Transcript Variable Expression

1-1 Variables and Expressions
 Variable: a letter that stands for a number
 Variable Expression: a mathematical phrase that uses
variables, numerals, and operation symbols
- Examples: c + 5
8-x
Writing Variable Expressions

1.
2.
3.
You can translate word phrases into variable
expressions.
- Examples:
Nine more than a number y
y+9
4 less than a number n
n-4
A number z times three
3z
Examples
1.
2.
3.
4.
5.
A number c divided by 12
5 times the quantity 4 plus a number d
The value in cents of 10 dimes
The number of gallons in 7 quarts
Four times the quantity 8 minus p
Examples - Answers
1.
2.
3.
4.
5.
A number c divided by 12
c/12
5 times the quantity 4 plus a number d
5(4 + d)
The value in cents of 10 dimes
.10(10)
The number of gallons in 7 quarts
7/4
Four times the quantity 8 minus p
4(8 - p)
1-2 Order of Operations
 The order in which you perform operations can
affect the value of an expression.
 P - Parenthesis ( ) [ ]
 E - Exponents x2
 M/D - Multiply/Divide x ÷
 A/S - Add/Subtract +  Please Excuse My Dear Aunt Sally OR
 PEMDAS
Simplifying Expression
4 + 15 x 3
=4 + 45
=49
2. 3 x 5 - 8 ÷ 4 + 6
1.
= 15 - 2
=
=
13
19
+6
+6
Examples
1.
4-1x2+6÷3
2.
5+6x4÷3-1
3.
21 + 15
3+6
4. [2 + (6 x 8)] - 1
Examples - Answers
1.
4-1x2+6÷3
= 4
2.
5+6x4÷3-1
= 12
3.
21 + 15
3+6
4. [2 + (6 x 8)] - 1
= 4
= 49
Using Grouping Symbols

Insert grouping symbols to make each sentence
true.
1.
7 + 4 x 6 = 66
2.
7 x 8 - 6 + 3 = 17
3.
3 + 8 - 2 x 5 = 45
Using Grouping Symbols - Answers

Insert grouping symbols to make each sentence
true.
1.
(7 + 4) x 6 = 66
2.
[7 x (8 - 6)] + 3 = 17
3.
[3 + (8 - 2)] x 5 = 45
Using <, >, or =
 15 x 3 - 2
15 x (3 - 2)
 12 ÷ 3 + 9 x 4
 (19 - 15) ÷ (3 + 1)
12 ÷ (3 + 9) x 4
19 - 15 ÷ 3 + 1
Using <, >, or = (Answers)
 15 x 3 - 2
15 x (3 - 2) >
 12 ÷ 3 + 9 x 4
 (19 - 15) ÷ (3 + 1)
12 ÷ (3 + 9) x 4 >
19 - 15 ÷ 3 + 1 <
1-3 Writing and Evaluating
Expressions
To EVALUATE a variable expression, you
first replace each variable with a number.
Then use order of operations to simplify.
Example: Evaluate 4y - 15 for y = 9.
4y - 15 = 4(9) - 15
= 36 - 15
= 21
Examples
Evaluate:
1. 4(t+3) + 1 for t = 8.
2.
63 - 5x for x = 7
3.
3x + 2
for x = 8
Examples - Answers
Evaluate:
1. 4(t+3) + 1 for t = 8
2.
63 - 5x for x = 7
3.
3x + 2
45
for x = 8 26
28
Replacing More Than One
Variable
Example:
Evaluate 3ab + c ⁄ 2 for a = 2, b = 5, and c = 10.
3ab + c ⁄ 2 = 3(2)(5) + 10 ⁄ 2
=6*5+5
= 30 + 5
= 35
Examples
Evaluate:
1.
(9 +y) ⁄ x
2.
2xy - z
3.
r+s
2
for x = 2 and y = 3.
for x = 4, y = 3, z = 1
for r = 10, s = 9
Examples - Answers
Evaluate:
1.
(9 +y) ⁄ x
2.
2xy - z
3.
r+s
2
for x = 2 and y = 3.
for x = 4, y = 3, z = 1
for r = 10, s = 9
6
23
9.5
1-4 Integers and Absolute Value
 Integers are whole numbers and their opposites. Zero
separates positive and negative numbers on a number
line. Zero is neither positive nor negative and is its
own opposite.
 Absolute value is a number’s distance from zero.
Integers
 32 degrees above zero = +32°F or 32°F
 32 degrees below zero = -32 °F
 A number line helps you compare positive and
negative numbers and arrange them in order.
Graphing on a number line
 Graph -1, 4, and -5 on a number line. Then order them
from least to greatest.
-5
Least to greatest
-1 0
4
-5, -1, 4
Finding
Absolute
Value
 Numbers that are the same distance from zero
on a number line but in opposite directions are
called opposites.
 Example: -3 and 3
 |-3| and |3| are both 3; this means they are both
3 units from zero
 Examples: 1. - |-10|
2. |-23|
3. 200 ft below sea level
4. A profit of $300
Use <, >, or =

Examples:
1.
-8
0
2. |-3|
|50|
3. |-10|
|10|
Use <, >, or = (Answers)

Examples:
1.
-8
0
<
2. |-3|
|50|
>
3. |-10|
|10|
=
1-5
Adding
Integers
 Adding Integer Rules:
Same Sign:
Sum of 2 positive integers = positive

Sum of 2 negative integers = negative
(Add the numbers and keep the same sign)

Different Signs:
1.
Find difference of the 2 numbers
2.
The sum has the sign of the integer with the greater
absolute value
(Subtract the numbers and take the sign of the larger
number)
Adding Integers with the same sign
Examples:
 -2 + (-2)
 -22 + (-16)
 (-25) + (-15)
Adding Integers with the same sign
(Answers)
Examples:
 -2 + (-2) =
 -22 + (-16) = -38
 (-25) + (-15) = -40
-4
Adding Integers with different signs
Examples:
1. 2 + (-6)
2. (-1) + 4
3. 7 + (-18)
4. 5 + (-3)
Adding Integers with different signs
(Answers)
Examples:
1. 2 + (-6) =
2. (-1) + 4 =
3. 7 + (-18) =
4. 5 + (-3) =
-4
3
-11
2
Using Order of Operations
1.
2.
3.
Examples
-12 + (-6) + 15 + (-2)
1 + (-3) + 2 + (-10)
-250 + 200 + (-100) + 220
Using Order of Operations Answers
1.
2.
3.
Examples
-12 + (-6) + 15 + (-2) =
1 + (-3) + 2 + (-10) =
-250 + 200 + (-100) + 220 = 70
-5
-10
1-6 Subtracting Integers
 To subtract integers, add its opposite.
(Leave the 1st number the same, change the
subtraction sign to an addition sign, and change
the next number to the opposite sign)
Example: 4 - 8 =
4 + (-8) = -4
Examples
1.
2.
3.
4.
5.
6.
-7 - (-3)
-2 - 3
87 - (-9)
16 - (-8)
-90 - (-80) -20
An airplane takes off, climbs 5,000 ft, and then
descends 700 ft. What is the airplane’s current
height?
Examples - Answers
1.
2.
3.
4.
5.
6.
-7 - (-3) =
-4
-2 - 3 =
-5
87 - (-9) =
96
16 - (-8) =
24
-90 - (-80) -20 =
-30
An airplane takes off, climbs 5,000 ft, and then
descends 700 ft. What is the airplane’s current
height? 4300 ft
1-7 Inductive Reasoning
1-8 Look for a Pattern
 Inductive Reasoning is making conclusions based on
pattern you observe.
 Conjecture is a conclusion you reach by inductive
reasoning.
Writing a Rule for Patterns



1.
2.
30, 25, 20, 15…
RULE: Start with 30 and subtract 5 repeatedly.
1, 3, 4, 12, 13…
RULE: Start with 1. Alternate multiplying 3
and adding 1.
Examples:
3, 9, 27, 81…
4, 9, 14, 19
Is each conjecture correct? If not,
give a counterexample?
 1. All birds can fly.
 2. Every square is a rectangle.
 3. The product of 2 numbers is never less than either
of the numbers.
Is each conjecture correct? If not,
give a counterexample? (Answers)
 1. All birds can fly.
No,
ostriches can’t fly
 2. Every square is a rectangle. Yes
 3. The product of 2 numbers is never less than either
of the numbers.
No, 1/2 x 1/2 = 1/4
1-9 Multiply/Divide Integers
 Rules:
 The product/quotient of 2 integers with the same sign is
positive.
 The product/quotient of 2 integers with different signs
is negative.
Examples
1.
2.
3.
4.
5.
-17 x 3
-65 ÷ -5
-6(-3) ÷ 2
-15(-3)
-9
-3 x 7(-2)
Examples - Answers
1.
2.
3.
4.
5.
-17 x 3 =
-65 ÷ -5 =
-6(-3) ÷ 2 =
-15(-3) =
-9
-3 x 7(-2) =
-51
13
9
-5
42
Find the average
1.
Temperature: 6°, -15°, -24°, 3°, -25°
2.
Stock Price Changes: $52, -$7, $20, -$63, -$82
Find the average - Answers
1.
Temperature: 6°, -15°, -24°, 3°, -25°
-11
2.
Stock Price Changes: $52, -$7,
$20, -$63, -$82
-16
1-10 The Coordinate Plane
 Coordinate plane: formed by the intersection of 2
number lines.
 X-axis: the horizontal number line
 Y-axis: the vertical number line
 The x-axis and y-axis divide the coordinate plane
into 4 quadrants
 Origin: where the axes intersect
 Ordered pair: gives the coordinates and location of
a point
Quadrant II
Quadrant I
Quadrant III
Quadrant IV
Y-axis
X-axis
Origin