United States
in 1900
$30.49
$8.51
Circle the best answer.
25.03
14.58
39.61
34–36
102
Math Journal 1, p. 102
240
207
6.
209.0
73.5
Unit 4 Division
INDEPENDENT
ACTIVITY
(Math Masters, p. 104)
40°
12
a.
Study Link 4 2
measurement of the missing angle.
50°
5. Solve.
Writing/Reasoning Have students write a response to
the following: Explain why your answer to Problem 4 is
correct. Sample answer: The sum of the measures of the
angles equals 180°. My answer is correct because 50 90 140,
and the missing angle measure is 40 because 50 90 40 180.
A.
In 1900, more than half of the
communities were rural.
B.
In 1900, 6 out of 10 communities in
the United States were rural.
C.
In 1900, more than of communities
4
in the United States were rural.
3
125
Home Connection Students practice the partial-quotients
division algorithm.
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Study Link Master
Name
3 Differentiation Options
Date
STUDY LINK
Time
Division
42
Here is the partial-quotients algorithm using a friendly numbers strategy.
7冄苶3
2苶7
苶
30
How many 7s are in 210? 30
The first partial quotient. 30 7 ⫽ 210
Subtract. 27 is left to divide.
⫺21
3
How many 7s are in 27? 3
The second partial quotient. 3 7 ⫽ 21
Subtract. 6 is left to divide.
6
33
Add the partial quotients: 30 ⫹ 3 ⫽ 33
→
Reviewing Divisibility Rules
⫺210
27
→
PARTNER
ACTIVITY
READINESS
5–15 Min
(Student Reference Book, p. 11)
Remainder Quotient
1.
Another way to rename 237 with multiples of 7 is
If the example had used this name for 237, what would the partial quotients have been?
10, 10, 10, and 3
2.
6冄1
苶6
苶6
苶
Answer:
4.
3.
214 / 5
5.
17冄4
苶0
苶8
苶
27 R4
Answer:
485 ⫼ 15
Answer:
PARTNER
ACTIVITY
Exploring Divisibility
Answer: 33 R6
237 ⫽ 70 ⫹ 70 ⫹ 70 ⫹ 21 ⫹ 6
To provide experience with identifying factors, have partners read
about divisibility on page 11 of the Student Reference Book and
complete the Check Your Understanding problems.
ENRICHMENT
22 23
Rename dividend (use multiples of the divisor):
237 ⫽ 210 ⫹ 21 ⫹ 6
32 R5
42 R4
24
Answer:
Practice
3,985
3,985 ⫺ 168; or 3,817 ⫽ 3,817; or 168
7. 52,517 ⫺ 281 ⫽ 52,236
Check: 281; or 52,236 ⫹ 52,236; or 281 ⫽ 52,517
6.
3,817 ⫹ 168 ⫽
Check:
5–15 Min
by the Digits
(Math Masters, p. 105)
To apply students’ understanding of factors, have them
explore divisibility from another perspective. Students
examine 3-digit numbers that meet certain divisibility
criteria. Then they use the same criteria to identify larger numbers.
Math Masters, p. 104
SMALL-GROUP
ACTIVITY
ELL SUPPORT
Supporting Math Vocabulary
15–30 Min
Development
To provide language support for division, have volunteers write a
division number model on chart paper in several different formats.
127 / 7 → 18 R1
127
7
Teaching Master
→ 18 R1
Name
127 7 → 18 R1
LESSON
42
Time
Divisibility by the Digits
Ms. Winters asked Vito and Jacob to make answer cards for a division puzzle.
They had to find numbers that met all of the following characteristics.
18 R1
2
7
71
For each number model, label and
underline the dividend in red (the
number being divided); label and
underline the divisor in blue (the
number the dividend is being
divided by); label and circle the
quotient in a third color; label and
circle the remainder in a fourth
color. Emphasize that both the
quotient and the remainder are
part of the answer. Display this
chart throughout all the division
lessons.
Date
Example:
A Division Problem
Dividend
◆ The first digit is divisible by 1.
1
◆ The first two digits are divisible by 2.
12
◆ The first three digits are divisible by 3.
120
◆ The first four digits are divisible by 4.
1,204
◆ The first five digits are divisible by 5.
12,040
◆ The first six digits are divisible by 6.
120,402
◆ The first seven digits are divisible by 7. 1,204,021
59 ⴜ 7 ⴝ 8 R3
1.
Quotient
◆ The first eight digits are divisible by 8.
12,049,216
◆ The first nine digits are divisible by 9.
120,402,162
Jacob knew that with divisibility rules, it should be easy. The boys started with 3-digit
numbers and found 123 and 242. Latoya checked their work. What should she tell them?
123 is correct because 1 is divisible by 1; 12
is correct because it is an even number; and
123 is correct because 1 ⫹ 2 ⫹ 3 ⫽ 6, which is
divisible by 3.
242 is not correct because 2 ⫹ 4 ⫹ 2 ⫽ 8,
which is not divisible by 3.
8 R3
7 59
Divisor
2.
Use the characteristics listed above to find as many puzzle numbers as you can. Record
them in the boxes below.
Sample answers:
59 / 7 ⴝ 8 R3
Puzzle Numbers
4-digit
Remainder
5-digit
6-digit
7-digit
8-digit
9-digit
1,472 14,725 147,252 1,472,527 14,725,272 147,252,726
1,624 16,240 162,408 1,624,084 16,240,840 162,408,402
Math Masters, p. 105
Lesson 4 2
241
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Objective
To provide practice with strategies for the
partial-quotients algorithm.
1
materials
Teaching the Lesson
Key Activities
Students play Divisibility Dash to practice recognizing multiples and using divisibility rules.
They practice finding partial quotients by using easy multiples of the divisor.
Math Journal 1, pp. 106 and 107
Student Reference Book,
pp. 22, 23, and 302
Study Link 4 3
Key Concepts and Skills
• Apply division facts and extended facts to identify partial quotients.
[Operations and Computation Goal 2]
• Use divisibility rules to identify multiples.
[Operations and Computation Goal 3]
• Use the partial-quotients algorithm for problems.
[Operations and Computation Goal 3]
Key Vocabulary
multiple • divisor • partial quotient • dividend
Ongoing Assessment: Informing Instruction See page 251.
Ongoing Assessment: Recognizing Student Achievement Use journal page 107.
Teaching Master
(Math Masters, p. 109; optional)
Teaching Aid Master
(Math Masters, p. 415)
Class Data Pad
calculator
Per partnership: 4 each of number
cards 0–9; 2 each of number cards
2, 3, 5, 6, 9 and 10 (from the
Everything Math Deck, if available)
See Advance Preparation
[Operations and Computation Goal 3]
2
Ongoing Learning & Practice
Students practice and maintain skills through Math Boxes and Study Link activities.
3
Students practice finding friendly numbers
using expanded notation and multiples.
Math Journal 1, p. 108
Study Link Master
(Math Masters, p. 110)
materials
Differentiation Options
READINESS
materials
EXTRA PRACTICE
Students use lists of multiples of the divisor
to solve division problems.
Teaching Masters
(Math Masters, pp. 111 and 112)
Per partnership: 4 each of number
cards 1–9 (from the Everything Math
Deck, if available)
See Advance Preparation
Additional Information
Advance Preparation For Part 1, you will need a coin for the calculator practice in the Mental
Math and Reflexes and 2 copies of the computation grid (Math Masters, page 415) for each
student.
For Part 3, prepare Math Masters, page 111 to provide individualized practice as needed.
248
Unit 4 Division
Technology
Assessment Management System
Journal page 107
See the iTLG.
EM07TLG1_G5_U04_L04.qxd 2/2/06 1:44 PM Page 249
Getting Started
Mental Math and Reflexes
Math Message
For calculator practice, write each problem on the board or a transparency.
Use a coin toss to determine whether students express the answer with a
whole-number remainder or a fraction remainder.
Write a 3-digit number that
is divisible by 6.
1
3
6
7 R1; 75
53
1
11
1
0
2 9 R3; 9 9 or 9 3
3
25
2
3
0 9 R5; 9
25 or 9 5
6
7
5 12 R3; 12 6 or 122
3
6
11 4 2 R3; 24
2
78 8 9 R6; 98 or 94
1
34 / 8 4 R2; 48 or 44
3
99 / 8 12 R3; 128
3
1
5
1
Study Link 4 3
Follow-Up
3
Allow students five minutes to compare
their answers and resolve any
differences. Circulate and assist.
680 / 50 13 R30;
30
135
0 or 13 5
Ask volunteers to explain the meaning of the fraction remainder. The divisor represents
how many are needed in a group or how many groups. The divisor is the denominator.
The remainder is the numerator; how many you have. The fraction represents
1
division— the remainder, 5, is one divided by 5.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Survey the class for their 3-digit numbers. Write student
responses on the Class Data Pad. Ask students how they might
check these numbers without actually dividing by 6. Most
students will refer to the divisibility rule for 6: A number is
divisible by 6 if it is divisible by 2 and 3. Check the numbers as a
class, and discuss students’ strategies for finding their numbers.
Introducing Divisibility Dash
WHOLE-CLASS
ACTIVITY
(Student Reference Book, p. 302)
Student Page
Games
Divisibility Dash
Materials number cards 0–9 (4 of each)
Playing Divisibility Dash provides students with practice
recognizing multiples and using divisibility rules in a context
that also develops speed. The variation is for 3-digit numbers.
Discuss the variation example on Student Reference Book, page
302, and demonstrate a turn by playing one hand as a class. Then
allow partners time to play at least 3 rounds of Divisibility Dash.
2 or 3
Skill
Recognizing multiples, using divisibility tests
The number cards 0–9
(4 of each) are the
draw cards. This set
of draw cards is also
called the draw pile.
Object of the game To discard all cards.
Directions
1. Shuffle the divisor cards and place them number-side down
on the table. Shuffle the draw cards and deal 8 to each
player. Place the remaining draw cards number-side down
next to the divisor cards.
The number cards
2, 3, 5, 6, 9, and 10
(2 of each) are the
divisor cards.
2. For each round, turn the top divisor card number-side up.
Players take turns. When it is your turn:
♦ Use the cards in your hand to make 2-digit numbers that
are multiples of the divisor card. Make as many 2-digit
numbers that are multiples as you can. A card used to
make one 2-digit number may not be used again to make
another number.
♦ Place all the cards you used to make 2-digit numbers in a
4. If the draw pile or divisor cards have all been used, they can
be reshuffled and put back into play.
5. The first player to discard all of his or her cards is the winner.
5
5
1
Andrew uses his cards to make
2 numbers that are multiples of 3:
7
1
5
8
Divisor card:
5
5
2
3
7
5
1
7
Andrew’s cards:
2
Example
5
1
Explain that another approach to finding partial quotients 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.
3. If a player disagrees that a 2-digit number is a multiple of
the divisor card, that player may challenge. Players use the
divisibility test for the divisor card value to check the
number in question. Any numbers that are not multiples of
the divisor card must be returned to the player’s hand.
5
(Math Masters, p. 415)
the divisor card, you must take a card from the draw pile.
Your turn is over.
7
Quotients Algorithm
discard pile.
♦ If you cannot make a 2-digit number that is a multiple of
8
WHOLE-CLASS
ACTIVITY
3
Reviewing the Partial-
Note
number cards: 2, 3, 5, 6, 9, and 10 (2 of each)
Players
He discards these 4 cards and holds the 2 and 8 for the next round of play.
Student Reference Book, p. 302
Lesson 4 4
249
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Student Page
1. Write the problem 61
,0
1
0
, drawing a vertical line to the right
of the problem.
Whole Numbers
Division Algorithms
Different symbols may be used to indicate division. For example,
94
“94 divided by 6” may be written as 94 6, 69
4
, 94 / 6, or 6.
Four ways to show
“123 divided by 4”
♦ The number that is being divided is called the dividend.
♦ The number that divides the dividend is called the divisor.
123 4
123 / 4
41
2
3
123
4
♦ The answer to a division problem is called the quotient.
♦ Some numbers cannot be divided evenly. When this happens,
the answer includes a quotient and a remainder.
Study the example below. To find the number of 6s in 1,010 first
find partial quotients and then add them. Record the partial
quotients in a column to the right of the original problem.
Write partial quotients in this column.
↓
100
Think: How many [6s] are in 1,010? At least 100.
The first partial quotient is 100. 100 ∗ 6 600
Subtract 600 from 1,010. At least 50 [6s] are left in 410.
410
300
50
The second partial quotient is 50. 50 ∗ 6 300
Subtract. At least 10 [6s] are left in 110.
110
60
10
●
So there are at least 100 [6s] but not more than 200 [6s].
Try 100.
Write 600 under 1,010. Write 100 to the right. 100 is the
first partial quotient.
1,010 / 6 ?
61
,0
1
0
600
Are there at least 100 [6s] in 1,010? Yes, because 100 6 600, which is less than 1,010. Are there at least 200 [6s] in
1,010? No, because 200 6 1,200, which is more than
1,010.
123 is the dividend.
4 is the divisor.
Partial-Quotients Method
In the partial-quotients method, it takes several steps to find
the quotient. At each step, you find a partial answer (called a
partial quotient). These partial answers are then added to
find the quotient.
Example
●
61
,0
1
0
– 600
100
The third partial quotient is 10. 10 ∗ 6 60
Subtract. At least 8 [6s] are left in 50.
50
48
8
2
168
↑
↑
Remainder
The first partial quotient,
100 6 600.
The fourth partial quotient is 8. 8 ∗ 6 48
2. Next find out how much is left to be divided. Subtract 600
from 1,010.
Subtract. Add the partial quotients.
Quotient
168 R2
The answer is 168 R2. Record the answer as 61
,0
1
0
or write 1,010 / 6 → 168 R2.
61
,0
1
0
– 600
Student Reference Book, p. 22
100
410
The first partial quotient,
100 6 600.
410 is left to divide.
3. Now find the number of 6s in 410. There are several ways to
do this:
Use a fact family and extended facts. 6 6 36;
60 6 360, so there are at least 60 [6s] in 410.
Teaching Master
Name
Date
LESSON
44
䉬
Time
61
,0
1
0
– 600
100
410
– 360
60
50
– 48
8
Easy Multiples
2
⫽
1,000 º
⫽
100 º
⫽
100 º
⫽
50 º
⫽
50 º
⫽
20 º
⫽
20 º
⫽
10 º
⫽
10 º
⫽
5 º
⫽
5º
⫽
1,000 º
⫽
1,000 º
⫽
100 º
⫽
100 º
⫽
50 º
⫽
50 º
⫽
20 º
⫽
20 º
⫽
10 º
⫽
10 º
⫽
5 º
⫽
5º
⫽
1,000 º
⫽
1,000 º
⫽
100 º
⫽
100 º
⫽
50 º
⫽
50 º
⫽
20 º
⫽
20 º
⫽
10 º
⫽
10 º
⫽
5 º
⫽
5º
⫽
1,000 º
Math Masters, p. 109
250
Unit 4 Division
The first partial quotient,
100 6 600.
410 is left to divide.
The second partial quotient,
60 6 360.
50 is left to divide.
The third partial quotient,
8 6 48.
2 is left to divide.
Or continue to use at least...not more than multiples with
easy numbers. For example, ask: Are there at least 100 [6s]
in 410? No, because 100 6 600. Are there at least 50
[6s]? Yes, because 50 6 300.
61
,0
1
0
– 600
100
410
– 300
50
110
The first partial quotient,
100 6 600.
410 is left to divide.
The second partial quotient,
50 6 300.
110 is left to divide.
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Subtract 300 from 410, and continue by asking: Are there
10 [6s] in 110? Yes, because 10 6 60. Are there 20 [6s]
in 110? No, because 20 6 120.
61
,0
1
0
– 600
100
410
– 300
50
110
– 60
10
50
– 48
8
2
The first partial quotient,
100 6 600.
410 is left to divide.
The second partial quotient,
50 6 300.
110 is left to divide.
The third partial quotient,
10 6 60.
50 is left to divide.
The fourth partial quotient,
8 6 48.
2 is left to divide.
4. When the subtraction leaves a number less than the divisor
(2 in this example), students should move to the final step and
add the partial quotients.
168 R2
61
,0
1
0
– 600
100
410
– 360
60
50
– 48
8
2
168
168 R2
61
,0
1
0
– 600
100
410
– 300
50
110
– 60
10
50
– 48
8
2
168
1,010 6 → 168 R2
Ongoing Assessment: Informing Instruction
Watch for students who use only multiples of 10. Encourage them to look for
larger multiples of the divisor, as appropriate. Suggest they first compile a list of
easy multiples of the divisor.
Student Page
Date
䉬
One way:
100 6 600
10 6 60
5 6 30
Remind students that listing the easy multiples in advance allows them to focus
on solving the division problem, rather than looking for multiples. Math Masters,
page 109 provides an optional form for writing multiples.
Use cards (1 through 9, 4 of each) to generate random 3- or 4-digit
dividends and 1- or 2-digit divisors for the class. Ask partners to
use the partial-quotients algorithm to solve these problems.
Circulate and assist.
Another way:
8冄1
苶8
苶5
苶
⫺80
105
⫺80
25
⫺24
1
200 6 1,200
20 6 120
The Partial-Quotients Algorithm
4 4
Example: 185 / 8 ∑ ?
Example: If the divisor is 6, students might make the following list:
50 6 300
Time
LESSON
10
10
8冄1
苶8
苶5
苶
⫺ 160
25
⫺24
1
Another way:
8冄1
苶8
苶5
苶
Rename 185 using
multiples of 8:
160 ⫹ 24 ⫹ 1
Think: 160 ⫽ 20 [8s]
24 ⫽ 3 [8s]
20 ⫹ 3 ⫽ 23 [8s] with
1 left over
20
3
23
3
23
The answer, 23 R1, is the same for each way.
Use the partial-quotients algorithm to solve these problems.
1. 64 ⫼ 8 ⫽
8
3. 2,628 ⫼ 36 ⫽
2. 749 / 7 ⫽
73
107
910
4. 8,190 / 9 ⫽
5. Raoul has 237 string bean seeds. He plants them in rows with 8 seeds in each row.
How many complete rows can he plant?
Estimate:
Solution:
8 ⴱ 30 ⫽ 240, or 240 ⫼ 8 ⫽ 30
29 rows
106
Math Journal 1, p. 106
Lesson 4 4
251
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Student Page
Date
After students have worked for a few minutes, look for
partnerships with solutions that have different partial-quotients
lists, and ask them to share their solutions with the class.
Emphasize the following:
Time
LESSON
The Partial-Quotients Algorithm
4 4
continued
Divide.
274 R1
65 R20
4,290 / 64 ➝ 67 R2
823 / 3 ➝
6.
2,815 ⫼ 43 ➝
7.
8.
Students should use the multiples that are easy for them. This
might sometimes require more steps, but it will make the work
go faster.
Regina put 1,610 math books into boxes.
Each box held 24 books. How many boxes did she use?
9.
Estimate:
Solution:
1,600 / 25 ⫽ 64, or 24 70 ⫽ 1,680
68 boxes
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 with a larger number being
subtracted from a smaller number. Students can use this
information to revise the multiple they used.
10. Make up a number story that can be solved with division.
Solve it using a division algorithm.
Answers vary.
Solution:
Answers vary.
Using the Partial-Quotients
INDEPENDENT
ACTIVITY
Algorithm
(Math Journal 1, pp. 106 and 107; Student Reference Book, pp. 22 and 23)
107
Math Journal 1, p. 107
Have students solve the problems on the journal pages, showing
their work on the computation grids. Encourage students to use
the Student Reference Book as needed. Circulate and assist.
NOTE Students will continue to practice the partial-quotients algorithm throughout
this unit and in Math Boxes and Ongoing Learning & Practice activities throughout
the year.
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 107
Problem 10
Use journal page 107, Problem 10 to assess students’ understanding of
division. Students are making adequate progress if they have written a number
story that can be solved using division.
[Operations and Computation Goal 3]
Student Page
Date
Time
LESSON
Math Boxes
44
1. Write ⬍ or ⬎.
2. Jamie bikes 18.5 mi per day. How
many miles will she ride in 13 days?
⬍ 0.70
1
ᎏᎏ ⬎ 0.21
4
3
0.38 ⬎ ᎏᎏ
10
2
0.6 ⬍ ᎏᎏ
3
90
0.95 ⬎ ᎏᎏ
100
3
a. ᎏᎏ
5
b.
c.
d.
e.
18.5 13 ⫽ m
m ⫽ 240.5
240.5 mi
Open sentence:
Solution:
Answer:
38–40
243
9 83
89
3. Write the prime factorization of 132.
Math Boxes 4 4
4. Without using a protractor, find the
2 2 3 11, or
22 3 11
2 Ongoing Learning & Practice
measurement of the missing angle.
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 108)
79°
120°
59
102°
°
207
12
5. Solve.
6. Fill in the circle next to the best answer.
a. 2.03 ⫺ 0.76 ⫽
b.
c.
61
1,198.49
d. 29.05 ⫹ 103.94 ⫽
1.27
Favorite 5th Grade Colors
⫽ 57.97 ⫹ 3.03
⫽ 691.23 ⫹ 507.26
132.99
blue
red
yellow green
1
A. More than ᎏᎏ of the students
2
chose blue.
B. 50% of the students chose
yellow or green.
C. More than 25% of the
34–36
108
Math Journal 1, p. 108
252
Unit 4 Division
students chose yellow
or red.
125
Mixed Review Math Boxes in this lesson are paired with
Math Boxes in Lessons 4-2 and 4-6. The skills in
Problems 5 and 6 preview Unit 5 content.
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7:20 PM
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Study Link Master
Study Link 4 4
INDEPENDENT
ACTIVITY
(Math Masters, p. 110)
Name
Date
STUDY LINK
Time
Division
44
䉬
Here is an example of the partial-quotients algorithm using an “at least...not
more than” strategy.
苶8
苶5
苶
8冄1
10
How many 8s are in 185? At least 10.
The first partial quotient. 10 º 8 80
Subtract. 105 is left to divide.
80
25
10
How many 8s are in 105? At least 10.
The second partial quotient. 10 º 8 80
Subtract. 25 is left to divide.
24
3
1
23
→
3 Differentiation Options
Add the partial quotients: 10 10 3 23
Remainder Quotient Answer: 23 R1
639 9
71
Answer:
PARTNER
ACTIVITY
Using Expanded Notation
How many 8s are in 25? At least 3.
The third partial quotient. 3 º 8 24
Subtract. 1 is left to divide.
Solve.
1.
READINESS
Begin estimating with multiples of 10.
80
105
→
Home Connection Students practice the partial-quotients
division algorithm.
22 23
3.
15–30 Min
954 18
4.
972 / 37
53
Answer:
1,990 / 24
Answer:
5.
2.
82 R22
Answer:
26 R10
Robert is making a photo album. 6 photos fit on a
page. How many pages will he need for 497 photos?
83
pages
Practice
to Find Multiples
6.
Check:
(Math Masters, p. 112)
7.
To explore using extended facts, have students write numbers in
expanded notation. Students then complete Math Masters, page
112 by using the expanded notation to find equivalent names.
2,814
2,814 2,746; 68
2,746 68 68; 2,746
3,296
Check: 165; 3,296 3,296; 165 3,461 165 3,461
Math Masters, p. 110
INDEPENDENT
ACTIVITY
EXTRA PRACTICE
Practicing Division
5–15 Min
(Math Masters, p. 111)
Use Math Masters, page 111 to create division problems
for individualized extra practice. Encourage students to
use multiplication to check their problems. Alternately,
have students create problems for partners to solve.
Teaching Master
Name
Date
LESSON
䉬
44
䉬
For each division problem, complete the list of multiples of the divisor.
Then divide.
冄2
苶3
苶4
苶5
苶6
苶6
苶
1.
Name
Date
LESSON
Division Practice
44
Teaching Master
Time
Answer:
Using Expanded Notation
◆ Work with a partner. Use a deck with 4 each of cards 1–9.
◆ Take turns dealing 4 cards and forming a 4-digit number.
◆ Write the number in standard notation and expanded notation.
2.
◆ Then write equivalent names for the value of each digit.
Answer:
Sample answers:
1,234
200 º
200 º
1.
Write a 4-digit number.
100 º
100 º
2.
Write the number in expanded notation.
50 º
50 º
20 º
20 º
10 º
10 º
5º
/
3.
200 º
1,000
200
30
4
Write equivalent names for the value of each digit.
1st digit
2nd digit
3rd digit
4th digit
2 º 500
10 º 100
600 400
2 º 100
50 º 4
8 º 25
3 º 10
15 º 2
6º5
2º2
3 1
4.
Answer:
3.
5º
Time
Answer:
200 º
100 º
100 º
50 º
50 º
20 º
20 º
10 º
10 º
5º
5º
Math Masters, p. 111
4.
Write a 4-digit number.
5.
Write the number in expanded notation.
6.
Write equivalent names for the value of each digit.
1st digit
2nd digit
3rd digit
4th digit
Math Masters, p. 112
Lesson 4 4
253