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Pre-Algebra Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
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Negative number – a number less than zero
Integers – the group of negative and positive numbers
Coordinate – number that corresponds to a point on a
number line
Inequality – any mathematical sentence containing <,>
signs that compares nmbers or quantities
Absolute Value – distance a number is from zero on the
number line
Example 1 Write Integers for Real-World Situations
Example 2 Compare Two Integers
Example 3 Order Integers
Example 4 Expressions with Absolute Value
Example 5 Algebraic Expressions with Absolute Value
Write an integer for each situation.
a. 32 feet under ground
Answer: The integer is –32.
b. 8 weeks after birth
Answer: The integer is +8.
c. a loss of 6 pounds
Answer: The integer is –6.
Write an integer for each situation.
a. a loss of 12 yards
Answer: The integer is –12.
b. 15 feet above sea level
Answer: The integer is +15.
c. the temperature
decreased 4 degrees
Answer: The integer is –4.
Use the integers graphed on the number line below.
Write two inequalities involving 7 and –4.
Answer: Since 7 is to the right of –4, write
Since –4 is to the left of 7, write
.
.
Use the integers graphed on the number line below.
Replace the  with <, >, or = in –2  3 to make a true
sentence.
Answer: –2 is less since it lies to the left of 3.
So write –2 < 3.
Use the integers graphed on the number line below.
a. Write two inequalities involving –4 and 1.
Answer:
b. Replace the  with <, >, or = in 6  –7 to make a true
sentence.
Answer:
Weather The high temperatures for the first seven
days of January were –8°, 10°, 2°, –3°, –11°, 0°, and 1°.
Order the temperatures from least to greatest.
Graph each integer on a number line.
Write the numbers as they appear from left to right.
Answer: The temperatures –11°, –8°, –3°, 0°, 1°, 2°, 10°
are in order from least to greatest.
Football The yards gained during the first six plays of
the football game were 5, –3, 12, –9, 6, and –1. Order
the yards from least to greatest.
Answer: The yards –9, –3, –1, 5, 6, and 12 are in order
from least to greatest.
Evaluate
.
The graph of 5 is 5 units from 0.
Answer: 5
Evaluate
.
The absolute value of –8 is 8.
The absolute value of –1 is 1.
Simplify.
Answer: 9
Evaluate
.
The absolute value of 6 is 6.
The absolute value of –4 is 4.
Simplify.
Answer: 2
Evaluate each expression.
a.
Answer: 9
b.
Answer: 5
c.
Answer: –2
Algebra Evaluate
if
.
Replace x with –2.
The absolute value of –2 is 2.
Simplify.
Answer: –6
Algebra Evaluate
Answer: –4
if
.
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Opposites – two numbers with the same
absolute value but with different signs
Additive inverse – an integer and its
opposite
Example 1 Add Integers on a Number Line
Example 2 Add Integers with the Same Sign
Example 3 Add Integers on a Number Line
Example 4 Add Integers with Different Signs
Example 5 Use Integers to Solve a Problem
Example 6 Add Three or More Integers
Find
.
3
4
Start at zero.
Move three units to the right.
From there, move four more units to the right.
Answer:
Find
Answer: –7
.
Find
.
Add
and
. Both numbers are
negative, so the sum is negative.
Answer: –9
Find
Answer: –11
.
Find
.
–11
7
Start at zero.
Move 7 units to the right.
From there, move 11 units to the left.
Answer:
Find
.
9
–2
Start at zero.
Move 2 units to the left.
From there, move 9 units to the right.
Answer:
Find each sum.
a.
Answer: 3
b.
Answer: –3
Find
.
To find –9 + 10, subtract
The sum is positive because
Answer: 1
from
.
Find
.
To find 8 + (–15), subtract
The sum is negative because
Answer: –7
from
Find each sum.
a.
Answer: –3
b.
Answer: 5
Weather On February 1, the temperature at dawn was
–22°F. By noon it had risen 19 degrees. What was the
temperature at noon?
Words
The temperature at dawn was –22°F. It had
risen 19 degrees by noon. What was the
temperature at noon?
Variable Let x
the temperature at noon.
Temperature
at dawn
Equation
–22
plus
increase
by noon
19
equals
temperature
at noon.
x
Solve the equation.
To find the sum, subtract
The sum is negative because
Answer: The temperature at noon was –3°F.
Hiking Dave started his hike at 32 feet below sea
level. During the hike he gained an altitude of 29 feet.
At what altitude did Dave complete his hike?
Answer: Dave completed his hike at 3 feet below
sea level.
Find
.
Commutative Property
Additive Inverse Property
Identity Property
of Addition
Answer: –4
Find
.
Commutative
Property
Associative
Property
or –4
Answer: –4
Simplify.
Find each sum.
a.
Answer: –9
b.
Answer: 10
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Example 1 Subtract a Positive Integer
Example 2 Subtract a Negative Integer
Example 3 Subtract Integers to Solve a Problem
Example 4 Evaluate Algebraic Expressions
Find
.
To subtract 14, add –14.
Simplify.
Answer: –5
Find
.
To subtract 8, add –8.
Simplify.
Answer: –18
Find each difference.
a.
Answer: –2
b.
Answer: –22
Find
.
To subtract –4, add 4.
Simplify.
Answer: 19
Find
.
To subtract –7, add 7.
Simplify.
Answer: –3
Find each difference.
a.
Answer: 10
b.
Answer: –7
Weather The table shows the record high and low
temperatures recorded in selected states. What is the
range for Wyoming?
State
Lowest Temp
°F
Highest Temp
°F
Utah
–69
117
Vermont
–50
–30
–48
–37
–54
–66
105
110
118
112
114
114
Virginia
Washington
West Virginia
Wisconsin
Wyoming
Explore You know the highest and lowest temperatures.
You need to find the range for Wyoming’s
temperatures.
Plan
Solve
To find the range, or difference, subtract the
lowest temperature from the highest
temperature.
To subtract –66, add 66.
Add 114 and 66.
Answer: The range for Wyoming is 180°.
Examine Think of a thermometer. The difference
between 114° above zero and 66° below zero
must be 114 + 66 or 180°. The answer appears
to be reasonable.
Weather The table shows the record high and low
temperatures recorded in selected states. What is the
range for Washington?
State
Lowest Temp Highest Temp Answer:
The range for
°F
°F
Washington
Utah
–69
117
is 166°.
Vermont
–50
105
Virginia
Washington
West Virginia
Wisconsin
Wyoming
–30
–48
–37
–54
–66
110
118
112
114
114
Evaluate
if
.
Write the expression.
Replace m with 4.
To subtract –2, add 2.
Add 4 and 2.
Answer: 6
Evaluate
if
and
.
Write the expression. Replace
x with –14 and y with –2.
To subtract –2, add 2.
Add –14 and 2.
Answer: –12
Evaluate
if
,
, and
.
Write the expression.
Replace p with –11, q with 6,
and r with –12.
Order of operations
Add –5 and 12.
Answer: 7
a. Evaluate
if
.
Answer: 2
b. Evaluate
if
and
.
Answer: –6
c. Evaluate
Answer: 0
if
,
, and
.
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Example 1 Multiply Integers with Different Signs
Example 2 Multiply Integers with the Same Sign
Example 3 Multiply More Than Two Integers
Example 4 Use Integers to Solve a Problem
Example 5 Simplify and Evaluate Algebraic Expressions
Find
.
The factors have different signs.
The product is negative.
Answer: –96
Find
.
The factors have different signs.
The product is negative.
Answer: –99
Find each product.
a.
Answer: –48
b.
Answer: –12
Find
.
The two factors have the same
sign. The product is positive.
Answer: 64
Find
Answer: 24
.
Find
.
Associative Property
Answer: 154
Find
Answer: –120
.
Multiple-Choice Test Item
A student missed only 4 problems on a test, each
worth 20 points. What is the total number of points
missed?
A
B
C
D
–5
–20
24
–80
Read the Test Item
The word missed means losing points, so the loss per
problem is –20. Multiply 4 times –20 to find the total
number of points lost.
Solve the Test Item
The product is negative.
Answer: The answer is D.
Football A football team loses 3 yards on each of 3
consecutive plays. Find the total loss.
Answer: –9
Simplify
.
Associative Property of Multiplication
Simplify.
Answer: –42k
Simplify
.
Commutative Property
of Multiplication
Answer: –40ab
Evaluate
if
and
.
Replace x with –4
and y with 9.
Associative Property
of Multiplication
The product of –3 and –4
is positive.
The product of 12 and 9
is positive.
Answer: 108
a. Simplify
.
Answer: –12c
b. Simplify
.
Answer: –35mn
c. Evaluate
Answer: –162
if
and
.
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Average (mean) – find the sum of the
numbers and then divide by the
number in the set
Example 1 Divide Integers with the Same Sign
Example 2 Divide Integers with Different Signs
Example 3 Evaluate Algebraic Expressions
Example 4 Find the Mean
Find
Answer: 7
.
The dividend and the divisor
have the same sign. The
quotient is positive.
Find
.
The dividend and the divisor
have the same sign.
The quotient is positive.
Answer: 12
Find each quotient.
a.
Answer: 5
b.
Answer: 16
Find
.
The signs are different. The
quotient is negative.
Answer: –18
Find
.
The signs are different. The
quotient is negative.
Simplify.
Answer: –7
Find each quotient.
a.
Answer: –9
b.
Answer: –9
Evaluate
if
and
.
Replace x with –4 and
y with –8.
The quotient of –24 and –8
is positive.
Answer: 3
Evaluate
Answer: –12
if
and
.
Sam had quiz scores of 89, 98, 96, 97, and 95. Find
the average (mean) of his quiz scores.
Find the sum of the quiz
scores. Divide by the
number of scores.
Simplify.
Answer: The average of Sam’s quiz scores is 95.
Find the average (mean) of 10, –12, 9, 15, –4, 0, –1,
and 7.
Find the sum of the set of integers.
Divide by the number in the set.
Simplify.
Answer: 3
a. Kyle had test scores of 89, 82, 85, 93, and 96. Find
the average (mean) of his test scores.
Answer: 89
b. Find the average (mean) of 8, –6, –12, 11, –4,
and –21.
Answer: –4
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Quadrants – the four regions within
the coordinate plane
Example 1 Write Ordered Pairs
Example 2 Graph Points and Name Quadrant
Example 3 Graph an Algebraic Relationship
Write the ordered pair that names point P.
The x-coordinate is 4.
The y-coordinate is –2.
Answer: The ordered
pair is (4, –2).
Write the ordered pair that names point Q.
The x-coordinate is –3.
The y-coordinate is –1.
Answer: The ordered
pair is (–3, –1).
Write the ordered pair that names point R.
The x-coordinate is 2.
The point lies on the x-axis,
so the y-coordinate is 0.
Answer: The ordered
pair is (2, 0).
Write the ordered pair that names each point.
a. M
Answer: (–2, –3)
b. N
Answer: (4, –1)
c. P
Answer: (0, –2)
Graph and label each point on a coordinate plane.
Then name the quadrant in which each point lies.
S (–1, –5)
Start at the origin.
Move 1 unit left.
Then move 5 units down and
draw a dot.
Point S is in Quadrant III.
S
Graph and label each point on a coordinate plane.
Then name the quadrant in which each point lies.
U (–2, 3)
Start at the origin.
Move 2 units left.
Then move 3 units up and
draw a dot.
Point U is in Quadrant II.
U
Graph and label each point on a coordinate plane.
Then name the quadrant in which each point lies.
T (0, –3)
Start at the origin.
Since the x-coordinate is 0, the
point lies on the y-axis.
Move 3 units down and
draw a dot.
Point T is not in any quadrant.
T
Graph and label each point on a coordinate plane.
Then name the quadrant in which each point lies.
a. A (3, –4)
Answer: Quadrant IV
b. B (–2, 1)
Answer: Quadrant II
B
C
c. C (–4, 0)
Answer: Not in any
quadrant
A
The difference between two integers is 4. If x
represents the first integer and y is subtracted from
it, make a table of possible values for x and y. Then
graph the ordered pairs and
x–y=4
describe the graph.
x
y
(x, y)
First, make a table. Choose
2 –2
(2, –2)
values for x and y that have
a difference of 4.
1 –3
(1, –3)
0
–4
(0, –4)
–1
–2
–5
–6
(–1, –5)
(–2, –6)
Then graph the ordered pairs on a coordinate plane.
x–y=4
x
2
1
y
–2
–3
(x, y)
(2, –2)
(1, –3)
0
–4
(0, –4)
–1
–2
–5
–6
(–1, –5)
(–2, –6)
The points on the graph are in a line that slants downward
to the left. The line crosses the y-axis at –4.
x–y=4
x
2
1
y
–2
–3
(x, y)
(2, –2)
(1, –3)
0
–4
(0, –4)
–1
–2
–5
–6
(–1, –5)
(–2, –6)
The sum of two integers is 3. If x represents the first
integer and y is added to it, make a table of possible
values for x and y. Then graph the ordered pairs and
describe the graph.
Answer: Values in table and graphs will vary from
student to student.
Explore online information about the
information introduced in this chapter.
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