alg1ch1

Foundations for Algebra

Chapter 1Part A

Essential Questions

• How can you represent quantities, patterns, and relationships?

• How are properties related to algebra?

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

Goal• To write algebraic

expressions.

Vocabulary

• Quantity• Variable• Algebraic expression• Numerical expression

Definition

• Quantity – A mathematical quantity is anything that can be measured or counted.– How much there is of something.– A single group, generally represented in an

expression using parenthesis () or brackets [].

• Examples: – numbers, number systems, volume, mass, length,

people, apples, chairs.– (2x + 3), (3 – n), [2 + 5y].

Definition• Variable – anything that can vary or change in value.

– In algebra, x is often used to denote a variable.– Other letters, generally letters at the end of the alphabet (p,

q, r, s, t, u, v, w, x, y, and z) are used to represent variables– A variable is “just a number” that can change in value.

• Examples:– A child’s height– Outdoor temperature– The price of gold

Definition

• Constant – anything that does not vary or change in value (a number).– In algebra, the numbers from arithmetic are constants.– Generally, letters at the beginning of the alphabet (a, b, c,

d)used to represent constants.

• Examples: – The speed of light– The number of minutes in a hour– The number of cents in a dollar– π.

Definition

• Term – any number that is added subtracted.– In the algebraic expression x + y, x and y are

terms.

• Example:– The expression x + y – 7 has 3 terms, x, y, and 7.

x and y are variable terms; their values vary as x and y vary. 7 is a constant term; 7 is always 7.

Definition

• Factor – any number that is multiplied.– In the algebraic expression 3x, x and 3 are

factors.

• Example: – 5xy has three factors; 5 is a constant factor, x and

y are variable factors.

Example: Terms and Factors

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

Definition

• Numerical Expression – a mathematical phrase that contains only constants and/or operations.

• Examples: 2 + 3, 5 ∙ 3 – 4, 4 + 20 – 7, (2 + 3) – 7, (6 × 2) ÷ 20, 5 ÷ (20 × 3)

Multiplication Notation

In expressions, there are many different ways to write multiplication.1) ab

2) a • b

3) a(b) or (a)b

4) (a)(b)

5) a ⤫ b

We are not going to use the multiplication symbol (⤫) any more. Why?Can be confused with the variable x.

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

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

Writing an algebraic expression with addition.

Two plus a number n

Writing an Algebraic Expression for a Verbal Phrase.

Orderof thewordingMatters

Writing an algebraic expression with addition.

Two more than a number

Writing an algebraic expression with subtraction.

The difference of seven and a number n

7 – n

Writing an algebraic expression with subtraction.

Eight less than a number

y – 8

Writing an algebraic expression with multiplication.

one-third of a number n.

Writing an algebraic expression with division.

The quotient of a number n and 3

Example

“Translating” a phrase into an algebraic expression:

Nine more than a number y

Can you identify the operation?

“more than” means add

Answer: y + 9

Example

4 less than a number n

Identify the operation?

“less than” means add

Answer: n – 4.

Why not 4 – n?????

Determine the order of the variables and constants.

Example

A quotient of a number x and12

Can you identify the operation?

“quotient” means divide

Determine the order of the variables and constants.

Answer: .

Why not ?????

Example

“Translating” a phrase into an algebraic expression, this one is harder……

5 times the quantity 4 plus a number c

Can you identify the operation(s)?

What does the word quantity mean?“times” means multiple and “plus” means add

that “4 plus a number c” is grouped using parenthesis

Answer: 5(4 + c)

Your turn:

1) m increased by 5.

2) 7 times the product of x and t.

3) 11 less than 4 times a number.

4) two more than 6 times a number.

5) the quotient of a number and 12.

1) m + 52) 7xt

3) 4n - 11

4) 6n + 2

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

3. Three times the difference of x and two

4. Three less than twice a number xA

5. Two more than three times a number x

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 +

n? __(2)

Term Number

Term Expression

Number Patterns

6(5) + 4

5(5) + 4

4(5) + 4

3(5) + 41

n? _____(5) + 4(n + 2)

How does the different part relate to the term number?

What’s the same?

What’s different?

Term Number Term Expression

Number Patterns

3 - 2(3)

3 - 2(2)

3 - 2(1)

3 - 2(0)1

n? 3 - 2(____)n - 1

How does the different part relate to the term number?

What’s the same?

What’s different?

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

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

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?

Common Difference Term NumbersX

X3 Number of Guests

We will add a column to our original table to do these

calculations:

Number of Guests

Bags of Spaghetti

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

What is the difference between all the numbers on the left and all the numbers on the right?

-1-1-1-1

We will now add another column to our table to do these calculations:

Number of Guests

Bags of Spaghetti

3n 3n – 1

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

Goal• To simplify expressions

involving exponents.

Vocabulary

• Power• Exponent• Base• Simplify

DefinitionA power expression has two parts, a base and an

exponent.

Power expression

ExponentBase

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.

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.

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

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.

29• 2

81 29• 2

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

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.

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

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

The product of three 4’s is 64.

b. –27; base –3

(–3)(–3)(–3)

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

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.

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.

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.

Work above the fraction bar

Simplify: Divide

Work below the fraction bar Grouping symbols

Expressions with fraction bars

Your Turn:Simplify the expression.

8 ÷ · 3 1 2

16 · 3

There are no groupingsymbols.

Divide.

Multiply.

5.4 – 32 + 6.2

5.4 – 9 + 6.2

–3.6 + 6.2

There are no groupingsymbols.

Simplify powers.

Subtract

–20 ÷ [–2(4 + 1)]

–20 ÷ [–2(5)]

–20 ÷ –10

There are two sets of groupingsymbols.

Perform the operations in theinnermost set.

Perform the operation insidethe brackets.

Divide.

Your Turn:

1. -3,236

3. 107

4. 16,996

Which of the following represents 112 + 18 - 33 · 5 in simplified form?

Your Turn:

Simplify 16 - 2(10 - 3)

Your Turn:

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

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.

Divide.

Your Turn:Simplify.2(–4) + 22 42 – 9

2(–4) + 22 42 – 9 –8 + 22 42 – 9

–8 + 22 16 – 9

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

Goal• To use the order of operations

to evaluate expressions.

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

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.

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.

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.

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

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:

2. -10

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

ALGEBRAIC MODEL

LABELS

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.

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.

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

Goal• To classify, graph, and

compare real numbers.• To find and estimate square

roots.

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

22 = 4

The square root of 4 is 2

32 = 9

42 = 16

52 = 25

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

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

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?

Example: Finding RootsFind the root.

Think: What number cubed equals –216?

(–6)(–6)(–6) = 36(–6) = –216 = –6

Your Turn:Find each root.

Your Turn:Find the root.

Think: What number to the fourth power equals 81?

Example: Finding Roots of Fractions

Find the root.

Think: What number squared

equals

Find the root.

Think: What number cubed equals

Find the root.

Think: What number squared

equals

Think: What number squared equals

Think: What number cubed equals

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.

Because 13 is closer to 16 than to 9, is closer to 4 than to 3.

You can use a guess-and-check method to estimate .

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.

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: , , .

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

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

Goal• To identify and use properties

of real numbers.

Vocabulary• Equivalent Expression• Deductive reasoning • Counterexample

Definition

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

show expressions that are equivalent for all real numbers.

Mathematical Properties

• Properties refer to rules that indicate a standard procedure or method to be followed.

• A proof is a demonstration of the truth of a statement in mathematics.

• Properties or rules in mathematics are the result from testing the truth or validity of something by experiment or trial to establish a proof.

• Therefore every mathematical problem from the easiest to the more complex can be solved by following step by step procedures that are identified as mathematical properties.

Commutative and Associative Properties

• Commutative Property – changing the order in which you add or multiply numbers does not change the sum or product.

• Associative Property – changing the grouping of numbers when adding or multiplying does not change their sum or product.

• Grouping symbols are typically parentheses (),but can include brackets [] or Braces {}.

Commutative Property of Addition - (Order)

Commutative Property of Multiplication - (Order)

For any numbers a and b , a + b = b + a

For any numbers a and b , a • b = b • a

45 + 5 = 5 + 45

6 • 8 = 8 • 6

50 = 50

48 = 48

Commutative Properties

Associative Property of Addition - (grouping symbols)

Associative Property of Multiplication - (grouping symbols)

For any numbers a, b, and c,

(a + b) + c = a + (b + c)

(ab)c = a (bc)

(2 + 4) + 5 = 2 + (4 + 5)

(2 • 3) • 5 = 2 • (3 • 5)

(6) + 5 = 2 + (9)

11 = 11

(6) • 5 = 2 • (15)

30 = 30

Associative Properties

Name the property that is illustrated in each equation.

A. 7(mn) = (7m)n

Associative Property of Multiplication

The grouping is different.

B. (a + 3) + b = a + (3 + b)

Associative Property of Addition

C. x + (y + z) = x + (z + y)

Commutative Property of Addition

The order is different.

Example: Identifying Properties

Name the property that is illustrated in each equation.

a. n + (–7) = –7 + n

b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3

c. (xy)z = (yx)z

Commutative Property of Multiplication

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 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)

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

The sum of a number and 0, the additive

identity, is the original number.

n + 0 = n 3 + 0 = 0

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

The product of a nonzero number and

its reciprocal, or multiplicative inverse,

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

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

A. 32n

B. 10 + 5b

Identify which property that justifies each of the following.

4 × (8 × 2) = (4 × 8) × 2

Associative Property of Multiplication

6 + 8 = 8 + 6

12 + 0 = 12

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

5 × 24 = 24 × 5

Commutative Property of Multiplication

-34 × 1 = -34

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

Goal• To find sums and differences

of real numbers.

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 + (–7)

Example: Adding & Subtracting on a Number Line

Start at 0. Move right to 3.

To subtract –6, move right 6 units.

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

3 – (–6) = 9

3 – (–6)

–(–6)

Example: Adding & Subtracting on a Number Line

–3 + 7 Start at 0. Move left to –3.

To add 7, move right 7 units.

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

–3 + 7 = 4

Your Turn:

–3 – 7 Start at 0. Move left to –3.

To subtract 7, move left 7 units.

–3–7

11 10 9 8 7 6 5 4 3 2 1 0

–3 – 7 = –10

Your Turn:

–5 – (–6.5) Start at 0. Move left to –5.

To subtract –6.5, move right 6.5 units.

8 7 6 5 4 3 2 1 0

–5 – (–6.5) = 1.5

– (–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

Use the sign of the number with the greater absolute value.

Different signs: subtract the absolute values.

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

–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.

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.

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

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

To subtract –247, add 247.Same signs: add the absolute

values.

–75 –247

The height of the iceberg is 322 feet.

What if…? The tallest known iceberg in the North Atlantic rose 550 feet above the ocean's surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet?

elevation at top of iceberg

minus elevation of the

Titanic

550 – (–12,468)

550 – (–12,468) = 550 + 12,468 To subtract –12,468, add 12,468.

= 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

Goal• To Find products and

quotients of real numbers.

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.

12The quotient of two numberswith the same sign is positive.

Example: Multiplying and Dividing Real Numbers

The quotient of two numberswith different signs is negative.

Multiply.

Example: Multiplying and Dividing Real Numbers

–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

Multiplicative inverses

Dividing by 5 is the same as multiplying by the

reciprocal of 5, .

Reciprocals

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

Helpful Hint

Example 2 Dividing by Fractions

Divide.

Example: Dividing with Fractions

To divide by , multiply by .

Multiply the numerators and multiply the denominators.

and have the same sign,

so the quotient is positive.

Divide.

Write as an improper fraction.

and have different signs,

so the quotient is negative.

Example: Dividing with Fractions

Divide.

and –9 have the same signs,

so the quotient is positive.

Your Turn:

Divide.

Multiply the numerators and multiply the denominators.

and have different signs,

so the quotient is negative.

Your Turn:

Check It Out! Example 2c

Divide.

To divide by multiply by .

The signs are different, so the quotient is negative.

• 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.

B. –22 ÷ 0

undefined

C. –8.45(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.

Example: Multiplying & Dividing with Zero

Multiply or divide.

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

A number is divided by zero.

A number is multiplied by zero.

Your Turn:

Find the distance traveled at a rate of 3 mi/h for 1 hour.To find distance, multiply rate by time.

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 13

= 15 4

• 43

Write and as improper fractions.34

3 1 13

15(4) 4(3)

= 6012

Multiply the numerators andmultiply the denominators.

and have the same sign, so

the quotient is positive.

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

Goal• To use the Distributive

Property to simplify expressions.

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-tile

This 1-by-1 square tile has an area of 1 square unit.

x-tile

This 1-by-x square tile has an area of x square units.

Area = 3(x + 2)

Area = 3(x ) + 3(2)

Model the Distributive Property using Algebra Tiles

MODELING THE DISTRIBUTIVE PROPERTY

x + 2+

The Distributive Property is used with Addition to Simplify Expressions.

The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

THE DISTRIBUTIVE PROPERTY

a(b + c) = ab + ac

(b + c)a = ba + ca

2(x + 5) 2(x) + 2(5)

2x + 10

(x + 5)2 (x)2 + (5)2 2x + 10

(1 + 5x)2 (1)2 + (5x)2 2 + 10x

y(1 – y) y(1) – y(y)

y – y

USING THE DISTRIBUTIVE PROPERTY

The product of a and (b + 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.

(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.

–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.

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

Section 1-7 Part 2

Goal• To use the Distributive

Property to simplify expressions.

Vocabulary• Term• Constant• Coefficient• Like Terms

The process of distributing the number on the outside of the parentheses to each term on the inside.a(b + c) = ab + ac and (b + c) a = ba + 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.

As a whole As two parts

Geometric Model for Distributive Property

Two ways to find the area of the rectangle.

As a whole As two partssame

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

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.

As a whole As two partssame

xy + xz

Write the product using the Distributive Property. Then simplify.

5(50 + 9)

5(50) + 5(9)

250 + 45

Rewrite 59 as 50 + 9.

Use the Distributive Property.

Multiply.

Example: Distributive Property with Mental Math

You can use the distributive property and mental math to make calculations easier.

9(50) - 9(2)

9(50 - 2)

450 - 18

Rewrite 48 as 50 - 2.

Multiply.

Subtract.

Example: Distributive Property with Mental Math

8(30 + 3)

8(30) + 8(3)

240 + 24

Multiply.

Your Turn:

12(98)

Rewrite 98 as 100 – 2.

Multiply.

Subtract.

12(100 – 2)

1200 – 24

12(100) – 12(2)

Your Turn:

7(30 + 4)

7(30) + 7(4)

210 + 28

Multiply.

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.

ax – bx = (a – b)x

Example

7x – 4x = (7 – 4)x

Combining Like Terms

Simplify the expression by combining like terms.

72p – 25p

72p and 25p are like terms.

Subtract the coefficients.

Example: Combining Like Terms

A variable without a coefficient has a coefficient of 1.

Write 1 as .

Add the coefficients.

and are like terms.

0.5m + 2.5n

0.5m and 2.5n are not like terms.

Do not combine the terms.

Caution!Add or subtract only the coefficients. 6.8y²

– y² ≠ 6.8

Simplify by combining like terms.

3a. 16p + 84p

16p + 84p

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.

+– 3y2x +– 3y2x

number

+– 3y2x

variable.

+– 3y2x

–1 is the coefficient of x.

3 is the coefficient of y2.

x is the variable.

y is the variable.

Each of these terms is the product of a number and a variable.

x2 x2y3 y3

Like terms have the same variable raised to the same power.

y2 – x2 + 3y3 – 5 + 3 – 3x2 + 4y3 + y

variable

power.Like terms

The constant terms –5 and 3 are also like terms.

Combine like terms.

SIMPLIFYING BY COMBINING LIKE TERMS

4x2 + 2 – x2 =

(8 + 3)x Use the distributive property.

= 11x Add coefficients.

8x + 3x =

Group like terms.

Rewrite as addition expression.

Multiply.

Combine like terms and simplify.

4x2 – x2 + 2

= 3x2 + 2

3 – 2(4 + x) =3 + (–2)(4 + x)

= 3 + [(–2)(4) + (–2)(x)]

= 3 + (–8) + (–2x)

= –5 + (–2x)

= –5 – 2x

–12x – 5x + x + 3a Commutative Property

Combine like terms.–16x + 3a

–12x – 5x + 3a + x1.

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

Goal• To solve equations using

tables and mental math.

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

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

(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)

2x = 600

ALGEBRAIC MODEL

At what speed would the jet have to fly to arrive in Chicago in 2 hours?

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

B. x/6 = 12

Assignment

Patterns, Equations, and Graphs

Section 1-9

Goal• To use tables, equations, and

graphs to describe relationships.

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

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

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

Example:Determine if the following ordered pairs satisfy the equation 2x + y = 5.

a.) (2, 1) b.) (3, – 4)

(2, 1) is a solution.

2x + y = 5

2(3) + (– 4) = 5

6 + (– 4) = 5

(3, – 4) is not a solution.

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)

y = 6x + 5

y = 6(0) + 5

y = 0 + 5

y = 55

(0, 5)

y = 6x + 5

y = 6(2) + 5

y = 12 + 5

y = 1717

(2, 17)

An engraver charges a setup fee of $10 plus $2 for every word engraved. Write a rule for the engraver’s fee. Write ordered pairs for the engraver’s fee when there are 5, 10, 15, and 20 words engraved.

Let y represent the engraver’s fee and x represent the number of words engraved.

Engraver’s fee is $10 plus $2 for each word

y = 10 + 2 · x

y = 10 + 2x

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

Writing Math

Number 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

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(–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

–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

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

alg1ch1

algebraic expression

algebraic expressions

number y y

number n

number systems

number of minutes

number line

number of cents

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