Opening Example
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Mine
Elevation (feet)
Above/Below Sea Level
Warm Springs Mine
+4500
Valley Mine
+1357
Salt Spring Mine
–260
Travertine Mine
–104
Galena Mine
+682
Copper Canyon Mine
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Miners who work deep underground must
consider the air they breathe. As a safety
engineer for the Diggem Mining Corporation,
Patsy monitors the safety of miners at the
mines owned by Diggem. Patsy’s safety report
includes a table of the elevations above or below
sea level where the workers are excavating. She
uses positive numbers with plus signs “+” for
elevations above sea level. She uses negative
numbers with minus signs “–” for elevations
below sea level.
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+4910
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You are familiar with whole
numbers, which include 0 and
the counting numbers 1, 2, 3,
4,… In this unit, you will learn
about numbers that can be
positive or negative, like the
numbers in Patsy’s table. The
positive and negative numbers in the table represent a set of numbers called
integers, which includes all the whole numbers and their opposites.
Quick Activity
This activity should
help you learn to use
a number line to
understand integers.
Your teacher may
ask for your
assistance to make a
number line on the
floor.
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When your teacher
invites you, go to the
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front of the room and draw a card. Read the signed number or
situation that is on your card aloud to the class. Then stand at
the place on the number line for that number.
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Quick Check
1. Which of the following are integers?
1⁄
4
7.45
2
–12
0
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15.1
5⁄
– 2.8
1,000
12
Write integers that model the following situations. Use positive
and negative signs.
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2.
A hot air balloon at 1515 feet above sea level
3.
A mine-shaft elevator at 270 feet below sea level
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INTEGERS Unit 1
4.
A temperature of 24 degrees below zero
Look at each pair of integers below. Which integer has the
larger value?
5.
–3
4
6.
–1
0
7.
–6
–8
8.
0
12
9.
10.
What is the value of |– 5|?
What is the value of |5|?
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Examples and Applications
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Example 1. Different ways to write
numbers
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Jenny drove to work on
Monday and parked on
the third underground
level of the parking
garage. She then went up
to her office located on the
fifth floor of her building.
You could indicate the
location of Jenny’s parked
car with the number – 3,
because it is 3 floors
below ground level. The
mark in front of the 3 is
called a negative sign.
– 3 is negative three.
Since Jenny’s office is located 5 floors above ground level, her
office location could be represented by the number +5. The
mark in front of the 5 is called a positive sign. The number +5
is read positive five. Numbers are sometimes written without
signs in front of them, such as the number 2. If a number does
INTEGERS Unit 1
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not have a sign in front of it, it always represents a positive
number. So, 2 could be read positive two as well as just two.
In numbers like – 3 and +5, the numerals 3 and 5 show the size
or amount of the number while the positive and negative signs
indicate the direction of the number from the 0 reference point
on a number line.
The number 0 is a special number since it is neither positive
nor negative. It is never written with a positive or negative sign.
Skill Practice
Write a number for each of the following situations using
integers.
1.
A temperature in Chicago of 4 degrees below 0
2.
A raise in salary of $1,000
3.
A student’s grade on a test if no answers are right
4.
A football team has a loss of 6 yards.
5.
A weight gain of 3 pounds
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Place and label the following on the number lines.
6.
–2
7.
8
8.
0
9.
–5
10.
6
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2
Example 2. The opposite of a number
On Friday, Tyler made
a deposit of $20 to his
checking account. Since
a deposit increases an
account balance, you
could represent the
deposit with the
number +20.
On Saturday, Tyler wrote a check for $20 for tickets to a
concert. The check will cause a decrease in his account balance
so you could represent the check amount with –20.
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The size or amount of each transaction is $20. However, if you
graphed the transactions on a number line, you would find that
the two amounts would be on opposite sides of the 0 reference
point. Numbers like +20 and –20 are called opposites. Two
numbers are opposites if they are the same size or amount but
are in different directions from the 0 reference point on a
number line.
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You already know that 0 is a special number that is neither
positive nor negative. Therefore, the number 0 is its own
opposite. In other words, the opposite of 0 is 0.
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Skill Practice
Write the opposites of the following numbers.
1.
146
2. – 75
3.
0
4.
1
5.
–1
Find and label on the number lines the opposites of the
following numbers.
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INTEGERS Unit 1
–9
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3
7.
2
8.
0
9.
6
10.
–3
Example 3. Integers and number lines
The First Bank building rises 8 floors above ground level. It
also has four basement-level floors. The elevators in the
building have a button for each floor.
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In the Opening Example, integers were
defined as all whole numbers and their
opposites.
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We can use a number line to picture
these numbers. The bank building
pictured here is like a vertical number
line.
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The floors above ground level are like the
positive integers. The floors below
ground level are like the negative
integers. And the ground floor is like
zero. These are all integers.
Like G for ground level, 0 is the starting
point and is called the origin. The
number 0 is neither negative nor
positive, but it is an integer.
The arrows on the number line say that the line extends
forever beyond that point. So, the arrow on top tells you that
beyond the number “8” are also the numbers 9, 10, 11, and on
and on. Above zero you have all the positive integers.
In the same way, below zero you have the negative integers
–1, –2, –3, and so on. The arrow says that you also have the
values of –6, –7, –8, and on and on.
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Number lines can also be horizontal. In that case, positive
numbers are to the right of zero and negative numbers to the
left.
Skill Practice
1.
Which of the following are integers?
–14
2.
0
–1
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–26
–1,839
0
–8
1⁄
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0
–31
5.
Copy the number line below. Label the points with
integers to the left and right of 0.
6.
Name some everyday objects that represent vertical and
horizontal number lines.
7.
Which of the following would be on the right side of 0?
10
8.
9.
4
–7
–8
–3
5
Which of the following would be above 0?
–2
10.
–4
Which of the following would be on the left side of 0?
8
1
–7
9
Which of the following would be below 0?
–6
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Which of the following are positive integers?
12
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0
Which of the following are negative integers?
–3⁄8
4.
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Which of the following represents the origin of a number
line?
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3.
1⁄
4
12.4
11
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–3
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4
Example 4. Comparing integers
Look at these noon temperatures in February for 5 cities.
Johnson Village
Pratt City
Apache Junction
Middletown
Moose Falls
–2 degrees
0 degrees
–5 degrees
+3 degrees
–7 degrees
Which city had the lowest temperature? Which had the highest
temperature?
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Moose Falls clearly had the lowest temperature because –7 is
below all the other temperatures. Middletown had the highest
temperature because +3 is above all the other temperatures.
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Skill Practice
Which of the two temperatures in each pair below is the lower
temperature?
1.
–5°
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–8°
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2.
0° –2°
3.
8°
3°
Put each set of temperatures on the thermometer. Circle the
highest and lowest temperatures.
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4.
12°
–18°
6°
–14°
5.
–10°
0°
–8°
1°
6.
20°
–4°
–10°
16°
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INTEGERS Unit 1
Circle the integer with the highest value.
7.
–4
2
–1
3
8.
18
–19
–18
8
9.
Are the following integers the same distance from 0?
5
10.
–5
Which of the following temperatures is the highest?
32°
5
0°
–14°
–64°
5°
Example 5. Symbols used to compare
integers
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In Example 4, you can see that the temperature at Apache
Junction is less than Pratt City’s 0°. You can write this using
the mathematical symbol < : –5° < 0°. Read it like this:
“negative 5 degrees is less than zero degrees.”
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The Middletown temperature was greater than Johnson
Village’s. You can write this: +3° > –2°. Read it like this: “plus
3 degrees is greater than negative 2 degrees.”
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Look at these:
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INTEGERS Unit 1
–7° < +3°
–5° < –2°
+3° > 0°
0° > –2°
Notice that the pointed end of the symbol
always points toward the lesser number.
When you read it, if you come to the pointed end of the symbol,
say “is less than.” If you come to the open end, say, “is greater
than.”
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Skill Practice
Write in words how each expression below would be read.
1.
6>4
2.
–2 < 3
Show which number in each pair below is the lesser by placing
the symbol < or > between the two numbers.
3.
–5
–3
4.
0
–1
True or False?
5.
6>0
6.
7 < –7
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Write two true statements for each expression.
7.
14 < 25
8.
–3 > –10
9.
6
a. 7 > –2
b. –2 < 7
c. both
d. neither
a. –10 > –2
b. 35 < –35
c. 0 > –6
d. –15 < –20
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Which is correct?
Example 6. The absolute value of a
number
Samantha delivers medicines to customers on a long stretch of
road in Montana. One system she uses to help her is a
“highway number line,” as shown here. The origin of the
number line is the store location, so it is placed at the 0 point.
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When Samantha makes a delivery to the Smiths at “2,” she
must drive 2 miles east from the store. If she delivers to the
Pauls at “–4,” she must drive 4 miles west.
When Samantha wants to know the driving distance, she can
ignore the sign of the number. She is interested in only the
number itself, not the sign.
This is the way absolute value works. So Samantha can make a
table of distances from the store using the absolute value of the
highway number line.
Location
Distance
(absolute value)
Bond
–6
6
Paul
–4
Customer
Smith
2
Rogers
5
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2
5
The way she can write this is to use the absolute value symbol:
two vertical bars surrounding the number.
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|2| = 2
The absolute value of 2 equals 2.
|–4| = 4
The absolute value of negative 4 equals 4.
Skill Practice
1.
⎜–10⎜ = ?
2.
⎜6⎜ = ?
For each pair, which number is farther from 0 on a number
line?
3.
5
–4
4.
–10
10
5.
8
15
Which absolute value is greater?
INTEGERS Unit 1
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6.
⎜8⎜
⎜–3⎜
7.
⎜–4⎜
⎜–7⎜
Show the absolute value of each number on the number line
below.
8.
–7
9.
2
Insert the absolute value symbol, ⎜⎜, around one of the
numbers in each pair to make the expression true.
10.
6 = –6
11.
–5 > 2
12.
0 < –8
Lab Activity
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You will participate in some group activities for
this unit that will help you practice the skills and
concepts learned. You will do things like make
measurements, collect data, and perform
calculations. For this unit, be prepared to take a
walk in the hall, play a beanbag toss game, or
calculate temperature differences among
different water samples. You will have fun and
learn at the same time!
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INTEGERS Unit 1
Standardized Test Problems
The following are problems you might find on a
standardized test or other exam.
Questions 1 through 3 refer to the following number line.
1.
2.
What is the value of point A on the number line?
a. –6
c. 0
b. –4
d. +3
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a. A
c. C
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b. B
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b. B > 0
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d. D
Which of the following is a true statement?
a. D < C
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What point corresponds to the origin of the number line?
c. 5 < E
d. –2 < D
Which of the following statements is true.
a. 0 is a positive integer.
b. Positive numbers are to the left of zero on the
number line.
c. On a number line, a smaller number is always to the
left of a larger number.
d. The set of integers contains only positive and
negative whole numbers.
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INTEGERS Unit 1
Which of the following is |–15|?
a. –15
c. 5
b. 0
d. 15
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Workplace Exercises
These problems are like the ones you might find in the
workplace.
1. Kara surveys land for a new office complex. She sets
up her surveying transit on the benchmark corner of
the property. She measures the elevations of the three
other corners from the benchmark. Kara finds that
corner A is 4 inches above the benchmark, corner B is
9 inches below the benchmark, and corner C is even
with the benchmark.
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a. In relation to the benchmark, what integer would
Kara use to record the elevation of each corner with
respect to the benchmark?
b. Show your integers on a number line. Label them as
A, B, and C, for each corner. Where would the
benchmark be? Label it, too.
c. Use the absolute values of Kara’s measurements to
find which corner is farthest above or below the
benchmark.
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2. At a space shuttle launch, the
flight director announces that the
time until launch is “T minus
45 seconds.”
a. Does the announcement of
“T minus 20” come before or
after the “T minus 45”
announcement?
b. Using an integer, how would
you represent the time at
which the launch takes place?
c. Using an integer, how would you represent the time
25 seconds after the launch has taken place?
3.
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Oscar is recovering from an illness. The hospital staff
watches his fluid intake and output. Whenever the
difference between the two exceeds 500 cc during a shift,
the head nurse calls the doctor immediately. The table
here shows Oscar’s fluid records for one day.
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Oscar’s Fluids for Saturday
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Shift
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B
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Intake
Output
Difference
1700 cc
1100 cc
+600 cc
1000 cc
1250 cc
–250 cc
1350 cc
1900 cc
–550 cc
a. Should the doctor have been called on any of these
shifts?
b. If so, on which shift(s) should the doctor have been
called?
c. Why might the doctor be concerned about the
difference between Oscar’s fluid intake and output?
4.
INTEGERS Unit 1
At the Hilltown Water Treatment Plant, Richelle opens
a valve that adds water to a tank at 40 gallons per
minute. A few minutes later, Jeremy opens a drain valve
to let the water out of the same tank at a rate of
32 gallons per minute.
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a. Express each flow rate as a positive or negative
integer.
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b. Is the water level in the tank rising or falling? Why?
5.
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House Location
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Betty is the construction foreman for Hank Hawthorne
Homes (HHH). She prepares a report on the last
5 houses built by HHH.
Scheduled
Completion
Date
Actual
Completion
Date
34 Primrose Lane
March 30
March 27
103 Gurley Avenue
March 10
March 14
542 Sweetbriar St.
February 12
February 12
100 Lilac Way
January 18
January 31
1024 Laurel Drive
February 25
February 19
a. Compare the scheduled completion dates to the
actual completion dates. Add a column of signed
integers to show the differences.
b. What numbers in your column will please the
company? What numbers will not please the
company?
c. What does the number 0 in your column mean to
Betty?
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Closing Check
1. Identify the integers in the group of numbers below.
4.72
6
3⁄
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–19
–3.1
2. Using the model of the number line below, answer the
following:
a. How would you explain why +5 is larger than –2?
b. What integer represents the origin of the number
line?
3.
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Rewrite the following integers so that they are in order,
from least to greatest.
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4.
a. – 2
1
c. – 8
–10
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e. 45
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Complete each expression by using the symbol <, >, or =
between the pair of numbers.
b. 0
|–5|
d. |–3|
|3|
–66
5.
Why do |–3| and |3| both equal 3? Using the concept
that answers this question:
a. Draw a number line.
b. Divide it into the appropriate number of segments.
c. Label each point so that you can place dots for all
integers whose absolute values are 2, 3, 4, 5, and 6.
6.
Arrange the following freezing points from warmest to
coldest.
water
0°C
helium
–272°C
lead
328°C
mercury
–39°C
oxygen
–219°C
sodium
98°C
tin
232°C
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7.
Place the opposites of the following two numbers on the
number line and label them: –7 4
8.
Write each of the following numbers with the correct
positive or negative sign.
a. a decrease of 12
b. a gain of 26
9.
Draw examples of a vertical and a horizontal number line.
Label the origin and points to 10 and their opposites.
10. Can you think of a number line that is neither vertical
nor horizontal? (Hint: There is at least one example in
your classroom.)
1. 34 × 72 = ?
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2. 1644 ÷ 12 = ?
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Mixed Review
3. Joan’s grandfather was born in the year 1912 and died in
the year 1994. About how old was he when he died?
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a. The longest song on a CD is 12 minutes.
b. It is common for CDs to have 11 or 12 songs on them.
c. Two songs on a CD are 9 minutes long.
d. Most CDs have 13 or more songs on them.
5.
Put these numbers in order from smallest to largest:
9,585
200
7,053
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