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Algebra I Notes – Unit Two: Variables

of 14 McDougal Littell: 1.1 – 1.3,1.5, 2.1, 2.6

Syllabus Objectives: 2.1 – The student will use order of operations to evaluate expressions.

2.2 – The student will evaluate formulas and algebraic expressions using rational numbers (with

and without technology).

Order of Operations (PEMDAS)

1. Parentheses – perform the operations within grouping symbols first

Note: When working inside grouping symbols, the order of operations rule begins again.

2. Exponents – evaluate powers

3. Multiply/Divide – multiply or divide in order from left to right

4. Add/Subtract – add or subtract in order from left to right

Ex: Evaluate the expression. 2

3 12 3 8 3

Step One: Parentheses 2

3 4 8 3

Step Two: Exponents (in brackets) 3 16 8 3

Step Three: Mult/Div (in brackets) 3 16 24

Step Four: Brackets (grouping symbols) 3 40

Step Five: Add/Sub 34

Ex: Evaluate the expression. 21

4 5 8 62

Step One: Exponents 1

4 5 8 362

Step Two: Mult/Div (left to right)

24 5 8 36

1

8 5 8 36

40 8 36

Step Three: Add/Sub (left to right) 32 36

4

Variable: a letter that represents a number

Variable (Algebraic) Expression: a collection of numbers, variables, and operations

Writing an Algebraic Expression (Translating Verbal Phrases)

In order to translate verbal phrases into algebra, we will use key words for each operation.



Examples of these key words are given:

Addition: sum, more than, plus, increased by

Subtraction: difference, less than, minus, decreased by

Multiplication: product, times, multiplied by

Division: quotient, divided by

Caution: Order is important for subtraction and division (these operations are NOT commutative)!

Ex: Translate the verbal phrases into algebraic expressions. (Use x for the variable.)

1. 2 more than a number 2x or 2 x

2. 5 less than a number 5x

3. 3 more than 5 times a number 5 3x or 3 5x

4. The sum of a number and 4, multiplied by 8 4 8x or 8 4x

5. 8 is decreased by the square of a number 2

8 x

6. The difference of a number and 2 divided by 3 2

3

x or 2 3x

7. 4 cubed divided by the sum of a number and 1

34

1x or

34 1x

8. 6 times the quantity of a number minus 10 6 10x or 10 6x

Evaluating Algebraic Expressions: to evaluate an algebraic expression, substitute the given value(s) in for

the variable(s) and simplify using the order of operations.

Ex: Evaluate 2

5 2 8 4x x when 2x .

Step One: Substitute 2x into the expression. 2

5 2 2 2 8 4

Step Two: Use the order of operations to simplify.

25 2 2 10 4 P

5 4 2 10 4 E

20 20 4 M/D

0 4 4 A/S



Ex: Evaluate 1 5

4 3 3 8 52 3

x y when 14x and 9y .

Step One: Substitute 14x and 9y into the expression. 1 5

4 14 3 3 8 9 52 3


14 14 3 3 8 15 5 P

2

14 14 3 3 7 5 P

2

14 14 3 3 12 P

2

7 256 3 12 56 36 16 36 M/D

2 7

52 A/S

Ex: Evaluate

26

33 15 8

6

x

x for 9x .

Step One: Substitute 9x into the expression.

26 9

33 9 15 8

6


6 63 9 15 8 P M

6

123 9 15 8 P(A)

6

3 9 15 2 8 P D

3 9 15 10 P(A)

27 150 M

177 A



Evaluating Formulas

Ex: A car traveled 150 miles in 3 hours. Find the average speed of the car using the formula

Average Speed = Distance

Time

d

t.

Solution: Average Speed = 150 mi

503 hr

Ex: Find the amount of time, in years, it would take for an amount of $600 invested at an annual

simple interest rate of 5% to earn $150. Use the formula I Prt .

600, 5% 0.05, 150, ?P r I t

Substitute values into the formula: 150 600 0.05I Prt t

Solve for t: 150 30 5t t 5 years

You Try:

1. Evaluate the expression

23a b a

ab when 1a and

1

2b .

2. Evaluate the expression 2

3 4x x when 2x .

3. Evaluate the expression 5 2 22 4x x when 25x .

QOD: Why is it important to have a specified order to performing operations?

Sample CCSD Common Exam Practice Question(s):

1. Simplify the expression:

28 16 8 6 2 6

A. –18

B. –10

C. 5

D. 12



2. Evaluate the expression 35 2 3 8y y when y = 6.

A. –13

B. –3

C. 11

D. 13

3. Evaluate the expression 2x – 8y + 9 when x = 3

2and y = 0.

A. –21

B. –3

C. 12

D. 15



Syllabus Objective: 2.3 – The student will use algebraic expressions to identify and describe the nth

term of a sequence.

Sequence: an ordered list of numbers

Finding the nth Term of a Sequence: look for a pattern within the sequence and write the pattern in terms

of n, which is the term number

Ex: Use the sequence 3, 6, 9, 12, …. Write the next three terms and write an expression for the

nth term.

Next three terms: We are adding 3 to each term to obtain the next term, so they are 15,18,21 .

nth term: When 1n , the term is 3; when 2n , the term is 6; when 3n , the term is 9;…

To obtain each term, we multiply the term number by 3. So the expression for the nth

term is 3n .

Ex: Use the sequence 12,8,4,0, 4,... Write the next three terms and write an expression for the

nth term.

Next three terms: We are subtracting 4 to each term to obtain the next term, so they are

8, 12, 16 .


To obtain each term, we multiply the term number by −4 and add 16. So the expression

for the nth term is 4 16n .

Challenge Ex: Use the sequence 4, 8, 16, …. Write the next three terms and write an expression

for the nth term.

Next three terms: We are multiplying each term by 2 to obtain the next term, so they are

32,64,128 .


To obtain each term, we raise 2 to the 1n power. So the expression for the nth term is

12

n.

You Try: Use the sequence 5,3,1, 1,... Write the next three terms and write an expression for the nth

term.

QOD: When a sequence is created by adding or subtracting a number to obtain the next term, where is

this number in the expression for the nth term?




Use the table below to write an equation that represents the value of y in terms of x.

A. y = x + 2

B. y = x + 3

C. y = 2x + 3

D. y = 3x + 2

Sample Nevada High School Proficiency Exam Questions (taken from 2009 released version H):

These are ONLY challenge exercises for Algebra 1. Look for patterns.

1. The first four terms of a sequence are shown below.

1 1 1 1

2 8 18 32

The sequence continues. What is the seventh term of the sequence?

A 1

256

B 1

98

C 1

64

D 1

60

2. The first five terms of a sequence are shown below. (Look for patterns of the first differences.)

4 10 28 82 244

The sequence continues. What is the sixth term of the sequence?

A 368

B 486

C 730

D 732

x 0 1 2 3 4 …

y 3 5 7 9 11 …

2

1The pattern is

2x



Syllabus Objective: 2.4 – The student will identify and apply real number properties using

variables, including distributive, commutative, associative, identity, inverse, and absolute value to

expressions or equations.

Real Number Properties

Commutative Property of Addition: a b b a

Real Number Example: 3 5 5 3 2

Commutative Property of Multiplication: a b b a

Real Number Example: 2 3 3 2 1

9 4 4 9 6

Associative Property of Addition: a b c a b c

Real Number Example: 4.2 5 3 4.2 5 3 3.8

Associative Property of Multiplication: a b c a b c

Real Number Example: 1 1

3 6 3 6 92 2

Identity Property of Addition: 0a a Note: The additive identity is 0.

Real Number Example: 3 0 3

Identity Property of Multiplication: 1a a Note: The multiplicative identity is 1.

Real Number Example: 6.2 1 6.2

Inverse Property of Addition: 0a a

Note: The opposite of a number is the additive inverse.

Real Number Example: 4 4 0

Inverse Property of Multiplication: 1

1aa

Note: The reciprocal of a number is the multiplicative inverse.

Real Number Example: 1

2 12



Zero Product Property: If 0a b , then either 0a or 0b .

Distributive Property: a b c a b b c

Real Number Example: Use the distributive property to multiply 12 32 .

We can rewrite 32 as 30 + 2 in order to use the distributive property so that we can perform the

multiplication in our heads. 12 30 2 12 30 12 2 360 24 384

Absolute Value: the distance a number is away from the origin on the number line

Ex: What two numbers have an absolute value of 4?

Solution: There are two numbers that are 4 units away from the origin, as shown in the number line.

They are 4 and 4 .

Notation: x x if 0x and x x if 0x

Ex: Evaluate the expression. 2 4 1

Note: Absolute value bars are like grouping symbols in the order of operations.

Evaluate the expression inside the absolute value bars. 2 3

Evaluate the absolute value and simplify. 2 3 6

Ex: Evaluate the variable expression 2 2

3 4x

x xy

when 1x and 2

3y .

Substitute 1x and 2

3y into the expression.

2 2 11 3 4 1

2

3



Use the order of operations to simplify.

2 2 11 3 5

2

3

21 15

2

3

31 15 2

2

1 15 3

16 3 19

You Try:

1. Identify the property shown. c d e d c e

2. Show that the distributive property does not work over absolute value bars by showing that

3 5 8 3 5 3 8 .

QOD: Determine if the statement is true or false. Explain your answer. “The reciprocal of any number is

greater than zero and less than 1.”

Sample Nevada High School Proficiency Exam Questions (taken from 2009 released version H):

1. Which equation illustrates the commutative property of addition?

A 4 3 3 4x x x x

B 3 4 0 3 4x x x x

C 4 3 4 3x x x x

D 3 4 3 4 3x x x x x

2. The equation below illustrates a property of real numbers.

3 7 9 21 27x x

Which property is illustrated by the equation?

A associative property

B commutative property

C distributive property

D identity property



3. Which equation shows the use of the inverse property of multiplication?

A 1

7 17

xx

B 7 1 7x x

C 7 0 0x

D 1 1

7 77 7

x x



Syllabus Objective: 2.5 – The student will simplify algebraic expressions by adding and subtracting

like terms.

Terms: algebraic expressions separated by a + or a – sign

Constant Term: a term that is a real number (no variable part)

Coefficient: the numerical factor of a term

Like Terms: terms that have the exact same variable part (must be the same letter raised to the same

exponent) Note: Only like terms can be added or subtracted (combined).

To add or subtract like terms, add or subtract the coefficients and keep the variable part.

Ex: Simplify 3 + 4.

This means we are adding 3 smiley faces plus 4 smiley faces. Therefore, we have a total of 7 smiley

faces, which we can write as 7.

Ex: Use the expression 2 2 2

1 3 4 5x xy x y to answer the following questions.

1. How many terms are in the expression?

Solution: There are 5 terms.

2. Name the constant term(s).

Solution: There is one constant term, 1.

3. How many like terms are in the expression?

Solution: There are 2 like terms. 2 2

3 and x x

4. What is the coefficient of the xy term? What is the coefficient of the fourth term?

Solution: The xy term has a coefficient of 4. The third term has a coefficient of 1.

5. Simplify the algebraic expression.

Solution: Combine like terms. 2 2 2 2 2

1 3 1 4 5 1 2 4 5x x xy y x xy y

6. How many terms are in the simplified form of the algebraic expression?

Solution: There are 4 terms.



Simplifying an Algebraic Expression

Ex: Simplify the expression 4 8 5 1x x .

Step One: Eliminate the parentheses using the distributive property. 4 32 5 1x x

Step Two: Combine like terms.

4 5 32 1

1 33

33

x x

x

x

Ex: Simplify the expression 2

5 2 3 8 4n n n

Step One: Eliminate the parentheses using the distributive property. 2 2

5 6 16 4n n n

Note: The expression in parentheses is being multiplied by 2n , which is what was

“distributed”.

Step Two: Combine like terms.

2 2

2

5 6 4 16

5 10 16

n n n

n n

Note: The answer may be written as 2

10 16 5n n by the commutative property.

Ex: Write a simplified expression for the perimeter of a rectangle with length 7x and width

2x . Note: The formula for the perimeter of a rectangle is 2 2P l w .

Substitute the length and width into the formula. 2 7 2 2x x

Simplify using the distributive property and combining like terms. 2 14 2 4

4 10

x x

x

You Try:

1. Simplify the expression 4 3 8x x .

2. Write a simplified expression for the area of a triangle with a base of 4x and a height of

2 1x .

QOD: Explain in your own words why only like terms can be added or subtracted.





5 3 4x x

A. 7x – 4

B. 7x – 32

C. 2x + 1

D. 2x – 7


2 25 2 6 4 6 1x x x x

A. 26 5x x

B. 23 10 5x x

C. 211 2 5x x

D. 211 6 5x x

3. Write an expression for the perimeter of the rectangle:

A. 2xy

B. 6xy

C. 4x + 2y

D. 2 24 2x y

2x

y

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