Download Subtraction of Positive and Negative Numbers

Subtraction of Positive
and Negative Numbers
Objective To develop a rule for subtracting positive
and negative numbers.
a
www.everydaymathonline.com
eToolkit
ePresentations
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Model differences of positive and negative
numbers with manipulatives. Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
[Operations and Computation Goal 1]
• Write and solve the equivalent addition
number model for signed number
subtraction problems. [Operations and Computation Goal 1]
• Use signed number subtraction patterns
to describe a rule for subtracting
signed numbers. [Patterns, Functions, and Algebra Goal 1]
• Write number sentences that model
signed-number addition and subtraction
problems. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Students use their
and
counters to
explore and describe a rule for subtracting
positive and negative numbers. They
practice applying the rule.
Materials
Playing High-Number Toss:
Decimal Version
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Comparing Elevations
Student Reference Book, p. 321
Math Masters, p. 511
per partnership: 4 each of number
cards 0–9 (from the Everything Math
Deck, if available)
Students compare decimals and
practice writing decimals in
scientific notation.
Student Reference Book, p. 381
Math Masters, p. 210
Students use addition and subtraction to
compare elevations of various places in
the world.
Ongoing Assessment:
Recognizing Student Achievement
5-Minute Math™, pp. 153 and 234
Students find a rule for in/out patterns
with negative numbers.
Use High-Number Toss Record Sheet. EXTRA PRACTICE
5-Minute Math
[Number and Numeration Goals 1 and 6]
Math Boxes 7 9
Math Journal 2, p. 241
Students practice and maintain skills
through Math Box problems.
Study Link 7 9
Math Masters, p. 209
Students practice and maintain skills
through Study Link activities.
Math Journal 2, pp. 237–240
Study Link 78
and
counters from Lesson 78 slate
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 100–102
584
Unit 7
Exponents and Negative Numbers
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Getting Started
Mental Math
and Reflexes
Math Message
Add positive and negative numbers.
Students may use
and
counters.
Suggestions:
Study Link 7 8
Follow-Up
Use your
and
cash
cards to help you complete page 237 in
your journal.
Partners compare answers and resolve
differences.
9 + (-3) 6
-32 + 17 -15
-18 + (-18) -36
1 Teaching the Lesson
WHOLE-CLASS
DISCUSSION
▶ Math Message Follow-Up
(Math Journal 2, p. 237)
Algebraic Thinking Ask students to show how they used their
and
counters to obtain their answers.
Use Problem 3 to remind students that if the same number of
and
counters are added to a balance, the balance will remain
the same. For Problem 5, 4
and 4
counters must be added
to the balance before the required counters can be taken away.
Then use change diagrams and number models to summarize
Problems 1–5.
Problem 1:
Change
Start
Student Page
End
Date
5
+5
?
LESSON
79
䉬
Time
Finding Balances
Math Message
Number model: 5 + (-5) = 0
Use your ⫹ and ⫺ cash card counters to model the following problems. Draw a
picture of the ⫹ and ⫺ counters to show how you found each balance.
Example:
Problem 2:
⫺ ⫺ ⫺
You have 3 ⫺ counters. Add 6 ⫹ counters.
Change
Start
5
1.
End
+7
⫹ ⫹ ⫹ ⫹ ⫹ ⫹
Balance ⫽ 3 ⫹ counters
You have 5 ⫹ counters. Add 5 ⫺ counters.
Balance ⫽
0
2.
3.
Number model: 5 + (-7) = -2
2 ⫺
counters
⫹ ⫹ ⫹ ⫹ ⫹
⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺
⫹ ⫹ ⫹ ⫹ ⫹
⫺ ⫺ ⫺ ⫺ ⫺
?
You have 5 ⫹ counters. Add 7 ⫺ counters.
Balance ⫽
counters
Show a balance of ⫺7 using 15 of your ⫹ and ⫺ counters.
⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺
⫹ ⫹ ⫹ ⫹
4.
You have 7 ⫺ counters. Take away
4 ⫺ counters.
Balance ⫽
3 ⫺
counters
⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺
5.
You have 7 ⫹ counters. Take away
4 ⫺ counters.
Balance ⫽
11 ⫹
counters
⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹
⫹ ⫹ ⫹ ⫹
⫺ ⫺ ⫺ ⫺
Math Journal 2, p. 237
Lesson 7 9
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Student Page
Date
Time
LESSON
79
䉬
Problem 3:
Adding and Subtracting Numbers
Change
You and your partner combine your ⫹ and ⫺ counters. Use the counters
to help you solve the problems.
1.
13 ⫹
⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
2.
5⫺
13 ⫹
⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫺ ⫺
⫺
⫺ ⫺
⫹
8
Start
11
⫺ ⫺
⫺
⫺ ⫺
8
⫹
Balance ⫽
If 4 ⫹ counters are added to
the container, what is the
new balance?
New balance ⫽
New balance ⫽
3.
15 ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
12 ⫹
8 ⫺ (⫺4) ⫽ 12
Number model:
4.
8⫹
15 ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫺
7
Change
⫹
8 ⫹ 4 ⫽ 12
12
Start
7
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
End
-4
?
Number model: (-7) - (-4) = -3
⫺
Balance ⫽
If 3 ⫹ counters are subtracted
from the container, what is the
new balance?
Balance ⫽
If 3 ⫺ counters are added to
the container, what is the
new balance?
New balance ⫽
New balance ⫽
10 ⫺
Number model: ⫺7 ⫺ 3 ⫽ ⫺10
?
Problem 4:
8⫹
7
+4
Number model: (-11) + 4 = -7
Balance ⫽
If 4 ⫺ counters are subtracted
from the container, what is the
new balance?
Number model:
End
5⫺
Problem 5:
Change
10 ⫺
Number model: ⫺7 ⫹ (⫺3) ⫽ ⫺10
Start
7
Math Journal 2, p. 238
End
-4
?
Number model: 7 - (-4) = 11
▶ Developing a Rule for Subtracting
PARTNER
ACTIVITY
Positive and Negative Numbers
(Math Journal 2, pp. 238 and 239)
Ask partners to pool their counters so they have 20 positive and
20 negative counters to use as they work through problems. Have
them complete problems 1–8 on journal pages 238 and 239.
When most students have finished, bring the class together to go
over the answers. For each problem, ask volunteers to draw a
change diagram and write a number model on the board. The
number models should be displayed in pairs, as follows:
Student Page
Date
Time
LESSON
79
䉬
5.
Adding and Subtracting Numbers
12 ⫹
7⫺
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫺
6.
⫺
⫺ ⫺
⫺
⫺ ⫺
5 ⫹
7⫺
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫺
continued
8 + 4 = 12
5 ⫹
Balance ⫽
If 6 ⫹ counters are added to
the container, what is the
new balance?
New balance ⫽
New balance ⫽
11 ⫹
5 ⫺ (⫺6) ⫽ 11
7.
10 ⫹
⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
Number model:
8.
16 ⫺
⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
10 ⫹
⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫹
⫹ ⫹
⫺
11 ⫹
5 ⫹ 6 ⫽ 11
⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
⫺ ⫺
⫺
Balance ⫽ 6
If 2 ⫹ counters are added to
the container, what is the
new balance?
New balance ⫽
New balance ⫽
4 ⫺
⫺6 ⫺ (⫺2) ⫽ ⫺4
Number model:
Problems 5 and 6:
5 - (-6) = 11
5 + 6 = 11
16 ⫺
Balance ⫽ 6
If 2 ⫺ counters are subtracted
from the container, what is the
new balance?
Number model:
Problems 1 and 2:
8 - (-4) = 12
⫺
⫺ ⫺
⫺
⫺ ⫺
Balance ⫽
If 6 ⫺ counters are subtracted
from the container, what is the
new balance?
Number model:
9.
12 ⫹
4 ⫺
⫺6 ⫹ 2 ⫽ ⫺4
Problems 3 and 4:
-7 - 3 = -10
-7 + (-3) = -10
Problems 7 and 8:
-6 - (-2) = -4
-6 + 2 = -4
Ask the class to look for similarities and differences between the
problems and number models for each of these pairs. Ask students
to write about what they notice. As you discuss their written
responses, include the following:
Similarities
Write a rule for subtracting positive and negative numbers.
Sample answer: To subtract a positive or negative
number b from number a, add the opposite of b to a.
Math Journal 2, p. 239
586
Unit 7
Where containers start with the same combination of
and
counters, the starting balances are the same, and the first
numbers in the number models are the same.
Exponents and Negative Numbers
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Where the same number of counters is added to or taken out of
the containers, the second numbers in the number models are
the same.
Where the new balances after the transactions are the same,
the results of the number model operations are the same.
Differences
Where the transactions for a pair of counters are opposites of
each other, the operations in the number models are opposites
of each other. One is subtraction, and the other is addition.
Where the counters that are subtracted or added have opposite
signs, the second numbers in the number models are opposites.
Refer students to Problem 1. Subtracting -4 from 8 has the same
effect as adding 4 to 8. In Problem 3, subtracting 3 has the same
effect as adding -3.
Help students describe a rule for subtracting positive and negative
numbers. Ask them to record the rule in their own words in
Problem 9 on journal page 239.
Have students use slates to practice subtracting positive and
negative numbers. Ask students to write the equivalent addition
number model. Suggestions:
●
6 minus 9 6 + (-9) = -3
●
6 minus -9 6 + 9 = 15
●
-6 minus 9 -6 + (-9) = -15
●
-6 minus -9 -6 + 9 = 3
Adjusting the Activity
Have students use larger numbers. Remind students of their rule for
subtracting positive and negative numbers. Suggestions:
• 52 minus (-25) 77
A U D I T O R Y
• -47 minus 22 -69
K I N E S T H E T I C
• -36 minus (-24) -12
T A C T I L E
V I S U A L
Student Page
Date
Time
LESSON
79
䉬
Subtraction Problems
Rewrite each subtraction problem as an addition problem. Then solve it.
▶ Subtracting Positive and
PARTNER
ACTIVITY
Negative Numbers
(Math Journal 2, p. 240)
Ask students to complete journal page 240 independently, then
compare answers with a partner and resolve differences. Circulate
and assist.
100 + (Ç45)
⫽
55
⫽
⫺145
160 ⫹ 80
⫽
240
9⫹2
⫽
11
⫺15 ⫺ (⫺30) ⫽
⫺15 ⫹ 30
⫽
15
6.
8 ⫺ 10 ⫽
8 ⫹ (⫺10)
⫽
⫺2
7.
⫺20 ⫺ (⫺7) ⫽
⫺20 ⫹ 7
⫽
⫺13
8.
0 ⫺ (⫺6.1) ⫽
0 ⫹ 6.1
⫽
6.1
9.
The Healthy Delights Candy Company specializes in candy that is wholesome.
Unfortunately, they have been losing money for several years. During the year
2006, they lost $12 million, ending the year with a total debt of $23 million.
1.
100 ⫺ 45 ⫽
2.
⫺100 ⫺ 45 ⫽
3.
160 ⫺ (⫺80) ⫽
4.
9 ⫺ (⫺2) ⫽
5.
10.
⫺100 ⫹ (⫺45)
a.
What was Healthy Delights’ total debt at the beginning of 2006?
b.
Write a number model that fits this problem.
$11 million
⫺11 ⫺ 12 ⫽ ⫺23, or
⫺11 ⫹ (⫺12) ⫽ ⫺23
In 2007, Healthy Delights is expecting to lose $8 million.
a.
What will Healthy Delights’ total debt be at the end of 2007?
b.
Write a number model that fits this problem.
$31 million
⫺23 ⫺ 8 ⫽ ⫺31, or
⫺23 ⫹ (⫺8) ⫽ ⫺31
Math Journal 2, p. 240
Lesson 7 9
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Student Page
Date
Time
LESSON
2 Ongoing Learning & Practice
Math Boxes
79
Make the following changes to the numeral
29,078.
1.
Solve.
2.
302 – m = 198
Change the digit
in the ones place to 4,
in the ten-thousands place to 6,
in the hundreds place to 2,
in the tens place to 9,
in the thousands place to 7.
Write the new numeral.
m=
104
I used the fact family to find
the missing number. I wrote
302 – 198 to get m.
6 7, 2 9 4
Standard
Notation
Scientific
Notation
3 ∗ 102
300
3,000
3 ∗ 103
4,000
4 ∗ 103
5 ∗ 102
500
7 ∗ 103
7,000
7,000
a.
(48 ÷ 6)+(2 ∗ 4) = 16
b.
(48 ÷(6 + 2))∗ 4 = 24
c.
45 = (54 –(24 6))– 5
d.
0 =((54 – 24) 6)– 5
e.
30 =(54 – 24)(6 – 5 )
High-Number Toss was introduced in Lesson 2-5. Have students
play the decimal version of the game as indicated on Student
Reference Book, page 321 and record their rounds on the Record
Sheet (Math Masters, page 511).
222 223
8
When Antoinette woke up on New Year’s
Day, it was –4°F outside. By the time the
parade started, it was 18°F. How many
degrees had the temperature risen by the
time the parade began?
5.
(Student Reference Book, p. 321; Math Masters, p. 511)
Insert parentheses to make each
sentence true.
4.
PARTNER
ACTIVITY
Decimal Version
219
4
Complete the table.
3.
▶ Playing High-Number Toss:
Explain how you got your answer.
Write < or >.
6.
1
_
a. 4
2
_
b. 7
22°
8
_
c. 9
7
_
d. 12
5
_
e. 12
<
<
>
>
<
High Number Ongoing Assessment:
Toss Record Sheet
Recognizing Student Achievement
3
_
8
2
_
5
7
_
8
3
_
6
5
_
11
92 203
66 67
Math Journal 2, p. 241
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Use the Record Sheet (Math Masters, page 511) for High Number Toss:
Decimal Version to assess students’ ability to write and compare decimals.
Students are making adequate progress if they have written and compared the
decimals correctly through thousandths. Some students may be able to find the
difference between the two decimals.
[Number and Numeration Goals 1 and 6]
▶ Math Boxes 7 9
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 241)
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 7-11. The skill in Problem 6
previews Unit 8 content.
▶ Study Link 7 9
Study Link Master
Name
Date
STUDY LINK
(Math Masters, p. 209)
Time
Addition and Subtraction Problems
79
Solve each problem. Be careful. Some
problems involve addition, and some
involve subtraction.
1.
-25 + (-16) =
3.
-4 - (-4) =
5.
29 - (-11) =
7.
-100 + 15 =
9.
1
1
4 _2 + (-2 _2 ) =
-41
0
40
-85
2
Reminder:
To subtract a number, you can
add the opposite of that number.
0 - (-43) =
4.
-4 - 4 =
6.
8.
10.
92–94
43
-8
20
9 - (-11) =
10 - 10.5 = -0.5
10 - (-10) = 20
2.
Temperature
before Change
Temperature
Change
Temperature after Change
40°
up 7°
40 + 7 =
10°
down 8°
10 - (-8) =
-15°
(15° below zero)
up 10°
-20°
(20° below zero)
down 10°
47°
ENRICHMENT
40 + (-7) =
10 - 8 =
-15 + 10 = -5°
-20 - 10 = -30°
▶ Comparing Elevations
2°
15 + 10 =
Practice
u = 65,664 13.
w = 30.841 15.
684 ∗ 96 = u
14.
32.486 - 1.645 = w
69 ÷ e = 23
9.45 - m = 3.99
INDEPENDENT
ACTIVITY
15–30 Min
(Student Reference Book, p. 381; Math Masters, p. 210)
20 - (-10) =
Find the number that each variable represents.
12.
Home Connection Students solve problems involving
addition and subtraction of positive and negative
numbers.
3 Differentiation Options
For each temperature change in the table, two number models are shown
in the Temperature after Change column. Only one of the number models
is correct. Cross out the incorrect number model. Then complete the correct
number model.
11.
INDEPENDENT
ACTIVITY
e=3
m = 5.46
To apply students’ understanding of adding and subtracting
signed numbers, have them compare the elevations of U.S.
locations that are above and below sea level. When they have
Math Masters, p. 209
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Unit 7
Exponents and Negative Numbers
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Teaching Master
finished the page, consider having them identify these locations on
the landform map of the United States on Student Reference Book,
page 381 and compare this information with the chart of elevation
along the 39th parallel on the Student Reference Book page.
Name
Date
LESSON
Time
Comparing Elevations
79
䉬
5,300
Denver, CO
This number line shows the elevation of several places. Elevation
measures how far above or below sea level a location is. For example,
an elevation of 5,300 for Denver means that Denver is 5,300 feet above
sea level. An elevation of ⫺280 for Death Valley means that some
point in Death Valley is 280 feet below sea level.
Fill in the table below. Use the example as a guide.
▶ 5-Minute Math
Example:
5–15 Min
Draw an arrow next to the number line. Start it at the elevation for
Denver (5,300 feet). End it at the elevation for Atlanta (1,000 feet). Use
the number line to find the length of the arrow (4,300 feet). Your final
elevation is lower, so report the change in elevation as 4,300 feet down.
Write a number model for the problem: 5,300 ⫺ 1,000 ⫽ 4,300.
If you start at Denver and travel to Atlanta, what is your change in elevation?
Solution:
4,300
EXTRA PRACTICE
SMALL-GROUP
ACTIVITY
2,400
Tucson, AZ
Algebraic Thinking To offer students more experience solving
“What’s My Rule?” problems involving negative numbers, see
5-Minute Math, pages 153 and 234.
Start at
Travel to
Change in Elevation
Number Model
Denver
Atlanta
Chicago
Tucson
4,300 feet down
5,300 ⫺ 1,000 ⫽ 4,300
600
Chicago, IL
0 Sea Level
⫺280
Death Valley, CA
1,800
feet
up
Death Valley Dead Sea
600 ⫹ 1,800 ⫽ 2,400
1,020 feet down
Dead Sea
Death Valley
⫺280 ⫹ (⫺1,020) ⫽ ⫺1,300
1,020 feet up
Tucson
Death Valley
⫺1,300 ⫹ 1,020 ⫽ ⫺280
2,680 feet down
Dead Sea
Atlanta
2,400 ⫹ (⫺2,680) ⫽ ⫺280
2,300 feet up
1,000
Atlanta, GA
⫺1,300 ⫹ 2,300 ⫽ 1,000
⫺1,300
Dead Sea
(Israel/Jordan)
Math Masters, p. 210
Student Page
Student Page
American Tour
Games
High-Number Toss: Decimal Version
Landform Map of the United States
Materials 䊐 number cards 0–9 (4 of each)
䊐 scorecard for each player
Ca
sc
a
Ro
in
Great Pla
de
Ran
ge
Geography
y
u
Central Plain
Rang
s
Atlantic Ocean
i
n
es
Ap
pa
lac
hia
n
a in
Mo
un
tain
s
Mo
39°
nt
a
evad
ley
Val
Central
N
Sierra
Coast
Pacific Ocean
Great
Basin
Co
Co
ast
al
as
ta
l
a
Pl
sR
ok
Br o
Alaska Ran
0
ge
ge
200
400 Miles
400
600 Kilometers
0
Feet above Sea Level
Sierra
Nevada
Mountains
0.
To tal:
Player 2 has the larger number and wins the round.
8,000 ft
6,000 ft
4,000 ft
Round 4
Player 1: 0 . 6 5 4
Player 2: 0 . 7 5 3
Rocky Mountains
14,000 ft
10,000 ft
0.
4. The winner’s score for a round is the difference between the
two players’ numbers. (Subtract the smaller number from
the larger number.) The loser scores 0 points for the round.
Elevation along the 39th Parallel
12,000 ft
0.
with the larger number wins the round.
Plains
Plains
200
Round 3
Round 2
♦ Players take turns doing this 2 more times. The player
Hills
Hills
100
2. Shuffle the cards and place them number-side
down on the table.
Score
writes the number on one of his or her blanks.
Plateaus
Plateaus
Miles 0
0.
♦ Player 2 draws the next card from the deck and
Widely spaced
spaced mountains
mountains
Widely
500 Miles
Round 1
Directions
1. Each player makes a scorecard like the one at
the right. Players fill out their own scorecards at
each turn.
writes the number on any of the 3 blanks on the
scorecard. It need not be the first blank—it can
be any of them.
500 Kilometers
Kilometers
Scorecard
Game 1
♦ Player 1 draws the top card from the deck and
Mountains
Mountains
Hawaii
0
200
0
Alaska
0
Decimal place value, subtraction, and
addition
3. In each round:
Plain
Gulf of Mexico
an
2
Skill
Object of the game To make the largest decimal
numbers possible.
s
ck
39°
Players
Since 0.753 ⫺ 0.654 ⫽ 0.099, Player 2 scores 0.099 point for the round.
Player 1 scores 0 points.
Appalachian
Mountains
Coast
Ranges
Mississippi
River
2,000 ft
Washington,
D.C.
0 ft
123 121 119 117 115 113 111 109 107 105 103 101 99
97
95
93
91
89
87
Longitude (degrees W)
Student Reference Book, p. 381
85 83
81
79
77
75
73
71
5. Players take turns starting a round. At the end of 4 rounds,
they find their total scores. The player with the larger total
score wins the game.
Student Reference Book, p. 321
Lesson 7 9
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Name
Date
Time
High-Number Toss: Decimal Version Record Sheet
Circle the winning number for each round. Fill in the Score column each time
you have the winning number.
Player 1
321
Player 2
(Name)
Round
Sample
Copyright © Wright Group/McGraw-Hill
1 2
4 3
(Name)
Player 1
0. 6
5
<, >, =
4
<
Player 2
0. 7
1
0.
0.
2
0.
0.
3
0.
0.
4
0.
0.
5
0.
0.
5
Score
3
0. 753
– 0.654
0.099
Total Score
511
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