Download Unit 2 Power Points

Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
Warm Up
Evaluate each algebraic expression for the given value
of the variables.
1. 7x + 4 for x = 6
46
2. 8y – 22 for y = 9
50
3. 12x + 8y for x = 7 and y = 4
86
4. y + 3z for y = 5 and z = 6
23
Problem of the Day
A farmer sent his two children out to count the
number of ducks and cows in the field. Jean
counted 50 heads. Charles counted 154 legs. How
many of each kind were counted?
23 ducks and 27 cows
Learn to translate words into numbers, variables,
and operations.
Although they are closely related, a Great Dane
weighs about 40 times as much as a Chihuahua. An
expression for the weight of the Great Dane could be
40c, where c is the weight of the Chihuahua.
When solving real-world problems, you will need to
translate words, or verbal expressions, into algebraic
expressions.
Turn composition notebook sideways and divide
page into 4 sections:
ADD
Subtract
Multiply
Divide
Operation
Verbal Expressions
• add 3 to a number
• a number plus 3
• the sum of a number and 3
• 3 more than a number
• a number increased by 3
Algebraic
Expressions
n+3
• subtract 12 from a number
• a number minus 12
• the difference of a number
and 12
• 12 less than a number
• a number decreased by 12
• take away 12 from a number
• a number less than 12
x – 12
Operation
Verbal Expressions
Algebraic
Expressions
• 2 times a number
• 2 multiplied by a number
2m or 2 • m
• the product of 2 and a
number
• 6 divided into a number
÷
• a number divided by 6
• the quotient of a number
and 6
a ÷6
or
a
6
Additional Example 1: Translating Verbal
Expressions into Algebraic Expressions
Write each phrase as an algebraic expression.
A. the quotient of a number and 4
quotient means “divide”
n
4
B. w increased by 5
increased by means “add”
w+5
Additional Example 1: Translating Verbal
Expressions into Algebraic Expressions
Write each phrase as an algebraic expression.
C. the difference of 3 times a number and 7
the difference of 3 times a number and 7
3•x
–7
3x – 7
D. the quotient of 4 and a number, increased by 10
the quotient of 4 and a number, increased by 10
4 + 10
n
Check It Out: Example 1
Write each phrase as an algebraic expression.
A. a number decreased by 10
decreased means “subtract”
n – 10
B. r plus 20
plus means “add”
r + 20
Check It Out: Example 1
Write each phrase as an algebraic expression.
C. the product of a number and 5
the product of a number and 5
n
•5
5n
D. 4 times the difference of y and 8
4 times the difference of y and 8
y – 8
4•
4(y – 8)
When solving real-world problems, you may need to
determine the action to know which operation to use.
Action
Operation
Put parts together
Add
Put equal parts together
Multiply
Find how much more
Subtract
Separate into equal parts
Divide
Additional Example 2A: Translating Real-World
Problems into Algebraic Expressions
Mr. Campbell drives at 55 mi/h. Write an algebraic
expression for how far he can drive in h hours.
You need to put equal parts together. This involves
multiplication.
55mi/h · h hours
=
55h miles
Additional Example 2B: Translating Real-World
Problems into Algebraic Expressions
On a history test Maritza scored 50 points on the essay.
Besides the essay, each short-answer question was worth 2
points. Write an expression for her total points if she
answered q short-answer questions correctly.
The total points include 2 points for each short-answer
question.
Multiply to put equal parts together.
2q
In addition to the points for short-answer questions, the total
points included 50 points on the essay.
Add to put the parts together: 50 + 2q
Check It Out: Example 2A
Julie Ann works on an assembly line building
computers. She can assemble 8 units an hour. Write an
expression for the number of units she can produce in h
hours.
You need to put equal parts together. This involves
multiplication.
8 units/h · h hours
=
8h
Check It Out: Example 2B
At her job Julie Ann is paid $8 per hour. In addition, she
is paid $2 for each unit she produces. Write an
expression for her total hourly income if she produces u
units per hour.
Her total wage includes $2 for each unit produced.
Multiply to put equal parts together.
2u
In addition the pay per unit, her total income includes $8 per
hour.
Add to put the parts together: 2u + 8.
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
Lesson Quiz
Write each phrase as an algebraic expression.
1. 18 less than a number
x – 18
2. the quotient of a number and 21
3. 8 times the sum of x and 15
x
21
8(x + 15)
4. 7 less than the product of a number and 5
5n – 7
5. The county fair charges an admission of $6 and then
charges $2 for each ride. Write an algebraic expression
to represent the total
cost after r rides at the fair.
6 + 2r
Lesson Quiz for Student Response Systems
1. Which of the following is an algebraic expression that
represents the phrase ‘15 less than a number’?
A. x – 15
B. x + 15
C. 15 – x
D. 15x
Lesson Quiz for Student Response Systems
2. Which of the following is an algebraic expression that
represents the phrase ‘the product of a number and
36’?
A. 36x
B. x
36
C.
36
x
D. x + 36
Lesson Quiz for Student Response Systems
3. Which of the following is an algebraic expression that
represents the phrase ‘5 times the sum of y and 17’?
A. 5(y + 17)
B. y + 17
C. 5y + 17
D. 5(y – 17)
Lesson Quiz for Student Response Systems
4. Which of the following is an algebraic expression that
represents the phrase ‘9 less than the product of a
number and 7’?
A. 7x + 9
B. 7x – 9
C. 9x + 7
D. 9x – 7
Lesson Quiz for Student Response Systems
5. A painter charges $675 for labor and $30 per gallon of
paint. Identify an algebraic expression that
represents the total cost of painting, if the painter
used x gallons of paint.
A. 30 + 675x
B. 675x
C. 675 + 30x
D. 30x
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
Warm Up
Evaluate each expression for y = 3.
1. 3y + y
12
2. 7y
21
3. 10y – 4y
18
4. 9y
27
5. y + 5y + 6y
36
Problem of the Day
Emilia saved nickels, dimes, and quarters in a jar.
She had as many quarters as dimes, but twice as
many nickels as dimes. If the jar had 844 coins, how
much money had she saved?
$94.95
Learn to simplify algebraic expressions.
Vocabulary
term
coefficient
In the expression 7x + 9y + 15, 7x, 9y, and 15 are called terms.
A term can be a number, a variable, or a product of numbers
and variables. Terms in an expression are separated by +
and –.
x
2
7x + 5 – 3y + y + 3
term
term
term
term
term
Coefficient
In the term 7x, 7 is called the coefficient.
A coefficient is a number that is
multiplied by a variable in an algebraic
expression. A variable by itself, like
y, has a coefficient of 1. So y = 1y.
Variable
Like terms are terms with the same variables raised to
the same exponents. The coefficients do not have to
be the same. Constants, like 5, 12 , and 3.2, are also
like terms.
Like Terms
Unlike
Terms
3x and 2x
5x2 and 2x
The exponents
are different.
w and
w
7
6a and 6b
The variables
are different
5 and 1.8
3.2 and n
Only one term
contains a
variable
Additional Example 1: Identifying Like Terms
Identify like terms in the list.
3t
5w2 7t
9v
4w2
8v
Look for like variables with like powers.
3t
5w2 7t
9v
4w2
8v
Like terms: 3t and 7t 5w2 and 4w2 9v and 8v
Helpful Hint
Use different shapes or colors to indicate sets of like terms.
Check It Out: Example 1
Identify like terms in the list.
2x
4y3 8x
5z
5y3 8z
Look for like variables with like powers.
2x
4y3
8x
5z
5y3 8z
Like terms: 2x and 8x 4y3 and 5y3 5z and 8z
Combining like terms is like grouping similar objects.
x
x
x
+
x
4x
x
x
+
x
x
5x
x
=
=
x
x
x
x
x
x
9x
To combine like terms that have variables, add or
subtract the coefficients.
x
x
x
Additional Example 2: Simplifying Algebraic
Expressions
Simplify. Justify your steps using the Commutative,
Associative, and Distributive Properties when necessary.
A. 6t – 4t
6t – 4t
6t and 4t are like terms.
2t
Subtract the coefficients.
B. 45x – 37y + 87
In this expression, there are no like terms
to combine.
Additional Example 2: Simplifying Algebraic
Expressions
Simplify. Justify your steps using the Commutative,
Associative, and Distributive Properties when necessary.
C. 3a2 + 5b + 11b2 – 4b + 2a2 – 6
3a2 + 5b + 11b2 – 4b + 2a2 – 6
(3a2 + 2a2) + (5b – 4b) + 11b2 – 6
5a2 + b + 11b2 – 6
Identify like
terms.
Group like
terms.
Add or subtract
the coefficients.
Check It Out: Example 2
Simplify. Justify your steps using the Commutative,
Associative, and Distributive Properties when necessary.
5y + 3y
5y + 3y
8y
5y and 3y are like terms.
Add the coefficients.
Check It Out: Example 2
Simplify. Justify your steps using the Commutative,
Associative, and Distributive Properties when necessary.
C. 4x2 + 4y + 3x2 – 4y + 2x2 + 5
4x2 + 4y + 3x2 – 4y + 2x2 + 5
Identify like
terms.
(4x2 + 3x2 + 2x2)+ (4y – 4y) + 5
Group like
terms.
9x2 + 5
Add or subtract
the coefficients.
Additional Example 3: Geometry Application
Write an expression for the perimeter of the triangle. Then
simplify the expression.
2x + 3
3x + 2
x
2x + 3 + 3x + 2 + x
(x + 3x + 2x) + (2 + 3)
6x + 5
Write an expression using
the side lengths.
Identify and group like terms.
Add the coefficients.
Check It Out: Example 3
Write an expression for the perimeter of the triangle. Then
simplify the expression.
2x + 1
2x + 1
x
x + 2x + 1 + 2x + 1
(x + 2x + 2x) + (1 + 1)
5x + 2
Write an expression using
the side lengths.
Identify and group like terms.
Add the coefficients.
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
Lesson Quiz: Part I
Identify like terms in the list.
1. 3n2 5n 2n3 8n
5n, 8n
2. a5 2a2 a3 3a 4a2
2a2, 4a2
Simplify. Justify your steps using the Commutative,
Associative, and Distributive Properties when necessary.
3. 4a + 3b + 2a
4. x2 + 2y + 8x2
6a + 3b
9x2 + 2y
Lesson Quiz: Part II
5. Write an expression for the perimeter of
figure.
2x + 3y
x+y
x+y
2x + 3y
6x + 8y
the given
Lesson Quiz for Student Response Systems
1. Identify the like terms in the list.
6a, 5a2, 2a, 6a3, 7a
A. 6a and 2a
B. 6a, 5a2, and 6a3
C. 6a, 2a, and 7a
D. 5a2 and 6a3
Lesson Quiz for Student Response Systems
2. Identify the like terms in the list.
16y6, 2y5, 4y2, 10y, 16y2
A. 16y6 and 16y2
B. 4y2 and 16y2
C. 16y6 and 2y2
D. 2y5 and 10y
Identify an expression for the perimeter of the given figure.
A. (4x + 5y)(x + 2y)
B. 4x + 5y
x + 2y
C. 10x + 14y
D. 5x + 7y
How do you write
equivalent
expressions using
the Distributive
Property?
In this lesson you will
learn how to write
equivalent expressions
by using the Distributive
Property.
How do you expand linear expressions
that involve multiplication, addition, and
subtraction?
For example, how do you expand
3(4 + 2x)?
Let’s
Review
Let’s
Review
Vocabulary:
Linear expression
Rational coefficient
Combine like terms
2v + 3 + 7v - 1
= 9v
+2
CoreReview
Lesson
Let’s
If we want to write an equivalent
expression for multiplying 3 times 6, we
can use the Distributive Property.
3(2 + 4) = (3 x 2) + (3 x 4)
CoreReview
Lesson
Let’s
Distributive Property Steps:
S1: Identify the term outside the ( ).
S2: Multiply the outside term by the 1st inside term.
S3: Bring down the correct operation.
S4: Multiply the outside term by the 2nd inside term.
S5: Continue multiplying the outside term by any other
inside terms.
S6: Combine like terms to simplify.
A Common
Let’s
Review Mistake
Forgetting to distribute to the second
term (number). For example,
3(2 + 4) ≠ (3 • 2) + 4
CoreReview
Lesson
Let’s
Expand 3(2x + 4)
2x
3
3(2x + 4) =
x
x
x
+
x
x
x
3(2x)
4
1 1 1 1
2x + 4
1 1 1 1
2x + 4
1 1 1 1
2x + 4
+ 3(4)
= 6x + 12
CoreReview
Lesson
Let’s
4(x + 2) =
x + 2
4
4x
8
4(x + 2) = 4x + 8
CoreReview
Lesson
Let’s
Expand and combine like terms:
11(3a - 2)
3a
11
33a
-2
-22
Guided
Practice
Let’s
Review
Simplify: 9(5k - 8)
Extension
Let’s
Review Activities
Use a diagram to show why
4(3y + 2) = 12y + 8.
Extension
Let’s
Review Activities
Simplify: 5(7y + 1) + 2(9y - 10) + 3(18 - 4y)
Quick
Quiz
Let’s
Review
1. Simplify: 7(3x-4) + 2(5 + 6x)
2. Simplify: 11(4 - 8w) + 6(-9w - 5)
How do you simplify
when there is a
negative term?
-3(2x -5)
In this lesson you will
learn how to simplify an
expression by
distributing a negative
term.
CoreReview
Lesson
Let’s
-4(x + 2) =
x + 2
-4
-4x
-8
-4(x + 2) = -4x - 8
CoreReview
Lesson
Let’s
-3(2x -5)
(-3)(2x) + (-3)(-5)
-6x + 15
CoreReview
Lesson
Let’s
Original Expression
Simplified
Expression
-4(x + 2)
-4x – 8
-3(2x – 5)
-6x + 15
When distributing a negative term, all signs of
the terms inside of the parentheses change.
In this lesson you have learned
how to simplify an expression by
distributing a negative term.
Guided
Practice
Let’s
Review
-3(2x + 4)
Extension
Let’s
Review Activities
A student simplified the following expression
but made a mistake in the process. Identify the
mistake and then explain why it was incorrect.
-4(x + y) = -4x + 4y
Quick
Quiz
Let’s
Review
-3(4x + 7)
-5(-x – 8)
Let’s
Review
Let’s
Review
1. 2(2x – 3) + 3(x + y) + 4(-4x – y) -7x – 6 - y
2. ½(6x – 12) 3x – 6
3. Solve (-23)(4)
-96
4. Solve 2/3 + (-5/8) 1/24
How do you reverse the
distributive property?
-4x + 12 = ?(? +?)
In this lesson you will learn to
reverse the distributive property
by factoring an expression.
Let’s
Review
Let’s
Review
The distributive property tells us that
5(x + 2) = 5x + 10
X
5
5x
+ 2
10
CoreReview
Lesson
Let’s
4x + 12
_____
4x
_____
_______
12
CoreReview
Lesson
Let’s
6x + 24
Correct:
6x + 24 = 6(x + 4)
What is wrong with this one?
6x + 24 = 3(2x + 8)
In this lesson you have learned
how to rewrite an expression by
factoring the expression.
Let’s
Review
Let’s
Review
6x - 9y + 18z
Let’s
Review
Let’s
Review
8a + 12b + 16c
Guided
Practice
Let’s
Review
-3x + 15
Guided
Practice
Let’s
Review
15m -30n + 75p
Quick
Quiz
Let’s
Review
-3x +18
5x - 20
Learn to solve one-step equations with integers.
Inverse Property of Addition
Words
Numbers
Algebra
The sum of a
3 + (–3) = 0 a + (–a ) = 0
number and its
opposite, or additive
inverse, is 0.
Additional Example 1A: Solving Addition and
Subtraction Equations
Solve each question. Check each answer.
–6 + x = –7
Additional Example 1B: Solving Addition and
Subtraction Equations
Solve each equation. Check each answer.
p + 5 = –3
Additional Example 1C: Solving Addition and
Subtraction Equations
Solve each equation. Check each answer.
y – 9 = –40
Check It Out: Example 1A
Solve each equation. Check each answer.
–3 + x = –9
Check It Out: Example 1B
Solve each equation. Check each answer.
q + 2 = –6
Check It Out: Example 1C
Solve each equation. Check each answer.
y – 7 = –34
Additional Example 2A: Solving Multiplication and
Division Equations
Solve each equation. Check each answer.
b =6
–5
Additional Example 2B: Solving Multiplication and
Division Equations
Solve each equation. Check each answer.
–400 = 8y
Check It Out: Example 2A
Solve each equation. Check each answer.
c = –24
4
Check It Out: Example 2B
Solve each equation. Check each answer.
–200 = 4x
Additional Example 3: Business Application
In 2003, a manufacturer made a profit of $300 million. This
amount was $100 million more than the profit in 2002. What
was the profit in 2002?
Let p represent the profit in 2002 (in millions of dollars).
This year’s profit
300
is = 100 million
100
More than
+
300 = 100 + p
–100 –100
200 = p
The profit was $200 million in 2002.
Last year’s profit
p
Check It Out: Example 3
This year the class bake sale made a profit of $243.
This was an increase of $125 over last year. How much
did they make last year?
Let x represent the money they made last year.
This year’s profit
243
is = 125 million
125
More than
+
243 = 125 + x
–125 –125
118 = x
The class earned $118 last year.
Last year’s profit
x
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
Lesson Quiz
Solve each equation. Check your answer.
1. –8y = –800
2. x – 22 = –18
3. –
y
=7
7
4. w + 72 = –21
100
4
–49
–93
5. Last year a phone company had a loss of $25
million. This year the loss is $14 million more
last year. What is this years loss?
$39 million
Lesson Quiz for Student Response Systems
1. Solve the equation.
y + 65= –20
A. y = 45
B. y = 85
C. y = –45
D. y = –85
Lesson Quiz for Student Response Systems
2. Solve the equation.
x – 25= –15
A. x = 10
B. x = 20
C. x = 35
D. x = 45
Lesson Quiz for Student Response Systems
3. Solve the equation.
–10y = –1000
A. y = –200
B. y = –100
C. y = 100
D. y = 200
Lesson Quiz for Student Response Systems
4. Solve the equation.
a
–—=6
9
A. a = 54
B. a = 15
C. a = –15
D. a = –54
Lesson Quiz for Student Response Systems
5. In an online test, Dick scored –34 points. This was
20 points less than his previous score. What was
his previous score?
A. 54 points
B. 14 points
C. –14 points
D. –54 points
Examples of Two-Step Equations
a)
3x – 5 = 16
b) y/4 + 3 = 12
c)
5n + 4 = 6
d) n/2 – 6 = 4
Steps for Solving
Two-Step Equations
1.
Solve for any Addition or Subtraction on the
variable side of equation by “undoing” the operation
from both sides of the equation.
2.
Solve any Multiplication or Division from variable
side of equation by “undoing” the operation from
both sides of the equation.
Opposite Operations
Addition  Subtraction
Multiplication  Division
Helpful Hints?
 Identify what operations are on the variable side.
(Add, Sub, Mult, Div)
 “Undo” the operation by using opposite operations.
 Whatever you do to one side, you must do to the
other side to keep equation balanced.
Ex. 1: Solve 4x – 5 = 11
4x – 5 =
+5
4x
=
4
x=5
15
+5 (Add 5 to both sides)
20 (Simplify)
4 (Divide both sides by 4)
(Simplify)
Try These Examples
1.
2x – 5 = 17
2.
3y + 7 = 25
3.
5n – 2 = 38
4.
12b + 4 = 28
Check your answers!!!
1.
x = 11
2.
y=6
3.
n=8
4.
b=2
Ready to Move on?
Ex. 2: Solve x/3 + 4 = 9
x/3 + 4 = 9
-4 -4
(Subt. 4 from both sides)
x/3 = 5 (Simplify)
(x/3)  3 = 5  3 (Mult. by 3 on both sides)
x = 15
(Simplify)
Try these examples!
1.
x/5 – 3 = 8
2.
c/7 + 4 = 9
3.
r/3 – 6 = 2
4.
d/9 + 4 = 5
Check your answers!!!
1.
x = 55
2.
c = 35
3.
r = 24
4.
d=9
Time to Review!




Make sure your equation is in the form Ax + B = C
Keep the equation balanced.
Use opposite operations to “undo”
Follow the rules:
1. Undo Addition or Subtaction
2. Undo Multiplication or Division
x5
x5
2
3 4 5 6
7
8
Inequalities and their Graphs
Objective:
with one variable
To write and graph simple inequalities
What is a good definition for Inequality?
An inequality is a statement that
two expressions are not equal
2
3 4 5 6
7
8
Inequalities and their Graphs
Terms you see and need to know to graph inequalities correctly
< less than
Notice
> greater than
open
circles
Inequalities and their Graphs
Terms you see and need to know to graph inequalities correctly
≤ less than or equal to
≥ greater than or equal to
Notice colored in circles
Inequalities and their Graphs
Let’s work a few together
x3
Notice: when variable is
on left side, sign shows
direction of solution
3
Inequalities and their Graphs
Let’s work a few together
x7
Notice: when variable is
on left side, sign shows
direction of solution
7
Inequalities and their Graphs
Let’s work a few together
p  2
Notice: when variable is
on left side, sign shows
direction of solution
-2
Color in
circle
Inequalities and their Graphs
Let’s work a few together
x 8
Color in circle
Notice: when variable is
on left side, sign shows
direction of solution
8
Try this one on your own
x  12
12
On your Own
s  12
12
On your Own
m2
2
Answer 1 through 5
1.
b  6
On your Own
2.
x  10
On your Own
3.
p  15
On your Own
4.
g  25
On your Own
5.
x  13
Answers: 1 through 5
1.
-6
3.
2.
10
4.
25
-15
5.
13
Answer 6 through 10
6.
b ≤ -7
On your Own
7.
a>8
On your Own
8.
q ≥ -5
On your Own
9.
s < 14
On your Own
10.
m≥8
Answers: 6 through 10
6.
7.
-7
8.
8
9.
-5
14
10.
8
Graphing
Inequalities
Excellent Job !!! Well Done