Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
1
GETTING THE IDEA
Understanding Positive and
Negative Integers
6.NS.5
LESSON
11
You have worked with positive integers. They are the set of counting numbers starting with 1.
{1, 2, 3, 4, 5, . . .}
The opposites of the positive integers make up the set of negative integers.
{. . . , 25, 24, 23, 22, 21}
Zero is also an integer. It is neither negative nor positive. So, the set of integers include zero,
the counting numbers, and their opposites.
{. . . , 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, . . .}
You can model integers on a number line. Remember that the number line continues without end in
both directions. The integers 24 and 4 are opposites. Notice that they are the same distance from 0.
503 421 35 4 12
Example 1
Use integers to label the points on the number line.
100 82 6
10 8 24
Strategy Identify positive integers and their opposites.
Step 1
Identify the scale of the number line.
Consecutive numbers on the number line have a difference of 2, so the scale is 2.
There are other integers between the tick marks that are not labeled. For example,
7 is halfway between 6 and 8.
102 Domain 2: The Number System
313NASE_Mth_G6_D2_PDF Round.indd 102 20/05/14 2:30 PM
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
Step 2
Label the positive integer.
The numbers to the right of zero are positive.
The point on the positive side of
the number line is halfway between 2 and 6.
The missing integer is 4.
100 82 6
10 8 24
4
Step 3
Label negative integer closest to zero.
The numbers to the left of zero are negative.
The point on the negative side of the number line
closest to zero is between 24 and 28.
Look at its opposite on the positive side of the number line. Halfway between
4 and 8 is 6. So, halfway between 24 and 28 is 26.
6
100 82 6
10 8 24
4
Step 4
Label the other negative integer.
The last point is between 28 and 210.
Counting down from zero, we would count as follows:
0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 210
The integer between 28 and 210 is 29.
9
6
100 82 6
10 8 24
4
Solution The completed number line is shown in Step 4.
Integers are used in many situations to show numbers above and below a baseline. You are likely
familiar with negative temperatures. On the Celsius scale, 0° is the temperature at which water freezes.
Positive temperatures are warmer than freezing, whereas negative temperatures are below freezing.
Integers can also show changes. When you withdrawal $50 from your bank account, the change in
your balance can be represented as 2$50. Bank accounts also use the terms credit, to indicate an
addition, and debit, to indication a subtraction or withdrawal.
Lesson 11: Understanding Positive and Negative Integers 103
313NASE_Mth_G6_D2_PDF Round.indd 103 20/05/14 2:30 PM
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
Example 2
Use numbers from the box to represent each situation.
A. a temperature of 15° below zero
B. a height of 15 miles above sea level
C. a deposit of $15 into your bank account
D. a drop in stock value of $15
15
215
Strategy Determine if the situation is above or below zero, or is a positive or a negative change.
Step 1
Look for key words.
The words above and below tell you the sign of the number. Any amount above 0
or a baseline is a positive number. Numbers below 0 or a baseline are negative.
A. a temperature of 15° below zero negative number 215
B. a height of 15 miles above sea level positive number 115
Step 2
Look for changes.
The situations described in parts C and D are changes in money amounts.
A deposit increases the balance in your bank account, so it can be represented
by a positive number. A drop in value is a decrease in the value, so it can be
represented by a negative number.
C. a deposit of $15 into your bank account positive number 115
D. a drop in stock value of $15 negative number 215
Solution The numbers that represent each situation are:
A. a temperature of 15° below zero 215
B. a height of 15 miles above sea level 15
C. a deposit of $15 into your bank account 15
D. a drop in stock value of $15 215
Example 3
Which integer is the opposite of 34?
Strategy Use the definition of opposite.
Step 1
Determine the sign.
A positive number is often shown without a sign. The numbers 134 and 34 have
the same value. So, 34 is a positive integer.
104 Domain 2: The Number System
313NASE_Mth_G6_D2_PDF Round.indd 104 20/05/14 2:30 PM
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
How do you know that 3 3 2 and 26 are the same distance from 0 on the number line?
The value of 3 3 2 is
.
Opposite integers are
distance from zero on the number line, but in
directions.
26 and
are opposite integers.
The points that represent 3 3 2 and 26 are the same distance from zero on the number line
because they are
integers.
Step 2
Write the opposite.
To find the opposite, change the sign. The opposite of a positive number is a
negative number.
Therefore, the opposite of 134 is 234.
Solution The opposite of 34 is 234.
Example 4
Tia hiked to a point 1,265 feet above sea level. Using scuba gear, Jose dove to a point 57 feet below
sea level. Write these altitudes as integers.
Strategy Determine if the points described are above or below 0 or a baseline.
Step 1
Look for key words.
Sea level is a baseline used for measuring altitude. Points on the earth and in
the ocean are often described as being above or below sea level. If you are
at sea level, your altitude is 0.
1,265 feet above sea level positive
57 feet below sea level negative
Step 2
Write integers to represent the situations.
1,265 feet above sea level 1,265
57 feet below sea level 257
Solution The integer 1,265 represents an altitude 1,265 feet above sea level. The integer
257 represents an altitude 57 feet below sea level.
Lesson 11: Understanding Positive and Negative Integers 105
2
COACHED EXAMPLE
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
313NASE_Mth_G6_D2_PDF Round.indd 105 20/05/14 2:30 PM
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
3
LESSON PRACTICE
Use numbers from the box to label points on the number line.
200 168 12
20 16
48
4
2
22
5
25
12
212
18
218
Select True or False for each statement.
A. Zero is a positive number. ○ True ○ False
B. 8 and 4 are opposite integers. ○ True ○ False
C. 58 and 1,248 are positive integers. ○ True ○ False
D. Opposite integers are the same distance from zero. ○ True ○ False
E. 238 is the opposite of 38. ○ True ○ False
For each integer, indicate with an “X” whether the number is negative, positive, or neither.
Integer Negative Positive Neither
5
0
228
497
2129
1
2
3
106 Domain 2: The Number System
313NASE_Mth_G6_D2_PDF Round.indd 106 20/05/14 2:30 PM
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
Circle the words that make the statements true.
The altitude, 37 feet
below sea level,
above sea level,
can be represented by a negative number.
A
debit
credit
to your account balance can be represented by a positive number.
A
discount
markup
on an item can be represented by a negative number.
Write the opposite integer for each integer shown.
A. 212
C. 219
B. 64 D. 2345
Keith is standing on a hill overlooking the ocean. The altitude is 50 feet above sea level. Jill dove
to a depth of 40 feet below sea level.
Part A
Write Keith’s and Jill’s altitudes using integers.
Keith’s altitude:
Jill’s altitude:
Part B
Who is at an altitude farther from sea level? Explain how you know.
4
5
6
Lesson 11: Understanding Positive and Negative Integers 107
313NASE_Mth_G6_D2_PDF Round.indd 107 20/05/14 2:30 PM
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
Use the numbers from the box to write the integer that describes each situation.
A. the temperature rises 20°
B. a debit of $20 to your bank account
C. an item is discounted by $20
D. a price increases by $20
E. 20 feet below sea level
F. a deposit of $20 in a bank account
20
220
The table shows transactions Rudo made with his checking account. Complete the table.
Record each transaction as a positive or a negative dollar amount.
Transaction Integer
Opened account and made a $500 deposit $500
Wrote a $25 check for soccer fees 2$25
Deposited babysitting earnings of $15
Used debit card to pay $7 for lunch
Used ATM to get $20
$2 ATM transaction fee charged
$1 interest earned
Ken says that 210 and 30 are opposite integers because they are both the same distance from
10 on the number line. Explain to Ken why his statement is not true. Use the number line to
model opposites of 210 and 30.
50030 402010 3050 40 1020
7
8
9
108 Domain 2: The Number System
313NASE_Mth_G6_D2_PDF Round.indd 108 20/05/14 2:30 PM
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC
Write the integer described by each expression.
A. the opposite of 4 3 2
B. the negative integer closest to 0 on the number line
C. the integer immediately to the left of 216 on the number line
D. the opposite of 212
E. the integer halfway between 220 and 216
Part A
Write the missing temperatures on the thermometer.
10
20
10
20
30
40
50
70
80
90
100
Part B
Explain how you knew where to place zero on the thermometer. Use opposite temperatures as
part of your explanation.
10
11
Lesson 11: Understanding Positive and Negative Integers 109
313NASE_Mth_G6_D2_PDF Round.indd 109 20/05/14 2:30 PM
