variables and expressions -...
TRANSCRIPT
Variables and Expressions
Section 1-1
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
• Algebraic Expression – a mathematical
phrase that may contain variables,
constants, and/or operations.
• Examples: 5x + 3, y/2 – 4, xy – 2x + y,
3(2x + 7), 2𝑎+𝑏
5𝑐
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.
Division Notation
Division, on the other hand, is written as:
1) 𝑥
3
2) x ÷ 3
In algebra, normally write division as a fraction.
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
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
Words for Operations - Examples
Words for Operations - Examples
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
Writing an Algebraic
Expression for a Verbal Phrase.
Order
of the
wording
Matters
Writing an algebraic expression with subtraction.
–
The difference of seven and a number n
7 n
7 – n
Writing an Algebraic
Expression for a Verbal Phrase.
Order
of the
wording
Matters
Writing an algebraic expression with subtraction.
8
Eight less than a number
–y
y – 8
Writing an Algebraic
Expression for a Verbal Phrase.
Order
of the
wording
Matters
Writing an algebraic expression with multiplication.
1/3
one-third of a number n.
· n
1
3n
Writing an Algebraic
Expression for a Verbal Phrase.
Order
of the
wording
Matters
Writing an algebraic expression with division.
The quotient of a number n and 3
n 3
Writing an Algebraic
Expression for a Verbal Phrase.
Order
of the
wording
Matters
3
n
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
“Translating” a phrase into an algebraic
expression:
4 less than a number n
Identify the operation?
“less than” means subtract
Answer: n – 4.
Why not 4 – n?????
Determine the order of the variables and constants.
Example
“Translating” a phrase into an algebraic
expression:
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 ?????
12
x
12
x
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)
12
x
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
j ÷ 6
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
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
Practice!
Pg. 6-7 # 1-19 odd, 21-24
Order of Operations and
Evaluating Expressions
Section 1-2 Part 1
Vocabulary
• Power
• Exponent
• Base
• Simplify
Definition
A power expression has two parts, a base and
an exponent.
103
Power expression
ExponentBase
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. 10 3 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.
29 2
9
= 4
81 29 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 andtwo 6’s.
Your Turn:
Evaluate the expression.
c.
2764
Use as a factor 3 times.34
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.
53621 53621
53621
5327
59
45
53621
5221
5221
1021
31
5327
Performing operations left to right onlyPerforming 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
218654 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. 218654
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).
2189
369
The order of operations must be followed each
time you rewrite the expression. 45
Divide
Multiply
Add
Order of Operations: Example 1
Evaluate without grouping symbols
6522
6522
6252
650
44
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).
The order of operations must be followed each
time you rewrite the expression.
Order of Operations: 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).
The order of operations must be followed each
time you rewrite the expression.
28432
28432
Grouping
symbols
6432
6163
648
8Divide
Order of Operations: 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).
The order of operations must be followed each
time you rewrite the expression.
)418(2
432
)418(2
432
Work above the
fraction bar
3
Simplify:
Divide
243
163
)418(2
48
)418(2
Work below the
fraction bar Grouping symbols
)14(2 Add
16
48
1648
Order of Operations: Example 4
Expressions with fraction bars
Your Turn:
Simplify the expression.
8 ÷ · 3 1 2
8 ÷ · 3 1 2
16 · 3
48
There are no groupingsymbols.
Divide.
Multiply.
Your Turn:Simplify the expression.
5.4 – 32 + 6.2
5.4 – 32 + 6.2
5.4 – 9 + 6.2
–3.6 + 6.2
2.6
There are no groupingsymbols.
Simplify powers.
Subtract
Add.
Your Turn:Simplify the expression.
–20 ÷ [–2(4 + 1)]
–20 ÷ [–2(4 + 1)]
–20 ÷ [–2(5)]
–20 ÷ –10
2
There are two sets of groupingsymbols.
Perform the operations in theinnermost set.
Perform the operation insidethe 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) – 3 3
5 + 2(–8)
(–2) – 3 3
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.
Practice
Pg. 12-13 # 1-20 all
Order of Operations and
Evaluating Expressions
Section 1-2 Part 2
Vocabulary
• Evaluate
Evaluating Expressions
• 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 · 2Simplify.= 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.
Example: Evaluating Algebraic
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
Example: Evaluating
Algebraic ExpressionsEvaluate 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:Evaluate the expression for the given value of x.
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:Evaluate the expression for the given value of x.
(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
What is the value of
5k3 if k = -4?
Your Turn:
1. 10
2. -10
3. -6
4. 6
t
mn2
What is the value of
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
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.
ALGEBRAIC
MODEL
LABELS
Example: Application
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.
Example: Application
ContinuedApproximately 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
To make 50 sleeping bags 4250 bottles are needed.
85(325) = 27,625To make 325 sleeping bags 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
To make 25 sweaters 1575 bottles are needed.
63(50) = 3150To make 50 sweaters 3150 bottles are needed.
Practice!
Pg. # 21-55 odd
Properties of Real Numbers
Section 1-3
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)
For any numbers a, b, and 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 Addition
Commutative Property of Multiplication
Associative Property of Addition
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
Commutative and Associative
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)
Commutative and Associative
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
1such that number one
exactly is there,0 , where, number nonzeroevery For
a
b
b
a
a
b
bab
a
3 3 4 3 4 12 Given the fraction ; then 1;
4 4 3 4 3 12
4the fraction is the reciprocal.
3
Together the two fractions are multiplicative
inverses that are equal to the product 1.
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.8
12
m
mn
8 8 1
12 12
m m
mn m n
8 1
12
m
m n
2 11
3 n
2
3n
Rewrite the numerator using the
Identity Property of Multiplication
Use the rule for multiplying fractionsa c ac
b d bd
Simplify the fractions
Simplify
Your Turn:
Simplify each expression.
A. 4(8n)
B. (3 + 5x) + 7
C.
A. 32n
B. 10 + 5b
C. 4y
8
2
xy
x
Identify which property
that justifies each of the
following.
4 (8 2) = (4 8) 2
Identify which property
that justifies each of the
following.
4 (8 2) = (4 8) 2
Associative Property of Multiplication
Identify which property
that justifies each of the
following.
6 + 8 = 8 + 6
Identify which property
that justifies each of the
following.
6 + 8 = 8 + 6
Commutative Property of Addition
Identify which property
that justifies each of the
following.
12 + 0 = 12
Identify which property
that justifies each of the
following.
12 + 0 = 12
Additive Identity Property
Identify which property
that justifies each of the
following.
5 + (2 + 8) = (5 + 2) + 8
Identify which property
that justifies each of the
following.
5 + (2 + 8) = (5 + 2) + 8
Associative Property of Addition
Identify which property
that justifies each of the
following.
5
9
9
51
Identify which property
that justifies each of the
following.
Multiplicative Inverse Property
5
9
9
51
Identify which property
that justifies each of the
following.
5 24 = 24 5
Identify which property
that justifies each of the
following.
5 24 = 24 5
Commutative Property of Multiplication
Identify which property
that justifies each of the
following.
-34 1 = -34
Identify which property
that justifies each of the
following.
-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:
Practice!
Adding and Subtracting Real
Numbers
Section 1-4, 1-5
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
Add or subtract using 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
Add or subtract using 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:
Add or subtract using a number line.
–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:
Add or subtract using a number line.
–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
Rules For Adding
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.
Same signs: add the absolute values.
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.
Same signs: add the absolute values.
Your Turn:
c. 52 + (–68)
(68 – 52 = 16)
–16Use the sign of the number with the
greater 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.
Additive Inverse Property
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.
Subtracting Real Numbers
Subtracting Real NumbersRules 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.
Use the sign of the number with the greater absolute value.–8
= 13 + (–21)
(21 – 13 = 8)
Your Turn:
–14 – (–12)
Subtract.
–14 – (–12) = –14 + 12
(14 – 12 = 2)
To subtract –12, add 12.
Use the sign of the number with the greater 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
icebergminus
elevation at bottom
of iceberg
75 – (–247)
75 – (–247) = 75 + 247
= 322
To subtract –247, add 247.
Same signs: add the absolute values.
–75 –247
Example: Application
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
icebergminus
elevation of the
Titanic
–
550 – (–12,468)
550 – (–12,468) = 550 + 12,468To 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.
Practice!
Multiplying and Dividing Real
Numbers
Section 1-6
Vocabulary
• Multiplicative Inverse
• Reciprocal
When you multiply two numbers, the signs of thenumbers you are multiplying determine whetherthe 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.
–5The product of two numbers
with different signs is negative.
A.
12The quotient of two numbers
with the same sign is positive.
B.
Example: Multiplying and
Dividing Real Numbers
The quotient of two numbers
with different signs is negative.
Multiply.
C.
Find the value of each expression.
Example: Multiplying and
Dividing Real Numbers
Find the value of each expression.
–7The quotient of two numbers
with different signs is negative.
a. 35 (–5)
44The product of two numbers
with the same sign is positive.
b. –11(–4)
c. –6(7)
–42The 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 ∙ = = 215
105
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.
To divide by , multiply by .
and have different signs,
so the quotient is negative.
Example: Dividing with
Fractions
Divide.
Write as an improper fraction.
To divide by , multiply by .
and –9 have the same signs,
so the quotient is positive.
Your Turn:
Divide.
To divide by , multiply by .
Multiply the numerators and
multiply the denominators.
and have different signs,
so the quotient is negative.
Your Turn:
Check It Out! Example 2c
Divide.
Write as an improper fraction.
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)
0The product of any number and 0 is
0.
A number is divided by zero.
A number is multiplied by zero.
Your Turn:
rate
33
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
Example: Application
33
4• 1
1
3= 15
4•
4
3Write and as improper fractions.
3
43 1 1
3
15(4)
4(3)=
60
12
= 5
Multiply the numerators and
multiply the denominators.
33
4and 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:
Practice!
The Distributive Property
Section 1-7 Part 1
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.
Distributive Property
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
For any numbers a, b, and c,
a(b + c) = ab + ac and (b + c)a = ba + bc;
a(b - c) = ab - ac and (b - c)a = ba - bc;
Find the sum (add) or
difference (subtract) of the
distributed products.
The Distributive Property
(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.
Distribute the –2.
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)
5(x 3)
5 3
5x 15
6(y 7)
6 7
6y 42
3(m 8)
3 8
3m 24
4(3 y)
4 3 4 y
12 4y
10(x 7)
10 x 10 7
10x 70
4( k 2)
4 k 4 2
4k 8
5 x
6 y
3 m
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.
31
23
3 9x x
21
2 10x 24x x
25 2x x
23y y
3 12x 4 20y
3 6x 2
2 6x x
Practice!
The Distributive Property
Section 1-7 Part 2
Vocabulary
• Term
• Constant
• Coefficient
• Like Terms
The Distributive Property
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
A w l
4 5 2
Geometric Model for Distributive
Property
Geometric Model for Distributive
Property
Two ways to find the area of the rectangle.
4
5 2
As a whole As two parts
A w l
4 5 2 4 5 4 2
4 5 4 2
same
4 5 2 4 5 4 2
Find the area of the rectangle in terms
of x, y and z in two different ways.
x
y z
As a whole As two parts
A w l
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
As a whole As two parts
A w l
x y z x y x z
x y x 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 MathYou 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.
Use the Distributive Property.
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
Rewrite 33 as 30 + 3.
Use the Distributive Property.
Multiply.
Add.
Write the product using the Distributive Property. Then
simplify.
Your Turn:
12(98)
1176
Rewrite 98 as 100 – 2.
Use the Distributive Property.
Multiply.
Subtract.
12(100 – 2)
1200 – 24
12(100) – 12(2)
Write the product using the Distributive Property. Then
simplify.
Your Turn:
7(34)
7(30 + 4)
7(30) + 7(4)
210 + 28
238
Rewrite 34 as 30 + 4.
Use the Distributive Property.
Multiply.
Add.
Write the product using the Distributive Property. Then
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
Distributive Property
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
Simplify the expression by combining like terms.
A variable without a coefficient has a
coefficient of 1.
Write 1 as .
Add the coefficients.
and are like terms.
Example: Combining Like
Terms
Simplify the expression by combining like 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.
Example: Combining Like
Terms
Caution!Add or subtract only the coefficients.
6.8y² – y² ≠ 6.8
2 2 26.8 5.8y y y
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.
Distribute the –2.
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 AdditionAssociative Property of AdditionCombine 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:
Practice!
An Introduction to Equations
Section 1-8
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
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
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.
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 perplate •
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 ofjet • Time =
Distance totravel
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
Practice!
Patterns, Equations, and Graphs
Section 1-9
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)
Start at the origin.
Move 5 units down.
•T(–4, 4)
• U(0, –5)
C. V (–2, –3)
Start at the origin.
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)
Start at the origin.
Move 2 units right and 3 units down.
Start at the origin.
Move 2 units up.
C. T(–2, 6)
Start at the origin.
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. HQuadrant 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. WQuadrant 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
Example: Application
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 ofWords
EngravedRule 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 ChargesOrdered
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 OutputOrdered
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 OutputOrdered
Pair
x y (x, y)
The points form a line.
y = x – 4; x = –4, –2, 0, 2, 41
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:
1 1 1 1, ,
4 9 16, 251,
2
1 1 or
6 36
Practice!