Foundations for Algebra
Chapter 1Part 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.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve 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.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced 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
•
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
•
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.
Orderof thewordingMatters
Writing an algebraic expression with addition.
2
Two more than a number
+x
x + 2
Writing an Algebraic Expression for a Verbal Phrase.
Orderof thewordingMatters
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.
Orderof thewordingMatters
Writing an algebraic expression with subtraction.
8
Eight less than a number
–y
y – 8
Writing an Algebraic Expression for a Verbal Phrase.
Orderof thewordingMatters
Writing an algebraic expression with multiplication.
1/3
one-third of a number n.
· n
Writing an Algebraic Expression for a Verbal Phrase.
Orderof thewordingMatters
Writing an algebraic expression with division.
The quotient of a number n and 3
n 3
Writing an Algebraic Expression for a Verbal Phrase.
Orderof thewordingMatters
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 add
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 ?????
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 + 52) 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 threeD
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 xA
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
CHALLENGEWrite 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.
PatternsMathematicians …• 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
Algebraic
Graphical
Number Patterns
22 + 22 + 2 + 22 + 2 + 2 +
24(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 FernandoI 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.
Shopping ListGuests = ?
Bags ofSpaghetti = ?
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 TermDifference 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
1234
25811
36912
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:
3nDoes this get us the answer
we want?
3n gives us: Bags of Spaghetti
3 26 59 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
1234
25811
36912
3n 3n – 1
25811
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
Assignment
•
Order of Operations and Evaluating Expressions
Section 1-2 Part 1
Goals
Goal• To simplify expressions
involving exponents.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.
Vocabulary
• Power• Exponent• Base• Simplify
DefinitionA 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 • 10103 = 1000
The base, 10, is the number that is used as a factor. 103 The exponent, 3, tells
how many times thebase, 10, is used as afactor.
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.
ss
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 andtwo 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 PowersWrite 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 OPERATIONS1. 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 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
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).
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.
Grouping symbols
Divide
Order of Operations: Example 3Evaluate 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.
Work above the fraction bar
Simplify: Divide
Work below the fraction bar Grouping symbols
Add
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.
Assignment
•
Order of Operations and Evaluating Expressions
Section 1-2 Part 2
Goals
Goal• To use the order of operations
to evaluate expressions.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.
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, andc = 2.
A. b – c
b – c = 7 – 2= 5
B. acac = 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 – 2Simplify. = 7
Substitute 9 for p and 3 for m.p ÷ m = 9 ÷ 3Simplify.
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 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: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
What is the value ofif 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 AVERBAL 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 ofbottles 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 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 make20, 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) = 4250To 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 ofbottles to make s sweaters.
To make one sweater, 63 twenty ounceplastic drink bottles must be recycled.
Your Turn: ContinuedTo make one sweater, 63 twenty ounceplastic drink bottles must be recycled.
Find the number of bottles needed to make12, 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) = 1575To make 25 sweaters 1575 bottles are needed.
63(50) = 3150To make 50 sweaters 3150 bottles are needed.
Assignment
•
Real Numbers and the Number Line
Section 1-3
Goals
Goal• To classify, graph, and
compare real numbers.• To find and estimate square
roots.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more 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
2
2
22 = 4
The square root of 4 is 2
32 = 9
3
3
The square root of 9 is 3
42 = 16
4
4
The square root of 16 is 4
5
5
52 = 25
The square root of 25 is 5
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 RootsFind each root.
Think: What number squared equals 81?
Think: What number squared equals 25?
Example: Finding RootsFind 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?
Think: What number squared 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.
Think: What number squared
equals
Example: Finding Roots of Fractions
Find the root.
B.
Think: What number cubed equals
Example: Finding Roots of Fractions
Find the root.
C.
Think: What number squared
equals
Your Turn:Find the root.
a.
Think: What number squared equals
Your Turn:Find the root.
b.
Think: What number cubed equals
Your Turn:Find the root.
c.
Think: What number squared 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.
Example: ApplicationAs 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 Continued5.0 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: , , .
ab12
71
910
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: , , π
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. π4. -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 unit.
– 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• Read and take notes on Sec. 1.5
Properties of Real Numbers
Section 1-4
Goals
Goal• To identify and use properties
of real numbers.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more 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)
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
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
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.
Identify which property that justifies each of the following.
Multiplicative Inverse Property
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 = b2³ = 8
c² = d3² = 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
•
• Read and take notes on Sec. 1.5• Read and take notes on Sec. 1.6
Adding and Subtracting Real Numbers
Section 1-5
Goals
Goal• To find sums and differences
of real numbers.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more 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.SubtractionTo 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 additiveinverses.
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 andthe bottom of the iceberg.
elevation at top of iceberg
minus 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 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.
Assignment
•
Multiplying and Dividing Real Numbers
Section 1-6
Goals
Goal• To Find products and
quotients of real numbers.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.
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 numberswith different signs is negative.
A.
12The quotient of two numberswith the same sign is positive.
B.
Example: Multiplying and Dividing Real Numbers
The quotient of two numberswith 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 numberswith different signs is negative.
a. 35 ÷ (–5)
44The product of two numberswith 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 ∙ = = 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 nonzeronumber 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 nonzeronumber 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
334
times
•
time
1 13
Find the distance traveled at a rate of 3 mi/h for 1 hour.To find distance, multiply rate by time.
34
13
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?
13
34
Example: Application
3 34
• 1 13
= 15 4
• 43
Write and as improper fractions.34
3 1 13
15(4) 4(3)
= 6012
= 5
Multiply the numerators andmultiply the denominators.
334
and have the same sign, so
the quotient is positive.
1 13
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:
Assignment
•
The Distributive Property
Section 1-7 Part 1
Goals
Goal• To use the Distributive
Property to simplify expressions.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more 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.
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;
The mailman property
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)
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.
Assignment
•
The Distributive Property
Section 1-7 Part 2
Goals
Goal• To use the Distributive
Property to simplify expressions.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.
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 + caa(b - c) = ab - ac and (b - c) a = ba - ca
Example 5(x + 7)
5 ∙ x + 5 ∙ 75x + 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
As a whole As two partssame
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
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 partssame
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.
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 propertyto 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
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:
Assignment
•
An Introduction to Equations
Section 1-8
Goals
Goal• To solve equations using
tables and mental math.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more 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 – 8Open, because there is a variable.
B. 16 – 7 = 4 + 5True, because both sides equal 9.
C. 32 ÷ 8 = 2 ∙ 3False, because both sides are not equal, 4 ≠ 6.
Your Turn:
Is the equation true, false, or open? Explain.
A. 17 + 9 = 19 + 6False, because both sides are not equal, 26 ≠ 25.
B. 4 ∙ 11 = 44True, because both sides equal 44.
C. 3x – 1 = 17Open, 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
SOLUTIONYou 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
Assignment
•
Patterns, Equations, and Graphs
Section 1-9
Goals
Goal• To use tables, equations, and
graphs to describe relationships.
RubricLevel 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more 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)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. EQuadrant ll
B. Fno quadrant (y-axis)
C. GQuadrant l
D. HQuadrant lll
•E
•F
•H
•Gx
y
Example: Locating Points
Name the quadrant in which each point lies.
A. Tno quadrant (y-axis)
B. UQuadrant l
C. VQuadrant 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 = 1515 = 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 + 5Output Input
Equation in Two Variables
Table of ValuesUse 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
–10
1
2
2(–2) + 1 = –3 (–2, –3)
(–1, –1)(0, 1)
(1, 3)
(2, 5)
2(–1) + 1 = –12(0) + 1 = 1
2(1) + 1 = 32(2) + 1 = 5
•
•
•
•
•
Input OutputOrdered
Pairx y (x, y)
The points form a line.
Example: Graphing Ordered Pairs
–4
–20
2
4
–2 – 4 = –6 (–4, –6)
(–2, –5)(0, –4)
(2, –3)
(4, –2)
–1 – 4 = –50 – 4 = –4
1 – 4 = –3
2 – 4 = –2
Input OutputOrdered
Pairx y (x, y)
The points form a line.
y = x – 4; x = –4, –2, 0, 2, 412
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:
Assignment