Alg1ch1

Foundations for Algebra
Chapter 1
Part A

Essential Questions
• How can you represent quantities, patterns,
and relationships?
• How are properties related to algebra?

Goals
Goal
• Write and evaluate
expressions with unknown
values.
• Use properties to simplify
expressions.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.

Content
• 1-1 Variables and Expressions
• 1-2 Part 1 Order of Operations
• 1-2 Part 2 Evaluating Expressions
• 1-3 Real Numbers and the Number Line
• 1-4 Properties of Real Numbers

Variables and Expressions
Section 1-1

Goals
Goal
• To write algebraic
expressions.
Rubric
goals.
simple problems.
Level 5 – Adapts and applies the
goals to different and more
complex problems.

Vocabulary
• Quantity
• Variable
• Algebraic expression
• Numerical expression

Definition
• Quantity – A mathematical quantity is anything
that can be measured or counted.
– How much there is of something.
– A single group, generally represented in an
expression using parenthesis () or brackets [].
• Examples:
– numbers, number systems, volume, mass, length,
people, apples, chairs.
– (2x + 3), (3 – n), [2 + 5y].

Definition
• Variable – anything that can vary or change in value.
– In algebra, x is often used to denote a variable.
– Other letters, generally letters at the end of the alphabet (p,
q, r, s, t, u, v, w, x, y, and z) are used to represent variables
– A variable is “just a number” that can change in value.
• Examples:
– A child’s height
– Outdoor temperature
– The price of gold

Definition
• Constant – anything that does not vary or change in value
(a number).
– In algebra, the numbers from arithmetic are constants.
– Generally, letters at the beginning of the alphabet (a, b, c,
d)used to represent constants.
• Examples:
– The speed of light
– The number of minutes in a hour
– The number of cents in a dollar
– π.

Definition
• Term – any number that is added subtracted.
– In the algebraic expression x + y, x and y are
terms.
• Example:
– The expression x + y – 7 has 3 terms, x, y, and 7.
x and y are variable terms; their values vary as x
and y vary. 7 is a constant term; 7 is always 7.

Definition
• Factor – any number that is multiplied.
– In the algebraic expression 3x, x and 3 are
factors.
• Example:
– 5xy has three factors; 5 is a constant factor, x and
y are variable factors.

Example: Terms and Factors
• The algebraic expression 5x + 3;
– has two terms 5x and 3.
– the term 5x has two factors, 5 and x.

Definition
• Numerical Expression – a mathematical
phrase that contains only constants and/or
operations.
• Examples: 2 + 3, 5 ∙ 3 – 4, 4 + 20 – 7, (2 + 3)
– 7, (6 × 2) ÷ 20, 5 ÷ (20 × 3)

Multiplication Notation
In expressions, there are many different ways to write
multiplication.
1) ab
2) a • b
3) a(b) or (a)b
4) (a)(b)
5) a ⤫ b
We are not going to use the multiplication symbol (⤫) any more. Why?
Can be confused with the variable x.

Translate Words into
Expressions
• To Translate word phrases into algebraic
expressions, look for words that describe
mathematical operations (addition,
subtraction, multiplication, division).

What words indicate a particular
operation?
Addition
• Sum
• Plus
• More than
• Increase(d) by
• Perimeter
• Deposit
• Gain
• Greater (than)
• Total
Subtraction
• Minus
• Take away
• Difference
• Reduce(d) by
• Decrease(d) by
• Withdrawal
• Less than
• Fewer (than)
• Loss of

Words for Operations - Examples

What words indicate a particular
operation?
Multiply
• Times
• Product
• Multiplied by
• Of
• Twice (×2), double (×2),
triple (×3), etc.
• Half (×½), Third (×⅓),
Quarter (×¼)
• Percent (of)
Divide
• Quotient
• Divided by
• Half (÷2), Third (÷3), Quarter
(÷4)
• Into
• Per
• Percent (out of 100)
• Split into __ parts

Writing an algebraic expression with addition.
2
Two plus a number n
+ n
2 + n
Writing an Algebraic Expression
for a Verbal Phrase.
Order
of the
wording
Matters

Writing an algebraic expression with addition.
2
Two more than a number
+x
x + 2
Order
of the
wording
Matters

Writing an algebraic expression with subtraction.
–
The difference of seven and a number n
7 n
7 – n
Order
of the
wording
Matters

Writing an algebraic expression with subtraction.
8
Eight less than a number
–y
y – 8
Order
of the
wording
Matters

Writing an algebraic expression with multiplication.
1/3
one-third of a number n.
· n
Order
of the
wording
Matters

Writing an algebraic expression with division.
The quotient of a number n and 3
n 3
Order
of the
wording
Matters

Example
“Translating” a phrase into an algebraic expression:
Nine more than a number y
Can you identify the operation?
“more than” means add
Answer: y + 9

Example
4 less than a number n
Identify the operation?
“less than” means add
Answer: n – 4.
Why not 4 – n?????
Determine the order of the variables and constants.

Example
A quotient of a number x and12
Can you identify the operation?
“quotient” means divide
Determine the order of the variables and constants.
Answer: .
Why not ?????

Example
“Translating” a phrase into an algebraic expression,
this one is harder……
5 times the quantity 4 plus a number c
Can you identify the operation(s)?
What does the word quantity mean?
“times” means multiple and “plus” means add
that “4 plus a number c” is grouped using
parenthesis
Answer: 5(4 + c)

Your turn:
1) m increased by 5.
2) 7 times the product of
x and t.
3) 11 less than 4 times a
number.
4) two more than 6 times
a number.
5) the quotient of a
number and 12.
1) m + 5
2) 7xt
3) 4n - 11
4) 6n + 2
5)

Your Turn:
a. 7x + 13
b. 13 - 7x
c. 13 + 7x
d. 7x - 13
Which of the following expressions represents
7 times a number decreased by 13?

Your Turn:
1. 28 - 3x
2. 3x - 28
3. 28 + 3x
4. 3x + 28
Which one of the following expressions represents 28
less than three times a number?

Your Turn:
1. Twice the sum of x and three
D
2. Two less than the product of 3 and x
E
3. Three times the difference of x and two
B
4. Three less than twice a number x
A
5. Two more than three times a number x
C
A.2x – 3
B.3(x – 2)
C.3x + 2
D.2(x + 3)
E.3x – 2
Match the verbal phrase and the expression

Translate an Algebraic
Expression into Words
• We can also start with an algebraic expression
and then translate it into a word phrase using
the same techniques, but in reverse.
• Is there only one way to write a given
algebraic expression in words?
– No, because the operations in the expression can
be described by several different words and
phrases.

Give two ways to write each algebra expression in words.
A. 9 + r B. q – 3
the sum of 9 and r
9 increased by r
the product of m and 7
m times 7
the difference of q and 3
3 less than q
the quotient of j and 6
j divided by 6
C. 7m D.
Example: Translating from
Algebra to Words

a. 4 - n b.
c. 9 + q d. 3(h)
4 decreased by n
the sum of 9 and q
the quotient of t and 5
the product of 3 and h
Give two ways to write each algebra expression in words.
Your Turn:
n less than 4 t divided by 5
q added to 9 3 times h

Your Turn:
1. 9 increased by twice a number
2. a number increased by nine
3. twice a number decreased by 9
4. 9 less than twice a number
Which of the following verbal expressions
represents 2x + 9?

Your Turn:
1. 5x - 16
2. 16x + 5
3. 16 + 5x
4. 16 - 5x
Which of the following expressions represents the
sum of 16 and five times a number?

Your Turn:
• 4(x + 5) – 2
• Four times the sum of x and 5 minus two
• 7 – 2(x ÷ 3)
• Seven minus twice the quotient of x and three
• m ÷ 9 – 4
• The quotient of m and nine, minus four
CHALLENGE
Write a verbal phrase that describes the expression

Your Turn:
• Six miles more than yesterday
• Let x be the number of miles for yesterday
• x + 6
• Three runs fewer than the other team scored
• Let x = the amount of runs the other team scored
• x - 3
• Two years younger than twice the age of your cousin
• Let x = the age of your cousin
• 2x – 2
Define a variable to represent the unknown and write the phrase
as an expression.

Patterns
Mathematicians …
• look for patterns
• find patterns in physical or
pictorial models
• look for ways to create
different models for
patterns
• use mathematical models to
solve problems
Numerical
Graphical

Number Patterns
2
2 + 2
2 + 2 + 2
2 + 2 + 2 + 2 4(2)
3(2)
2(2)
1(2)1
2
3
4
n?
__(2)
Term
Number
n
2
4
6
8
Term Expression

Number Patterns
6(5) + 4
5(5) + 4
4(5) + 4
3(5) + 41
2
3
4
n? _____(5) + 4(n + 2)
How does
the
different
part relate
to the term
number?
What’s the
same?
What’s
different?
19
24
29
34
Term
Number Term Expression

Number Patterns
3 - 2(3)
3 - 2(2)
3 - 2(1)
3 - 2(0)1
2
3
4
n? 3 - 2(____)n - 1
How does
the
different
part relate
to the term
number?
What’s the
same?
What’s
different?
3
1
-1
-3
Term
Number Term Expression

Writing a Rule to Describe a
Pattern
• Now lets try a real-life problem.

Bonjouro! My name is Fernando
I am preparing to cook a GIGANTIC
home-cooked Italian meal for my family.
The only problem is I don’t know yet how
many people are coming. The more people
that come, the more spaghetti I will need to
buy.

From all the meals I have cooked
before I know:
For 1 guest I will need 2 bags of spaghetti,
For 2 guests I will need 5 bags of spaghetti,
For 3 guests I will need 8 bags of spaghetti,
For 4 guests I will need 11 bags of spaghetti.

Here is the table of how
many bags of spaghetti I will
need to buy:
Number of
Guests
Bags of
Spaghetti
1
2
3
4
2
5
8
11

The numbers in the ‘spaghetti’ column make a
pattern:
2 5 8 11
What do we need to add on each time to get to
the next number?
+ 3 + 3 + 3

We say there is a
COMMON DIFFERENCE
between the numbers.
We need to add on the same number
every time.
What is the common difference for
this sequence?
3

Now we know the common difference
we can start to work out the
MATHEMATICAL RULE.
The mathematical rule is the algebraic
expression that lets us find any value in
our pattern.

We can use our common difference to help us
find the mathematical rule.
We always multiply the common difference by
the TERM NUMBER to give us the first step of
our mathematical rule.
What are the term numbers in my case are?
NUMBER OF GUESTS

So if we know that step one of finding the
mathematical rule is:
Common Term
Difference Numbers
then what calculations will we do in this
example?
X
Common Difference Term NumbersX
X3 Number of Guests

We will add a column to our original table
to do these calculations:
Number of
Guests
(n)
Bags of
Spaghetti
1
2
3
4
2
5
8
11
3
6
9
12
3n

We are trying to find a
mathematical rule that will take
us from:
Number of Guests
Number of Bags of Spaghetti
At the moment we have:
3n
Does this get us the answer we
want?

3n gives us: Bags of Spaghetti
3 2
6 5
9 8
12 11
What is the difference between all the
numbers on the left and all the
numbers on the right?
-1
-1
-1
-1
-1

We will now add another column to our
table to do these calculations:
Number of
Guests
(n)
Bags of
Spaghetti
1
2
3
4
2
5
8
11
3
6
9
12
3n 3n – 1
2
5
8
11

Does this new column get us to where
we are trying to go?
So now we know our mathematical
rule:
3n –1

Your Turn:
• The table shows how the cost of renting a scooter
depends on how long the scooter is rented. What is
a rule for the total cost? Give the rule in words and
as an algebraic expression.
Hours Cost
1 $17.50
2 $25.00
3 $32.50
4 $40.00
5 $47.50
Answer:
Multiply the number of
hours by 7.5 and add 10.
7.5n + 10

Order of Operations and
Evaluating Expressions
Section 1-2 Part 1

Goals
Goal
• To simplify expressions
involving exponents.
Rubric
goals.
simple problems.
complex problems.

Vocabulary
• Power
• Exponent
• Base
• Simplify

Definition
A power expression has two parts, a base and an
exponent.
103
Power expression
Exponent
Base

Power
In the power expression 103, 10 is called the base and
3 is called the exponent or power.
103 means 10 • 10 • 10
103 = 1000
The base, 10, is the number that
is used as a factor. 103 The exponent, 3, tells
how many times the
base, 10, is used as a
factor.

Definition
• Base – In a power expression, the base is the
number that is multiplied repeatedly.
• Example:
– In x3, x is the base. The exponent says to
multiply the base by itself 3 times; x3 = x ⋅ x ⋅ x.

Definition
• Exponent – In a power expression, the exponent
tells the number of times the base is used as a
factor.
• Example:
– 24 equals 2 ⋅ 2 ⋅ 2 ⋅ 2.
– If a number has an exponent of 2, the number is often
called squared. For example, 42 is read “4 squared.”
– Similarly, a number with an exponent of is called
“cubed.”

When a number is raised to the second power, we usually
say it is “squared.” The area of a square is s  s = s2, where
s is the side length.
s
s
When a number is raised to the third power, we usually say
it is “cubed.” The volume of a cube is s  s  s = s3, where s
is the side length.
s
s
s
Powers

There are no easy geometric models for numbers raised to exponents greater than 3, but
you can still write them using repeated multiplication or with a base and exponent.
3 to the second power, or 3
squared
3  3  3  3  3
Multiplication Power ValueWords
3  3  3  3
3  3  3
3  3
33 to the first power
3 to the third power, or 3 cubed
3 to the fourth power
3 to the fifth power
3
9
27
81
243
31
32
33
34
35
Reading Exponents
Powers

Caution!
In the expression –5², 5 is the base because the
negative sign is not in parentheses. In the
expression (–2)³, –2 is the base because of the
parentheses.

Definition
• Simplify – a numerical expression is
simplified when it is replaced with its single
numerical value.
• Example:
– The simplest form of 2 • 8 is 16.
– To simplify a power, you replace it with its
simplest name. The simplest form of 23 is 8.

Example: Evaluating Powers
Simplify each expression.
A. (–6)3
(–6)(–6)(–6)
–216
Use –6 as a factor 3 times.
B. –102
–1 • 10 • 10
–100
Think of a negative sign in
front of a power as
multiplying by a –1.
Find the product of –1 and
two 10’s.

Example: Evaluating Powers
Simplify the expression.
C.
2
9
 2
9
=
4
81
2
9
 2
9
Use as a factor 2 times.2
9

Your Turn:
Evaluate each expression.
a. (–5)3
(–5)(–5)(–5)
–125
Use –5 as a factor 3 times.
b. –62
–1  6  6
–36
Think of a negative sign in
front of a power as
multiplying by –1.
Find the product of –1 and
two 6’s.

Your Turn:
Evaluate the expression.
c.
27
64
Use as a factor 3 times.3
4

Example: Writing Powers
Write each number as a power of the given base.
A. 64; base 8
8  8
82
The product of two 8’s is 64.
B. 81; base –3
(–3)(–3)(–3)(–3)
(–3)4
The product of four –3’s is 81.

Your Turn:
Write each number as a power of a given base.
a. 64; base 4
4  4  4
43
The product of three 4’s is 64.
b. –27; base –3
(–3)(–3)(–3)
–33
The product of three (–3)’s is –27.

Order of Operations
Rules for arithmetic and algebra
expressions that describe what
sequence to follow to evaluate an
expression involving more than
one operation.

Order of Operations
Is your answer 33 or 19?
You can get 2 different answers depending on
which operation you did first. We want
everyone to get the same answer so we
must follow the order of operations.
Evaluate 7 + 4 • 3.

Remember the phrase
“Please Excuse My Dear Aunt Sally”
or PEMDAS.
ORDER OF OPERATIONS
1. Parentheses - ( ) or [ ]
2. Exponents or Powers
3. Multiply and Divide (from left to right)
4. Add and Subtract (from left to right)

The Rules
Step 1: First perform operations that are within grouping
symbols such as parenthesis (), brackets [], and braces {},
and as indicated by fraction bars. Parenthesis within
parenthesis are called nested parenthesis (( )). If an
expression contains more than one set of grouping symbols,
evaluate the expression from the innermost set first.
Step 2: Evaluate Powers (exponents) or roots.
Step 3: Perform multiplication or division operations in order
by reading the problem from left to right.
Step 4: Perform addition or subtraction operations in order by
reading the problem from left to right.

Method 1 Method 2
Performing operations left to right only
Performing operations using order of
operations
The rules for order
of operations exist
so that everyone
can perform the
same consistent
operations and
achieve the same
results. Method 2 is
the correct method.
Can you imagine what
it would be like if
calculations were
performed differently
by various financial
institutions or what if
doctors prescribed
different doses of
medicine using the
same formulas and
achieving different
results?
Order of Operations

Follow the left to right rule: First solve any
multiplication or division parts left to right. Then
solve any addition or subtraction parts left to right.
A good habit to develop while learning order of operations is to
underline the parts of the expression that you want to solve
first. Then rewrite the expression in order from left to right and
solve the underlined part(s).
The order of operations must be followed
each time you rewrite the expression.
Divide
Multiply
Add
Order of Operations:
Example 1
Evaluate without grouping
symbols

Exponents (powers)
Multiply
Subtract
Follow the left to right rule: First solve
exponent/(powers). Second solve multiplication or
division parts left to right. Then solve any addition or
subtraction parts left to right.
A good habit to develop while learning order of operations is
to underline the parts of the expression that you want to solve
first. Then rewrite the expression in order from left to right and
solve the underlined part(s).
Example 2
Expressions with powers

Exponents (powers)
Multiply
Subtract
Follow the left to right rule: First solve parts inside
grouping symbols according to the order of
operations. Solve any exponent/(Powers). Then
solve multiplication or division parts left to right.
Then solve any addition or subtraction parts left to
right.
A good habit to develop while learning
order of operations is to underline the parts
of the expression that you want to solve
first. Then rewrite the expression in order
from left to right and solve the underlined
part(s).
Grouping
symbols
Divide
Example 3
Evaluate with grouping symbols

Exponents (powers)
Multiply
Subtract
Follow the left to right rule: Follow the order of
operations by working to solve the problem above
the fraction bar. Then follow the order of operations
by working to solve the problem below the fraction
bar. Finally, recall that fractions are also division
problems – simplify the fraction.
A good habit to develop while learning order of operations is to underline the parts of the expression
that you want to solve first. Then rewrite the expression in order from left to right and solve the
underlined part(s).
Work above the
fraction bar
Simplify:
Divide
Work below the
fraction bar Grouping symbols
Add
Example 4
Expressions with fraction bars

Your Turn:
8 ÷ · 3
1 2
8 ÷ · 3
1 2
16 · 3
48
There are no grouping
symbols.
Divide.
Multiply.

Your Turn:
5.4 – 32 + 6.2
5.4 – 32 + 6.2
5.4 – 9 + 6.2
–3.6 + 6.2
2.6
There are no grouping
symbols.
Simplify powers.
Subtract
Add.

Your Turn:
–20 ÷ [–2(4 + 1)]
–20 ÷ [–2(4 + 1)]
–20 ÷ [–2(5)]
–20 ÷ –10
2
There are two sets of grouping
symbols.
Perform the operations in the
innermost set.
Perform the operation inside
the brackets.
Divide.

Your Turn:
1. -3,236
2. 4
3. 107
4. 16,996
Which of the following represents 112 + 18 - 33 · 5 in
simplified form?

Your Turn:
1. 2
2. -7
3. 12
4. 98
Simplify 16 - 2(10 - 3)

Your Turn:
1. 72
2. 36
3. 12
4. 0
Simplify 24 – 6 · 4 ÷ 2

Caution!
Fraction bars, radical symbols, and absolute-value
symbols can also be used as grouping symbols.
Remember that a fraction bar indicates division.

Your Turn:
Simplify.
5 + 2(–8)
(–2) – 33
5 + 2(–8)
(–2) – 33
5 + 2(–8)
–8 – 3
5 + (–16)
– 8 – 3
–11
–11
1
The fraction bar acts as a grouping symbol.
Simplify the numerator and the denominator
before dividing.
Evaluate the power in the denominator.
Multiply to simplify the numerator.
Add.
Divide.

Your Turn:
Simplify.
2(–4) + 22
42 – 9
2(–4) + 22
42 – 9
–8 + 22
42 – 9
–8 + 22
16 – 9
14
7
2
The fraction bar acts as a grouping symbol.
Simplify the numerator and the denominator
before dividing.
Multiply to simplify the numerator.
Evaluate the power in the denominator.
Add to simplify the numerator. Subtract to
simplify the denominator.
Divide.

Order of Operations and
Section 1-2 Part 2

Goals
Goal
• To use the order of operations
to evaluate expressions.
Rubric
goals.
simple problems.
complex problems.

• In Part 1 of this lesson, we simplified
numerical expressions with exponents and
learned the order of operations.
• Now, we will evaluate algebraic expressions
for given values of the variable.

Definition
• Evaluate – To evaluate an expression is to
find its value.
• To evaluate an algebraic expression,
substitute numbers for the variables in the
expression and then simplify the expression.

Example: Evaluating Algebraic
Expressions
Evaluate each expression for a = 4, b =7, and
c = 2.
A. b – c
b – c = 7 – 2
= 5
B. ac
ac = 4 ·2
= 8
Substitute 7 for b and 2 for c.
Simplify.
Substitute 4 for a and 2 for c.
Simplify.

Your Turn:
Evaluate each expression for m = 3, n = 2, and
p = 9.
a. mn
b. p – n
c. p ÷ m
Substitute 3 for m and 2 for n.mn = 3 · 2
Simplify.= 6
Substitute 9 for p and 2 for n.p – n = 9 – 2
Simplify.= 7
Substitute 9 for p and 3 for m.p ÷ m = 9 ÷ 3
Simplify.

Expressions
Evaluate the expression for the given value
of x.
10 – x · 6 for x = 3
First substitute 3 for x.10 – x · 6
10 – 3 · 6 Multiply.
10 – 18 Subtract.
–8

Expressions
Evaluate the expression for the given value of x.
42(x + 3) for x = –2
First substitute –2 for x.42(x + 3)
42(–2 + 3)
Perform the operation inside
the parentheses.42(1)
Evaluate powers.16(1)
Multiply.16

Your Turn:
14 + x2 ÷ 4 for x = 2
14 + x2 ÷ 4
First substitute 2 for x.14 + 22 ÷ 4
Square 2.14 + 4 ÷ 4
Divide.14 + 1
Add.15

Your Turn:
(x · 22) ÷ (2 + 6) for x = 6
(x · 22) ÷ (2 + 6)
First substitute 6 for x.(6 · 22) ÷ (2 + 6)
Square two.(6 · 4) ÷ (2 + 6)
Perform the operations inside the
parentheses.
(24) ÷ (8)
Divide.3

Your Turn:
1. -62
2. -42
3. 42
4. 52
What is the value of
-10 – 4x if x = -13?

Your Turn:
1. -8000
2. -320
3. -60
4. 320
5k3 if k = -4?

Your Turn:
1. 10
2. -10
3. -6
4. 6
if n = -8, m = 4, and t = 2 ?

Example: Application
A shop offers gift-wrapping services at three price levels. The
amount of money collected for wrapping gifts on a given day
can be found by using the expression 2B + 4S + 7D. On
Friday the shop wrapped 10 Basic packages B, 6 Super
packages S, and 5 Deluxe packages D. Use the expression to
find the amount of money collected for gift wrapping on
Friday.

Example - Solution:
2B + 4S + 7D
First substitute the value for
each variable.
2(10) + 4(6) + 7(5)
Multiply.20 + 24 + 35
Add from left to right.44 + 35
Add.79
The shop collected $79 for gift wrapping on Friday.

Your Turn:
Another formula for a player's total number of bases is Hits + D + 2T + 3H. Use
this expression to find Hank Aaron's total bases for 1959, when he had 223 hits,
46 doubles, 7 triples, and 39 home runs.
Hits + D + 2T + 3H = total number of bases
First substitute values for each
variable.
223 + 46 + 2(7) + 3(39)
Multiply.223 + 46 + 14 + 117
Add.400
Hank Aaron’s total number of bases for 1959 was 400.

USING A VERBAL MODEL
Use three steps to write a mathematical model.
WRITE A
VERBAL MODEL.
ASSIGN
LABELS.
WRITE AN ALGEBRAIC
MODEL.
Writing algebraic expressions that represent real-life
situations is called modeling.
The expression is a mathematical model.

A PROBLEM SOLVING PLAN USING MODELS
Writing an Algebraic Model
Ask yourself what you need to know to solve the
problem. Then write a verbal model that will give
you what you need to know.
Assign labels to each part of your verbal
problem.
Use the labels to write an algebraic model based
on your verbal model.
VERBAL
MODEL
problem.
ALGEBRAIC
MODEL
LABELS

Write an expression for the number of
bottles needed to make s sleeping bags.
The expression 85s models the number of
bottles to make s sleeping bags.
Approximately eighty-five 20-ounce plastic
bottles must be recycled to produce the fiberfill
for a sleeping bag.

Continued
Approximately eighty-five 20-ounce plastic
bottles must be recycled to produce the fiberfill
for a sleeping bag.
Find the number of bottles needed to make
20, 50, and 325 sleeping bags.
Evaluate 85s for s = 20, 50, and 325.
s 85s
20
50
325
85(20) = 1700
To make 20 sleeping bags
1700 bottles are needed.
85(50) = 4250
4250 bottles are needed.
85(325) = 27,625
27,625 bottles are needed.

Your Turn:
Write an expression for the number of
bottles needed to make s sweaters.
The expression 63s models the number of
bottles to make s sweaters.
To make one sweater, 63 twenty ounce
plastic drink bottles must be recycled.

Your Turn: Continued
To make one sweater, 63 twenty ounce
plastic drink bottles must be recycled.
Find the number of bottles needed to make
12, 25 and 50 sweaters.
Evaluate 63s for s = 12, 25, and 50.
s 63s
12
25
50
63(12) = 756
To make 12 sweaters 756
bottles are needed.
63(25) = 1575
bottles are needed.
63(50) = 3150
bottles are needed.

Real Numbers and the Number
Line
Section 1-3

Goals
Goal
• To classify, graph, and
compare real numbers.
• To find and estimate square
roots.
Rubric
goals.
simple problems.
complex problems.

Vocabulary
• Square Root
• Radicand
• Radical
• Perfect Square
• Set
• Element of a Set
• Subset
• Rational Numbers
• Natural Numbers
• Whole Numbers
• Integers
• Irrational Numbers
• Real Numbers
• Inequality

Opposite of squaring a number is taking the
square root of a number.
A number b is a square root of a number a if b2
= a.
In order to find a square root of a, you need a #
that, when squared, equals a.
Square Roots

The principal (positive) square root is noted as
The negative square root is noted as
Principal Square Roots
Any positive number has two real square roots, one
positive and one negative, √x and -√x
√4 = 2 and -2, since 22 = 4 and (-2)2 = 4

Radical expression is an expression containing a
radical sign.
Radicand is the expression under a radical sign.
Note that if the radicand of a square root is a
negative number, the radical is NOT a real number.
Radicand

Square roots of perfect square radicands
simplify to rational numbers (numbers that
can be written as a quotient of integers).
Square roots of numbers that are not perfect
squares (like 7, 10, etc.) are irrational
numbers.
IF REQUESTED, you can find a decimal
approximation for these irrational numbers.
Otherwise, leave them in radical form.
Perfect Squares

Perfect Squares
The terms of the following sequence:
1, 4, 9, 16, 25, 36, 49, 64, 81…
12,22,32,42, 52 , 62 , 72 , 82 , 92…
These numbers are called the Perfect
Squares.

The small number to the left of the root is the
index. In a square root, the index is understood to
be 2. In other words, is the same as .
Writing Math

Roots
A number that is raised to the third power to form a
product is a cube root of that product. The symbol
indicates a cube root. Since 23 = 8,
= 2. Similarly, the symbol indicates a fourth
root: 24 = 16, so = 2.

Example: Finding Roots
Find each root.
Think: What number squared equals 81?
Think: What number squared equals 25?

Example: Finding Roots
Find the root.
C.
Think: What number cubed equals
–216?
(–6)(–6)(–6) = 36(–6) = –216= –6

Your Turn:
Find each root.
a.
b.
Think: What number squared
equals 4?
equals 25?

Your Turn:
Find the root.
c.
Think: What number to the fourth
power equals 81?

Example: Finding Roots of Fractions
Find the root.
A.
equals

Find the root.
B.
Think: What number cubed equals

Find the root.
C.
equals

Your Turn:
Find the root.
a.
equals

Your Turn:
Find the root.
b.
Think: What number cubed
equals

Your Turn:
Find the root.
c.
equals

Roots and Irrational Numbers
Square roots of numbers that are not perfect squares,
such as 15, are irrational numbers. A calculator can
approximate the value of as 3.872983346...
Without a calculator, you can use square roots of
perfect squares to help estimate the square roots of
other numbers.

As part of her art project, Shonda will need to
make a paper square covered in glitter. Her
tube of glitter covers 13 in². Estimate to the
nearest tenth the side length of a square with
an area of 13 in².
Since the area of the square is 13 in², then each
side of the square is in. 13 is not a perfect
square, so find two consecutive perfect squares
that is between: 9 and 16. is between
and , or 3 and 4. Refine the estimate.

Example: Application Continued
Because 13 is closer to 16 than to 9,
is closer to 4 than to 3.
3 4
You can use a guess-and-check
method to estimate .

Example: Application Continued
3.63 3.7 4
Guess 3.6: 3.62 = 12.96
too low
is greater than 3.6.
Guess 3.7: 3.72 = 13.69
too high
is less than 3.7.
Because 13 is closer to 12.96 than to
13.69, is closer to 3.6 than to 3.7.  3.6

The symbol ≈ means “is approximately equal to.”
Writing Math

Your Turn:
What if…? Nancy decides to buy more wildflower
seeds and now has enough to cover 26 ft2.
Estimate to the nearest tenth the side length of a
square garden with an area of 26 ft2.
Since the area of the square is 26 ft², then each
side of the square is ft. 26 is not a perfect
square, so find two consecutive perfect squares
that is between: 25 and 36. is between
and , or 5 and 6. Refine the estimate.

Solution Continued
5.0 5.02 = 25 too low
5.1 5.12 = 26.01 too high
Since 5.0 is too low and 5.1 is too high, is between
5.0 and 5.1. Rounded to the nearest tenth,
 5.1.
The side length of the square garden is  5.1 ft.

•A set is a collection of objects.
–These objects can be anything: Letters, Shapes, People,
Numbers, Desks, cars, etc.
–Notation: Braces ‘{ }’, denote “The set of …”
•The objects in a set are called elements of the set.
•For example, if you define the set as all the fruit
found in my refrigerator, then apple and orange would
be elements or members of that set.
•A subset of a set consists of elements from the given
set. A subset is part of another set.
Sets:

Definitions: Number Sets
• Natural numbers are the counting numbers: 1,
2, 3, …
• Whole numbers are the natural numbers and
zero: 0, 1, 2, 3, …
• Integers are whole numbers and their opposites:
–3, –2, –1, 0, 1, 2, 3, …
• Rational numbers can be expressed in the
form , where a and b are both integers and b
≠ 0: , , .
a
b
1
2
7
1
9
10

Definitions: Number Sets
• Terminating decimals are rational numbers in
decimal form that have a finite number of digits: 1.5,
2.75, 4.0
• Repeating decimals are rational numbers in decimal
form that have a block of one or more digits that
repeat continuously: 1.3, 0.6, 2.14
• Irrational numbers cannot be expressed in the form
a/b. They include square roots of whole numbers that
are not perfect squares and nonterminating decimals
that do not repeat: , , 

Rational or Not Rational?
1. 3.454545…
2. 1.23616161…
3. 0.1010010001…
4. 0.34251
5. π
Rational
Rational
Irrational
Rational
Irrational

All numbers that can be represented on a number line are called real
numbers and can be classified according to their characteristics.
Number Sets

Number Sets - Notation
• Ν Natural Numbers - Set of positive integers {1,2,3,…}
• W Whole Numbers - Set of positive integers & zero
{0,1,2,3,…}
• Z Set of integers {0,±1,±2,±3,…}
• Q Set of rational numbers
{x: x=a/b, b≠0 ∩ aєΖ, bєΖ}
• Q Set of irrational numbers
{x: x is not rational}
• R Set of real numbers (-∞,∞)

Example: State all numbers sets to
which each number belongs?
1. 2/3
2. √4
3. π
4. -3
5. √21
6. 1.2525…
1. Rational, real
2. Natural, integer,
rational, real
3. Irrational, real
4. Integer, rational, real
5. Irrational, real
6. Rational, real

-5 50 10-10
Number Lines
• A number line is a line with marks on it that are
placed at equal distances apart.
• One mark on the number line is usually labeled
zero and then each successive mark to the left or
to the right of the zero represents a particular unit
such as 1 or ½.
• On the number line above, each small mark
represents ½ unit and the larger marks represent 1

– 4 – 3 – 2 – 1 0 1 2 3 4
| | | | | | | | |
Negative numbers Positive numbers
Zero is neither negative
nor positive
Whole Numbers
Integers
Rational Numbers on a Number Line

Definition
• Inequality – a mathematical sentence that
compares the values of two expressions
using an inequality symbol..
• The symbols are:
– <, less than
– ≤, less than or equal to
– >, Greater than
– ≥, Greater than or equal to

Comparing the position of two numbers on
the number line is done using inequalities.
a < b means a is to the left of b
a = b means a and b are at the same location
a > b means a is to the right of b
Inequalities can also be used to describe the sign
of a real number.
a > 0 is equivalent to a is positive.
a < 0 is equivalent to a is negative.

Comparing Real Numbers
• We compare numbers in order by their location on the
number line.
• Graph –4 and –5 on the number line. Then write two
inequalities that compare the two numbers.
• Put –1, 4, –2, 1.5 in increasing order
0 1 2 3 4 5 6 7 8 9 10 11-3 -2 -1-4-5-6-7-8-9-10
–4 > –5 or –5 < –4
0 1 2 3 4 5 6 7 8 9 10 11-3 -2 -1-4-5-6-7-8-9-10
Since –5 is farther left, we say
–2, –1, 1.5, 4Left to right

• Write the following set of numbers in increasing
order:
–2.3, –4.8, 6.1, 3.5, –2.15, 0.25, 6.02
Your Turn:
–4.8, –2.3, –2.15, 0.25, 3.5, 6.02, 6.1
0 1 2 3 4 5 6 7 8 9 10 11-3 -2 -1-4-5-6-7-8-9-10

Comparing Real Numbers
• To compare real numbers rewrite all the
numbers in decimal form.
• To convert a fraction to a decimal,
• Write each set of numbers in increasing order.
a. b.
• YOU TRY c and d!
c. –3, -3.2, -3.15, -3.001, 3 d.
Divide the numerator by the denominator

Example: Comparing Real Numbers
You can write a set of real numbers in order from greatest
to least or from least to greatest.
To do so, find a decimal approximation for each number in
the set and compare.
Write in order from least
to greatest. Write each number as a decimal.

Solution:
or about 2.4495
or about 2.4444
Answer: The numbers arranged in order from least to
greatest are

Your turn:
Write in order from least
to greatest.
Answer:
Do this in your notes, PLEASE!

Your Turn:
• What is the order of from
least to greatest?
• Answer:

Assignment
•
• Read and take notes on Sec. 1.4

Properties of Real Numbers
Section 1-4

Goals
Goal
• To identify and use properties
of real numbers.
Rubric
goals.
simple problems.
complex problems.

Vocabulary
• Equivalent Expression
• Deductive reasoning
• Counterexample

Definition
• Equivalent Expression – Two algebraic
expressions are equivalent if they have the
same value for all values of the variable(s).
– Expressions that look difference, but are equal.
– The Properties of Real Numbers can be used to
show expressions that are equivalent for all real
numbers.

Mathematical Properties
• Properties refer to rules that indicate a standard procedure or
method to be followed.
• A proof is a demonstration of the truth of a statement in
mathematics.
• Properties or rules in mathematics are the result from testing
the truth or validity of something by experiment or trial to
establish a proof.
• Therefore every mathematical problem from the easiest to
the more complex can be solved by following step by step
procedures that are identified as mathematical properties.

Commutative and Associative
Properties
• Commutative Property – changing the order in which you
add or multiply numbers does not change the sum or product.
• Associative Property – changing the grouping of numbers
when adding or multiplying does not change their sum or
product.
• Grouping symbols are typically parentheses (),but can
include brackets [] or Braces {}.

Commutative
Property of
Addition - (Order)
Commutative
Property of
Multiplication -
(Order)
For any numbers a and b , a + b = b + a
For any numbers a and b , a  b = b  a
45 + 5 = 5 + 45
6 8 = 8 6
50 = 50
48 = 48
Commutative Properties

Associative Property
of Addition - (grouping
symbols)
Associative Property
of Multiplication -
(grouping symbols)
For any numbers a, b, and c,
(a + b) + c = a + (b + c)
(ab)c = a (bc)
(2 + 4) + 5 = 2 + (4 + 5)
(2 3) 5 = 2 (3 5)
(6) + 5 = 2 + (9)
11 = 11
(6)  5 = 2  (15)
30 = 30
Associative Properties

Name the property that is illustrated in each equation.
A. 7(mn) = (7m)n
Associative Property of Multiplication
The grouping is different.
B. (a + 3) + b = a + (3 + b)
Associative Property of Addition
The grouping is different.
C. x + (y + z) = x + (z + y)
Commutative Property of Addition
The order is different.
Example: Identifying Properties

Name the property that is illustrated in each equation.
a. n + (–7) = –7 + n
b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3
c. (xy)z = (yx)z
Commutative Property of Multiplication
The order is
different.
The grouping is
different.
The order is
different.
Your Turn:

Note!
The Commutative and Associative
Properties of Addition and Multiplication
allow you to rearrange an expression.

Commutative and associative properties are very helpful
to solve problems using mental math strategies.
Solve: 18 + 13 + 16 + 27 + 22 + 24 Rewrite the problem by grouping numbers that can
be formed easily. (Associative property)
This process may change the order in which the
original problem was introduced. (Commutative
property)
(18 + 22) + (16 + 24) + (13 + 27)
(40) + (40) + (40) = 120
Properties

Commutative and associative properties are very helpful
to solve problems using mental math strategies.
Solve: 4 7 25
Rewrite the problem by changing the order in
which the original problem was introduced.
(Commutative property)
4 25 7
(4 25) 7
(100) 7 = 700
Group numbers that can be formed easily.
(Associative property)
Properties

Identity and Inverse
Properties
• Additive Identity Property
• Multiplicative Identity Property
• Multiplicative Property of Zero
• Multiplicative Inverse Property

Additive Identity Property
For any number a, a + 0 = a.
The sum of any number and zero is equal to that
number.
The number zero is called the additive identity.
If a = 5 then 5 + 0 = 5

Multiplicative Identity Property
For any number a, a  1 = a.
The product of any number and one is equal to that
number.
The number one is called the multiplicative identity.
If a = 6 then 6  1 = 6

Multiplicative Property of Zero
For any number a, a  0 = 0.
The product of any number and zero is equal to
zero.
If a = 6, then 6  0 = 0

Multiplicative Inverse Property
Two numbers whose product is 1 are called multiplicative
inverses or reciprocals.
Zero has no reciprocal because any number times 0 is 0.

Identity and Inverse Properties
Property Words Algebra Numbers
Additive
Identity
Property
The sum of a number
and 0, the additive
identity, is the original
number.
n + 0 = n 3 + 0 = 0
Multiplicative
Identity
Property
The product of a
number and 1, the
multiplicative identity,
is the original number.
n  1 = n
Additive
Inverse
Property
The sum of a number
and its opposite, or
additive inverse, is 0.
n + (–n) = 0 5 + (–5) = 0
Multiplicative
Inverse
Property
The product of a
nonzero number and
its reciprocal, or
multiplicative inverse,
is 1.

Example: Writing Equivalent
Expressions
A. 4(6y)
Use the Associative Property of
Multiplication4(6y) = (4•6)y
Simplify=24y
B. 6 + (4z + 3)
6 + (4z + 3) = 6 + (3 + 4z)
= (6 + 3) + 4z
= 9 + 4z
Use the Commutative
Property of Addition
Use the Associative
Property of Addition
Simplify

Example: Writing Equivalent
Expressions
C.
Rewrite the numerator using the Identity
Property of Multiplication
Use the rule for multiplying fractions
Simplify the fractions
Simplify

Your Turn:
Simplify each expression.
A. 4(8n)
B. (3 + 5x) + 7
C.
A. 32n
B. 10 + 5b
C. 4y

Identify which property that
justifies each of the following.
4  (8  2) = (4  8)  2

4  (8  2) = (4  8)  2
Associative Property of Multiplication

6 + 8 = 8 + 6

12 + 0 = 12

12 + 0 = 12
Additive Identity Property

5 + (2 + 8) = (5 + 2) + 8

Multiplicative Inverse Property

5  24 = 24  5

5  24 = 24  5
Commutative Property of Multiplication

-34  1 = -34

-34  1 = -34
Multiplicative Identity Property

Deductive Reasoning
Deductive Reasoning – a form of argument in
which facts, rules, definitions, or properties are
used to reach a logical conclusion (i.e. think
Sherlock Holmes).

Counterexample
• The Commutative and Associative Properties are
true for addition and multiplication. They may not
be true for other operations.
• A counterexample is an example that disproves a
statement, or shows that it is false.
• One counterexample is enough to disprove a
statement.

Caution!
One counterexample is enough to disprove
a statement, but one example is not enough
to prove a statement.

Statement Counterexample
No month has fewer than 30 days.
February has fewer than 30 days, so the
statement is false.
Every integer that is divisible by 2 is
also divisible by 4.
The integer 18 is divisible by 2 but is
not by 4, so the statement is false.
Example: Counterexample

Find a counterexample to disprove the statement “The Commutative
Property is true for raising to a power.”
Find four real numbers a, b, c, and d such that
a³ = b and c² = d, so a³ ≠ c².
Try a³ = 2³, and c² = 3².
a³ = b
2³ = 8
c² = d
3² = 9
Since 2³ ≠ 3², this is a counterexample. The statement is false.
Example: Counterexample

Find a counterexample to disprove the statement “The
Commutative Property is true for division.”
Find two real numbers a and b, such that
Try a = 4 and b = 8.
Since , this is a counterexample.
The statement is false.
Your Turn:

Assignment
•

Adding and Subtracting Real
Numbers
Section 1-5

Goals
Goal
• To find sums and differences
of real numbers.
Rubric
goals.
simple problems.
complex problems.

Vocabulary
• Absolute value
• Opposite
• Additive inverses

The set of all numbers that can be represented on a
number line are called real numbers. You can use a
number line to model addition and subtraction of real
numbers.
Addition
To model addition of a positive number, move right.
To model addition of a negative number, move left.
Subtraction
To model subtraction of a positive number, move left.
To model subtraction of a negative number, move
right.
Real Numbers

Add or subtract using a number line.
Start at 0. Move left to –4.
11 10 9 8 7 6 5 4 3 2 1 0
+ (–7)
–4 + (–7) = –11
To add –7, move left 7 units.
–4
–4 + (–7)
Example: Adding & Subtracting
on a Number Line

Start at 0. Move right to 3.
To subtract –6, move right 6 units.
-3 -2 -1 0 1 2 3 4 5 6 7 8 9
+ 3
3 – (–6) = 9
3 – (–6)
–(–6)
Example: Adding & Subtracting on
a Number Line

–3 + 7 Start at 0. Move left to –3.
To add 7, move right 7 units.
-3 -2 -1 0 1 2 3 4 5 6 7 8 9
–3
+7
–3 + 7 = 4
Your Turn:

–3 – 7
Start at 0. Move left to –3.
To subtract 7, move left 7 units.
–3
–7
11 10 9 8 7 6 5 4 3 2 1 0
–3 – 7 = –10
Your Turn:

–5 – (–6.5) Start at 0. Move left to –5.
To subtract –6.5, move right 6.5 units.
8 7 6 5 4 3 2 1 0
–5
–5 – (–6.5) = 1.5
1 2
– (–6.5)
Your Turn:

Definition
• Absolute Value – The distance between a
number and zero on the number line.
– Absolute value is always nonnegative since
distance is always nonnegative.
– The symbol used for absolute value is | |.
• Example:
– The |-2| is 2 and the |2| is 2.

The absolute value of a number is the distance from
zero on a number line. The absolute value of 5 is
written as |5|.
5 units 5 units
210123456 6543- - - - - -
|5| = 5|–5| = 5
Absolute Value on the Number Line

Add.
Use the sign of the number with the greater
absolute value.
Different signs: subtract the
absolute values.
A.
B. –6 + (–2)
(6 + 2 = 8)
–8 Both numbers are negative, so the sum is
negative.
Same signs: add the absolute values.
Example: Adding Real Numbers

Add.
–5 + (–7)
–12 Both numbers are negative, so the
sum is negative.
a.
(5 + 7 = 12)
–13.5 + (–22.3)b.
(13.5 + 22.3 = 35.8)
–35.8 Both numbers are negative, so the
sum is negative.
Your Turn:

c. 52 + (–68)
(68 – 52 = 16)
–16
absolute value.
Different signs: subtract the
absolute values.
Add.
Your Turn:

Definition
• Additive Inverse – The negative of a
designated quantity.
– The additive inverse is created by multiplying
the quantity by -1.
• Example:
– The additive inverse of 4 is -1 ∙ 4 = -4.

Opposites
• Two numbers are opposites if their sum is 0.
• A number and its opposite are additive
inverses and are the same distance from zero.
• They have the same absolute value.

Subtracting Real Numbers
• To subtract signed numbers, you can use
additive inverses.
• Subtracting a number is the same as adding
the opposite of the number.
• Example:
– The expressions 3 – 5 and 3 + (-5) are
equivalent.

A number and its opposite are additive inverses.
To subtract signed numbers, you can use additive
inverses.
11 – 6 = 5 11 + (–6) = 5
Additive inverses
Subtracting 6 is the same
as adding the inverse of 6.
Subtracting a number is the same as adding the
opposite of the number.

Rules For Subtracting

Subtract.
–6.7 – 4.1
–6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1.
Same signs: add absolute values.
–10.8 Both numbers are negative, so the sum
is negative.
(6.7 + 4.1 = 10.8)
Example: Subtracting Real
Numbers

Subtract.
5 – (–4)
5 − (–4) = 5 + 4
9
To subtract –4, add 4.
Same signs: add absolute values.(5 + 4 = 9)
Both numbers are positive, so the sum is
positive.
Example: Subtracting Real
Numbers

On many scientific and graphing calculators, there is
one button to express the opposite of a number and a
different button to express subtraction.
Helpful Hint

Subtract.
13 – 21
13 – 21 To subtract 21, add –21.
Different signs: subtract absolute values.
absolute value.–8
= 13 + (–21)
(21 – 13 = 8)
Your Turn:

–14 – (–12)
Subtract.
–14 – (–12) = –14 + 12
(14 – 12 = 2)
absolute value.
–2
Different signs: subtract absolute values.
Your Turn:

An iceberg extends 75 feet above the sea. The
bottom of the iceberg is at an elevation of –247
feet. What is the height of the iceberg?
Find the difference in the elevations of the top of the iceberg and
the bottom of the iceberg.
elevation at top of
iceberg
minus
elevation at bottom
of iceberg
75 – (–247)
75 – (–247) = 75 + 247
= 322
Same signs: add the absolute
values.
–75 –247
The height of the iceberg is 322 feet.

What if…? The tallest known iceberg in the
North Atlantic rose 550 feet above the ocean's
surface. How many feet would it be from the top
of the tallest iceberg to the wreckage of the
Titanic, which is at an elevation of –12,468 feet?
elevation at top of
iceberg
minus
elevation of the
Titanic
–
550 – (–12,468)
550 – (–12,468) = 550 + 12,468
To subtract –12,468,
add 12,468.
Same signs: add the
absolute values.
= 13,018
550 –12,468
Your Turn:
Distance from the top of the iceberg to the Titanic is 13,018 feet.

Multiplying and Dividing Real
Numbers
Section 1-6

Goals
Goal
• To Find products and
quotients of real numbers.
Rubric
goals.
simple problems.
complex problems.

Vocabulary
• Multiplicative Inverse
• Reciprocal

When you multiply two numbers, the signs of the
numbers you are multiplying determine whether
the product is positive or negative.
Factors Product
3(5) Both positive
3(–5) One negative
–3(–5) Both negative
15 Positive
–15 Negative
15 Positive
This is true for division also.
Multiplying Real Numbers

Rules for Multiplying and
Dividing

Find the value of each expression.
–5
The product of two numbers
with different signs is negative.
A.
12
The quotient of two numbers
with the same sign is positive.
B.
Example: Multiplying and
Dividing Real Numbers

Multiply.
C.
Example: Multiplying and
Dividing Real Numbers

–7
a. 35  (–5)
44
The product of two numbers
with the same sign is positive.
b. –11(–4)
c. –6(7)
–42
The product of two numbers with different
signs is negative.
Your Turn:

Reciprocals
• Two numbers are reciprocals if their product is 1.
• A number and its reciprocal are called
multiplicative inverses. To divide by a number, you
can multiply by its multiplicative inverse.
• Dividing by a nonzero number is the same as
Multiplying by the reciprocal of the number.

10 ÷ 5 = 2 10 ∙ = = 2
1
5
10
5
Multiplicative inverses
Dividing by 5 is the same as multiplying by the
reciprocal of 5, .
Reciprocals

You can write the reciprocal of a number by
switching the numerator and denominator. A whole
number has a denominator of 1.
Helpful Hint

Example 2 Dividing by Fractions
Divide.
Example: Dividing with
Fractions
To divide by , multiply by .
Multiply the numerators and
multiply the denominators.
and have the same sign,
so the quotient is positive.

Divide.
Write as an improper fraction.
and have different signs,
so the quotient is negative.
Example: Dividing with
Fractions

Divide.
and –9 have the same signs,
so the quotient is positive.
Your Turn:

Divide.
and have different signs,
so the quotient is negative.
Your Turn:

Check It Out! Example 2c
Divide.
To divide by multiply by .
The signs are different, so the
quotient is negative.

Zero
• No number can be multiplied by 0 to give a product
of 1, so 0 has no reciprocal.
• Because 0 has no reciprocal, division by 0 is not
possible. We say that division by zero is undefined.
• The number 0 has special properties for
multiplication and division.

Multiply or divide if possible.
A.
15
0
B. –22  0
undefined
C. –8.45(0)
0
Zero is divided by a nonzero number.
The quotient of zero and any nonzero
number is 0.
A number is divided by zero.
Division by zero is undefined.
A number is multiplied by zero.
The product of any number and 0 is 0.
0
Example: Multiplying & Dividing
with Zero

Multiply or divide.
a.
0
Zero is divided by a nonzero number.
The quotient of zero and any nonzero
number is 0.
b. 0 ÷ 0
undefined A number divided by 0 is undefined.
c. (–12.350)(0)
0
The product of any number and 0 is
0.
A number is divided by zero.
A number is multiplied by zero.
Your Turn:

rate
3
3
4
times

time
1 1
3
Find the distance traveled at a rate of 3 mi/h for 1 hour.
To find distance, multiply rate by time.
3
4
1
3
The speed of a hot-air balloon is 3 mi/h. It travels
in a straight line for 1 hours before landing. How
many miles away from the liftoff site will the
balloon land?
1
3
3
4

3 3
4
• 1 1
3
= 15
4
• 4
3
Write and as improper fractions.
3
4
3 1 1
3
15(4)
4(3)
= 60
12
= 5
3
3
4
and have the same sign, so
the quotient is positive.
1 1
3
The hot-air balloon lands 5 miles from the liftoff site.
Example: Continued

What if…? On another hot-air balloon trip, the wind
speed is 5.25 mi/h. The trip is planned for 1.5 hours.
The balloon travels in a straight line parallel to the
ground. How many miles away from the liftoff site
will the balloon land?
5.25(1.5) Rate times time equals distance.
= 7.875 mi Distance traveled.
Your Turn:

The Distributive Property
Section 1-7 Part 1

Goals
Goal
• To use the Distributive
Property to simplify
expressions.
Rubric
goals.
simple problems.
complex problems.

Vocabulary
• Distributive Property

Distributive Property
• To solve problems in mathematics, it is often
useful to rewrite expressions in simpler form.
• The Distributive Property, illustrated by the
area model on the next slide, is another
property of real numbers that helps you to
simplify expressions.

You can use algebra tiles to model algebraic expressions.
1
1 1-tile
This 1-by-1 square tile has
an area of 1 square unit.
x-tile
x
1
This 1-by-x square tile has
an area of x square units.
3
x + 2
Area = 3(x + 2)
3
2
3
x
Area = 3(x ) + 3(2)
Model the Distributive Property using Algebra Tiles
MODELING THE DISTRIBUTIVE PROPERTY
x + 2
+

The Distributive Property is used with Addition to Simplify
Expressions.
The Distributive Property also works with subtraction because
subtraction is the same as adding the opposite.

THE DISTRIBUTIVE PROPERTY
a(b + c) = ab + ac
(b + c)a = ba + ca
2(x + 5) 2(x) +
2(5)
2x + 10
(x + 5)2 (x)2 + (5)2 2x + 10
(1 + 5x)2 (1)2 + (5x)2 2 +
10x
y(1 – y) y(1) – y(y) y – y 2
USING THE DISTRIBUTIVE PROPERTY
=
=
=
=
=
=
=
=
The product of a and (b + c):

Distributive
Property
a(b + c) = ab + ac and (b + c)a = ba + bc;
a(b - c) = ab - ac and (b - c)a = ba - bc;
The mailman property
Find the sum (add) or
difference (subtract) of the
distributed products.

(y – 5)(–2) = (y)(–2) + (–5)(–2)
= –2y + 10
–(7 – 3x) = (–1)(7) + (–1)(–3x)
= –7 + 3x
= –3 – 3x
(–3)(1 + x) = (–3)(1) + (–3)(x)
Simplify.
Distribute the –3.
Simplify.
Simplify.
–a = –1 • a
USING THE DISTRIBUTIVE PROPERTY
Remember that a factor must multiply each term of an expression.
Forgetting to distribute the negative sign when multiplying by a negative
factor is a common error.

1)
2)
3)
4)
5)
6)
Your Turn: Simplify

Your turn:
1. 2(x + 5) = 5. (x - 4)x =
2. (15+6x) x = 6. y(2 - 6y) =
3. -3(x + 4) = 7. (y + 5)(-4) =
4. -(6 - 3x) = 8.

Section 1-7 Part 2

Vocabulary
• Term
• Constant
• Coefficient
• Like Terms

The process of distributing the number on the
outside of the parentheses to each term on
the inside.
a(b + c) = ab + ac and (b + c) a = ba + ca
a(b - c) = ab - ac and (b - c) a = ba - ca
Example
5(x + 7)
5 ∙ x + 5 ∙ 7
5x + 35

Two ways to find the area of the rectangle.
4
5 2
As a whole As two parts
Geometric Model for Distributive
Property

Geometric Model for Distributive
Property
Two ways to find the area of the rectangle.
4
5 2
same

Find the area of the rectangle in terms
of x, y and z in two different ways.
x
y z

Your Turn: Find the area of the rectangle in
terms of x, y and z in two
different ways.
x
y z
same
xy + xz

Write the product using the Distributive Property. Then simplify.
5(59)
5(50 + 9)
5(50) + 5(9)
250 + 45
295
Rewrite 59 as 50 + 9.
Use the Distributive Property.
Multiply.
Add.
Example: Distributive Property
with Mental Math
You can use the distributive property and mental math to make
calculations easier.

9(48)
9(50) - 9(2)
9(50 - 2)
450 - 18
432
Rewrite 48 as 50 - 2.
Multiply.
Subtract.
Write the product using the Distributive Property. Then
simplify.
Example: Distributive Property
with Mental Math

8(33)
8(30 + 3)
8(30) + 8(3)
240 + 24
264
Multiply.
Add.
simplify.
Your Turn:

12(98)
1176
Rewrite 98 as 100 – 2.
Multiply.
Subtract.
12(100 – 2)
1200 – 24
12(100) – 12(2)
simplify.
Your Turn:

7(34)
7(30 + 4)
7(30) + 7(4)
210 + 28
238
Multiply.
Add.
simplify.
Your Turn:

Find the difference mentally.
Find the products mentally.
The mental math is easier if you
think of $11.95 as $12.00 – $.05.
Write 11.95 as a difference.
You are shopping for CDs.
You want to buy six CDs
for $11.95 each.
Use the distributive property
to calculate the total cost
mentally.
6(11.95) = 6(12 – 0.05)
Use the distributive property.= 6(12) – 6(0.05)
= 72 – 0.30
= 71.70
The total cost of 6 CDs at $11.95 each is $71.70.
MENTAL MATH CALCULATIONS
SOLUTION

Definition
• Term – any number that is added or
subtracted.
– In the algebraic expression x + y, x and y are
terms.
• Example:
– The expression x + y – 7 has 3 terms, x, y, and 7.
x and y are variable terms; their values vary as x
and y vary. 7 is a constant term; 7 is always 7.

Definition
• Coefficient – The numerical factor of a term.
• Example:
– The coefficient of 3x2 is 3.

Definition
• Like Terms – terms in which the variables
and the exponents of the variables are
identical.
– The coefficients of like terms may be different.
• Example:
– 3x2 and 6x2 are like terms.
– ab and 3ab are like terms.
– 2x and 2x3 are not like terms.

Definition
• Constant – anything that does not vary or change in
value (a number).
– In algebra, the numbers from arithmetic are constants.
– Constants are like terms.

The terms of an expression are the parts to be added
or subtracted. Like terms are terms that contain the
same variables raised to the same powers. Constants
are also like terms.
4x – 3x + 2
Like terms Constant
Example:

A coefficient is a number multiplied by a variable.
Like terms can have different coefficients. A variable
written without a coefficient has a coefficient of 1.
1x2 + 3x
Coefficients
Example:

Like terms can be combined. To combine like
terms, use the Distributive Property.
Notice that you can combine like terms by adding or
subtracting the coefficients. Keep the variables and
exponents the same.
= 3x
ax – bx = (a – b)x
Example
7x – 4x = (7 – 4)x
Combining Like Terms

Simplify the expression by combining like terms.
72p – 25p
72p – 25p
47p
72p and 25p are like terms.
Subtract the coefficients.
Example: Combining Like
Terms

A variable without a coefficient has a
coefficient of 1.
Write 1 as .
Add the coefficients.
and are like terms.
Terms

0.5m + 2.5n
0.5m + 2.5n
0.5m + 2.5n
0.5m and 2.5n are not like terms.
Do not combine the terms.
Terms

Caution!
Add or subtract only the coefficients. 6.8y²
– y² ≠ 6.8

Simplify by combining like terms.
3a. 16p + 84p
16p + 84p
100p
16p + 84p are like terms.
Add the coefficients.
3b. –20t – 8.5t2
–20t – 8.5t2 20t and 8.5t2 are not like terms.
–20t – 8.5t2 Do not combine the terms.
3m2 + m3 3m2 and m3 are not like terms.
3c. 3m2 + m3
Do not combine the terms.3m2 + m3
Your Turn:

SIMPLIFYING BY COMBINING LIKE TERMS
Each of these terms is the product of a number and a
variable.
terms
+– 3y2x +– 3y2x
number
+– 3y2x
variable.
+– 3y2x
–1 is the
coefficient of x.
3 is the
coefficient of y2.
x is the
variable.
y is the
variable.
Each of these terms is the product of a number and a
variable.
x2 x2y3 y3
Like terms have the same variable raised to the same power.
y2 – x2 + 3y3 – 5 + 3 – 3x2 + 4y3 + y
variable power.Like terms
The constant terms –5 and 3 are also like terms.

Combine like terms.
SIMPLIFYING BY COMBINING LIKE TERMS
4x2 + 2 – x2 =
(8 + 3)x Use the distributive property.
= 11x Add coefficients.
8x + 3x =
Group like terms.
Rewrite as addition expression.
Multiply.
Combine like terms
and simplify.
4x2 – x2 + 2
= 3x2 + 2
3 – 2(4 + x) =3 + (–2)(4 + x)
= 3 + [(–2)(4) + (–2)(x)]
= 3 + (–8) + (–2x)
= –5 + (–2x)
= –5 – 2x

–12x – 5x + x + 3a Commutative Property
Combine like terms.–16x + 3a
–12x – 5x + 3a + x1.
2.
3.
Procedure Justification
Simplify −12x – 5x + 3a + x. Justify each step.
Your Turn:

Simplify 14x + 4(2 + x). Justify each step.
14x + 4(2) + 4(x) Distributive Property
Multiply.
Commutative Property of
Addition
Associative Property of
Addition
Combine like terms.
14x + 8 + 4x
(14x + 4x) + 8
14x + 4x + 8
18x + 8
14x + 4(2 + x)1.
2.
3.
4.
5.
6.
Statements Justification
Your Turn:

An Introduction to Equations
Section 1-8

Goals
Goal
• To solve equations using
tables and mental math.
Rubric
goals.
simple problems.
complex problems.

Vocabulary
• Equation
• Open sentence
• Solution of an equation

Definition
• Equation – A mathematical sentence that states one expression is
equal to a second expression.
• mathematical sentence that uses an equal sign (=).
• (value of left side) = (value of right side)
• An equation is true if the expressions on either side of the equal
sign are equal.
• An equation is false if the expressions on either side of the equal
sign are not equal.
• Examples:
• 4x + 3 = 10 is an equation, while 4x + 3 is an expression.
• 5 + 4 = 9 True Statement
• 5 + 3 = 9 False Statement

Equation or Expression
In Mathematics there is a difference between a phrase
and a sentence. Phrases translate into expressions;
sentences translate into equations or inequalities.
ExpressionsPhrases
Equations or InequalitiesSentences

Definition
• Open Sentence – an equation that contains
one or more variables.
– An open sentence is neither true nor false until
the variable is filled in with a value.
• Examples:
– Open sentence: 3x + 4 = 19.
– Not an open sentence: 3(5) + 4 = 19.

Example: Classifying
Equations
Is the equation true, false, or open? Explain.
A. 3y + 6 = 5y – 8
Open, because there is a variable.
B. 16 – 7 = 4 + 5
True, because both sides equal 9.
C. 32 ÷ 8 = 2 ∙ 3
False, because both sides are not equal, 4 ≠ 6.

Your Turn:
Is the equation true, false, or open? Explain.
A. 17 + 9 = 19 + 6
False, because both sides are not equal, 26 ≠ 25.
B. 4 ∙ 11 = 44
True, because both sides equal 44.
C. 3x – 1 = 17
Open, because there is a variable.

Definition
• Solution of an Equation – is a value of the
variable that makes the equation true.
– A solution set is the set of all solutions.
– Finding the solutions of an equation is called
solving the equation.
• Examples:
– x = 5 is a solution of the equation 3x + 4 = 19,
because 3(5) + 4 = 19 is a true statement.

Example: Identifying Solutions
of an Equation
Is m = 2 a solution of the equation 6m
– 16 = -4?
6m – 16 = -4
6(2) – 16 = -4
12 – 16 = -4
-4 = -4 True statement, m = 2 is a solution.

Your Turn:
Is x = 5 a solution of the equation 15
= 4x – 4?
No, 15 ≠ 16. False statement, x = 5 is not a
solution.

A PROBLEM SOLVING PLAN USING MODELS
Procedure for Writing an Equation
problem.
VERBAL
MODEL
problem.
ALGEBRAIC
MODEL
LABELS

Writing an Equation
You and three friends are having a dim sum lunch at a
Chinese restaurant that charges $2 per plate. You order lots
of plates. The waiter gives you a bill for $25.20, which
includes tax of $1.20. Write an equation for how many
plates your group ordered.
Understand the problem situation
before you begin. For example,
notice that tax is added after the
total cost of the dim sum plates is
figured.
SOLUTION

LABELS
VERBAL
MODEL
Writing an Equation
Cost per
plate •
Number of
plates = Bill Tax–
Cost per plate =2
Number of plates =p
Amount of bill = 25.20
Tax = 1.20
(dollars)
(dollars)
(dollars)
(plates)
25.20 1.20–2 =p
2p = 24.00
The equation is 2p = 24.
ALGEBRAIC
MODEL

Your Turn:
JET PILOT A jet pilot is flying from Los Angeles, CA to Chicago, IL at
a speed of 500 miles per hour. When the plane is 600 miles from
Chicago, an air traffic controller tells the pilot that it will be 2 hours
before the plane can get clearance to land. The pilot knows the
speed of the jet must be greater then 322 miles per hour or the jet
could stall.
Write an equation to find at what
speed would the jet have to fly
to arrive in Chicago in 2 hours?

LABELS
VERBAL
MODEL
Solution
Speed of
jet • Time =
Distance to
travel
Speed of jet = x
Time = 2
Distance to travel =600
(miles per hour)
(miles)
(hours)
600=
2x = 600
ALGEBRAIC
MODEL
At what speed would the jet have to fly to arrive in Chicago in 2 hours?
2 x
SOLUTION You can use the formula (rate)(time) = (distance) to write a verbal model.

Example: Use Mental Math to
Find Solutions
• What is the solution to the equation? Use
mental math.
• 12 – y = 3
– Think: What number subtracted from 12 equals 3.
– Solution: 9.
– Check: 12 – (9) = 3, 3 = 3 is a true statement,
therefore 9 is a solution.

Your Turn:
What is the solution to the equation? Use mental
math.
A. x + 7 = 13
6
B. x/6 = 12
72

Patterns, Equations, and Graphs
Section 1-9

Goals
Goal
• To use tables, equations, and
graphs to describe
relationships.
Rubric
goals.
simple problems.
complex problems.

Vocabulary
• Solution of an equation
• Inductive reasoning

The coordinate plane is formed by
the intersection of two perpendicular
number lines called axes. The point
of intersection, called the origin, is
at 0 on each number line. The
horizontal number line is called the
x-axis, and the vertical number line
is called the y-axis.
Review: Graphing in the
Coordinate Plane

Points on the coordinate plane are described using ordered
pairs. An ordered pair consists of an x-coordinate and a y-
coordinate and is written (x, y). Points are often named by a
capital letter.
The x-coordinate tells how many units to move left or right from the
origin. The y-coordinate tells how many units to move up or down.
Reading Math
Graphing in the Coordinate
Plane

Graph each point.
A. T(–4, 4)
Start at the origin.
Move 4 units left and 4 units
up.
B. U(0, –5)
Move 5 units down.
•
T(–4, 4)
• U(0, –5)
C. V (–2, –3)
Move 2 units left and 3 units
down.
•
V(–2, −3)
Example: Graphing in the
Coordinate Plane

Graph each point.
A. R(2, –3)
B. S(0, 2)
Move 2 units right and 3 units
down.
Move 2 units up.
C. T(–2, 6)
Move 2 units left and6 units up.
•
R(2, –3)
S(0,2)
T(–2,6)
Your Turn:

The axes divide the
coordinate plane into
four quadrants. Points
that lie on an axis are not
in any quadrant.
Graphing in the Coordinate
Plane

Name the quadrant in which each point lies.
A. E
Quadrant ll
B. F
no quadrant (y-axis)
C. G
Quadrant l
D. H
Quadrant lll
•E
•F
•H
•G
x
y
Example: Locating Points

Name the quadrant in which each point lies.
A. T
no quadrant (y-axis)
B. U
Quadrant l
C. V
Quadrant lll
D. W
Quadrant ll
•T
•W
•V
•U
x
y
Your Turn:

The Rectangular Coordinate System
SUMMARY: The Rectangular Coordinate System
• Composed of two real number lines – one horizontal (the x-axis) and
one vertical (the y-axis). The x- and y-axes intersect at the origin.
• Also called the Cartesian plane or xy-plane.
• Points in the rectangular coordinate system are denoted (x, y) and are
called the coordinates of the point. We call the x the x-coordinate and
the y the y-coordinate.
• If both x and y are positive, the point lies in quadrant I; if x is
negative, but y is positive, the point lies in quadrant II; if x is negative
and y is negative, the point lies in quadrant III; if x is positive and y is
negative, the point lies in quadrant IV.
• Points on the x-axis have a y-coordinate of 0; points on the y-axis have
an x-coordinate of 0.

Equation in Two Variables
An equation in two variables, x and y, is a statement in which the
algebraic expressions involving x and y are equal. The expressions
are called sides of the equation.
Any values of the variables that make the equation a true statement
are said to be solutions of the equation.
x + y = 15 x2 – 2y2 = 4 y = 1 + 4x
x + y = 15
The ordered pair (5, 10) is a solution of the equation.
5 + 10 = 15
15 = 15

Solutions to Equations
2x + y = 5
2(2) + (1) = 5
4 + 1 = 5
5 = 5
Example:
Determine if the following ordered pairs satisfy the equation
2x + y = 5.
a.) (2, 1) b.) (3, – 4)
(2, 1) is a solution.
True
2x + y = 5
2(3) + (– 4) = 5
6 + (– 4) = 5
2 = 5
(3, – 4) is not a solution.
False

An equation that contains two variables can be used as a
rule to generate ordered pairs. When you substitute a
value for x, you generate a value for y. The value
substituted for x is called the input, and the value
generated for y is called the output.
y = 10x + 5
Output Input
Equation in Two Variables

Table of Values
Use the equation y = 6x + 5 to complete the table and list
the ordered pairs that are solutions to the equation.
x y (x, y)
– 2
0
2
y = 6x + 5
x = – 2
y = 6(– 2) + 5
y = – 12 + 5
y = – 7
(– 2, – 7)
– 7
y = 6x + 5
x = 0
y = 6(0) + 5
y = 0 + 5
y = 55
(0, 5)
y = 6x + 5
x = 2
y = 6(2) + 5
y = 12 + 5
y = 1717
(2, 17)

An engraver charges a setup fee of $10 plus $2 for every
word engraved. Write a rule for the engraver’s fee. Write
ordered pairs for the engraver’s fee when there are 5, 10,
15, and 20 words engraved.
Let y represent the engraver’s fee and x represent the
number of words engraved.
Engraver’s fee is $10 plus $2 for each word
y = 10 + 2 · x
y = 10 + 2x

The engraver’s fee is determined by the number of
words in the engraving. So the number of words is
the input and the engraver’s fee is the output.
Writing Math

Number of
Words
Engraved
Rule Charges
Ordered
Pair
x (input) y = 10 + 2x y (output) (x, y)
y = 10 + 2(5)5 20 (5, 20)
y = 10 + 2(10)10 30 (10, 30)
y = 10 + 2(15)15 40 (15, 40)
y = 10 + 2(20)20 50 (20, 50)
Example: Solution

What if…? The caricature artist increased his fees. He now
charges a $10 set up fee plus $20 for each person in the
picture. Write a rule for the artist’s new fee. Find the artist’s
fee when there are 1, 2, 3 and 4 people in the picture.
y = 10 + 20x
Let y represent the artist’s fee and x represent the number of
people in the picture.
Artist’s fee is $10 plus $20 for each person
y = 10 + 20 · x
Your Turn:

Number of
People in
Picture
Rule Charges
Ordered
Pair
x (input) y = 10 + 20x y (output) (x, y)
y = 10 + 20(1)1 30 (1, 30)
y = 10 + 20(2)2 50 (2, 50)
y = 10 + 20(3)3 70 (3, 70)
y = 10 + 20(4)4 90 (4, 90)
Solution:

When you graph ordered pairs generated by
a function, they may create a pattern.
Graphing Ordered Pairs

Generate ordered pairs for the function using
the given values for x. Graph the ordered pairs
and describe the pattern.
y = 2x + 1; x = –2, –1, 0, 1, 2
–2
–1
0
1
2
2(–2) + 1 = –3 (–2, –3)
(–1, –1)
(0, 1)
(1, 3)
(2, 5)
2(–1) + 1 = –1
2(0) + 1 = 1
2(1) + 1 = 3
2(2) + 1 = 5
•
•
•
•
•
Input Output
Ordered
Pair
x y (x, y)
The points form a line.
Example: Graphing Ordered Pairs

–4
–2
0
2
4
–2 – 4 = –6 (–4, –6)
(–2, –5)
(0, –4)
(2, –3)
(4, –2)
–1 – 4 = –5
0 – 4 = –4
1 – 4 = –3
2 – 4 = –2
Input Output
Ordered
Pair
x y (x, y)
The points form a line.
y = x – 4; x = –4, –2, 0, 2, 4
1
2
Your Turn: Generate ordered pairs for the
function using the given values for x. Graph the
ordered pairs and describe the pattern.

Definition
• Inductive Reasoning – is the process of
reaching a conclusion based on an observed
pattern.
– Can be used to predict values based on a pattern.

Inductive Reasoning
• Moves from specific observations to broader
generalizations or predictions from a pattern.
• Steps:
1. Observing data.
2. Detect and recognizing patterns.
3. Make generalizations or predictions from those patterns.
Observation
Pattern
Predict

Make a prediction about the next number based on the
pattern.
2, 4, 12, 48, 240
Answer: 1440
Find a pattern:
2 4 12 48 240
×2
The numbers are multiplied by 2, 3, 4, and 5.
Prediction: The next number will be multiplied by 6. So, it will
be (6)(240) or 1440.
×3 ×4 ×5
Example: Inductive Reasoning

Make a prediction about the next number based on the
pattern.
Answer: The next number will be
Your Turn:

Alg1ch1

More Related Content

What's hot (20)

Similar to Alg1ch1 (20)

Recently uploaded (20)

Alg1ch1