UNIT 5

EXPRESSIONS

NAME:______________________

CLASS:_____________________

TEACHER: Ms. Schmidt _

Powers and Exponents

Solve the following 8 x 4 x 2 =

What are each of the digits above called?

1. Evaluate: 2 x 2 x 2 x 2

What is another way we can write the example of number 2?

24

Vocabulary:

Factors-

_____________________________________________________________________________

Exponential Form-

_________________________________________________________________________

Base -

_____________________________________________________________________________

Exponent-

_____________________________________________________________________________

Write each of the following as a product of the same factor.

2. 106 6. 54

3. 62 7. 106

4. 41

Write in exponential form.

8. 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9

9. 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3

Powers and Exponents

Evaluate:

10. 35 =

11. 43 =

12. 104 =

13. 107 =

**Hint-

If 81 is the product and the base is 3, what is the exponent?

13. 3? = 81

*Think- how many times do I need to multiply the base by itself to get 81?

Classwork:

Write each of the following as the product of the same factor.

1. 82 2. 97

3. 43 4. 14

5. 710 6. 1012

Write in exponential form.

7. 3 ∙ 3 ∙ 3 ∙ 3 8. 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5

9. 12 ∙ 12 ∙ 12 ∙ 12 ∙ 12 10. 6 ∙ 6 ∙ 6 ∙ 6 ∙ 6

Evaluate.

11. 105 12. 73

13. 85 14. 1010

Find the exponent in each of the following

15. 3? = 27 16. 4? = 64 17. 5? = 125 18. 6? = 1296

Order of Operations

Vocabulary

Numerical Expression- ________________________________________________________

Some mathematical expressions involve several operations. Does the order in which these

operations are done make a difference?

6 + 3 x 4 6 + 3 x 4

9 x 4 = 36 6 + 12 = 18 Which is correct, and why?

***Please Excuse My Dear Aunt Sally!

Solve each of the following below.

1. 1 + 7 x 3 2. (1 + 7) x 3

Do they have the same answer? Are parenthesis important?

Order of Operations

Evaluate each of the following numerical expressions.

3. 21 – (18 + 2) 4. 15 + 5 x 3 – 2 5. 32 – 3 x 7 + 4

6. 36 ÷ 9 x 2 7. 12 ÷ 24 8. 28 ÷ (3 – 1)

9. 1

4𝑥 (

1

5+

1

2) −

1

10 10. (

1

4)2𝑥

1

2−

1

10 11. (.25)2 +

1

2 x 2

12. . (3.4+4.4)×(4.8÷0.4) 13. (8.5−5.9)×(4.8÷1.6) 14. 18( ½) – 9

15. 6 + 52- 12 + 4 16. 5 x ( 2-1)2+ 17 17. (27 ÷ 3)2 – 3

Order of Operations

18. On his math test, Steve scored 5 points on each of 5 questions, 2 points on each of 2

questions and 3 points on each of 4 questions. To find the number of points he scored, evaluate

the expression below:

55 + 22 + (3 x 4)

19. Robert and Lily solved the expression below. Robert got an answer of 94 and Lily got an

answer of 64.

80 ÷ (2 x 1)3 + 54

a) Who is correct?

b) Explain why it is important to use the proper order of operations to solve.

20. 12 − (8

5+ 3 ÷

2

3 ) 21. 4 − (

4

3𝑥

5

4+

11

6 )

Properties

Properties of Operations

Commutative Property- numbers can be added or multiplied in any order to give you the same

answer.

Ex:

Associative Property- numbers are grouped differently to give you the same answer.

Ex:

Identity Property of Addition- when you add anything with zero is equals itself.

Ex:

Identity Property of Multiplication- when you multiply anything with one it equals itself.

Ex:

Zero Property- anything multiplied by zero gives you a product of zero

Ex:

Distributive Property- this is when you break down one of the numbers, multiply the factor’s

addends by a number, then add the products together.

Ex:

Using your notes, name the property for each below.

1. 32 x 1 = 32

2. 10 + 6+ 2 = 6 + 2 + 10

3. 8 x 14 = (8 x 14)+(8 x 4)

4. (3 x 5) x7 = 3x (5 x7)

5. 0 + 12 = 128 + 0 = 8

6. mt = tm

7. 11 + ( 4 + 8) = (11 + 4) +8

8. 5 x (9 + 7) = (5x9) + (5x7)

Properties

Properties of Addition

Property Example:

Commutative

Associative

Identity

Properties of Multiplication

Property Example:

Commutative

Associative

Identity

Zero

Distributive

Find the missing number in the equations and name the property.

9. 4 x (7 x 2) = (4 x 7) x ____ 10. 11 + 7 = 7 + ____

11. 11 x ____ = 11 12. 3 x 2 x 8 = 8 x 3 x ___

13. 8 x ___ = 0 14. 9 + ___ = 9

15. 7 + (4 + 9) = ( __ + 4) + 9 16. 3 ( 4 + 1) = (__ x 4) + ( __ x 1)

Properties

Fill in each of the table with an example of each of the properties!

Properties of Addition

Property Example:

Commutative

Associative

Identity

Properties of Multiplication

Property Example:

Commutative

Associative

Identity

Zero

Distributive

Distributive Property

Distribute – to give something to each group!

( + )

Distributive Property

There are two different ways we can show hot items are distributed using mathematical

expressions. Both ways will give the same result in the end, they will be equal!

Example: If there are 4 students and each brought 2 folders and one notebook to class.

We can find how many supplies they brought two different ways.

4( 2 + 1) 4( 2 + 1)

2(3) 8 + 4

12 12

Order of Operations- Distributive Property 4( 2+ 1) = 4(2) = 4 (1)

solve parenthesis first.

Use the Distributive Property to complete the table

Expression Rewrite Expression Evaluate

1) 2 (4 + 1)

2) 7 (8 + 4)

3) 9( 3 + 9)

4) 2 ( 4 + 2 + 6)

5) 3 ( n + 5)

6) (a + 9)

7) 5 ( x + y + z)

Distributive Property

8) Each day, you run on a treadmill for 10 minutes and lift weights for 15 minutes. Write

and solve an expression using the distributive property to find out how many minutes of

exercise you do in 5 days.

Use the following information for number 9.

Prices

Museum Exhibit

Child (under 8) $8 $4

Student $12 x

9) A class of 20 students visits an art museum and a special exhibit while there. Use the

distributive property to write and simplify an expression for the cost.

Use the Distributive Property to solve each of the following.

10) 5 x 24 11) 6 x 81 12) 7 x 93

Use the Distributive Property to solve each of the following area models.

13) 2 6 14) 2 10

2 4

Distributive Property

Use the Distributive Property to complete the table

Expression Rewrite Expression Evaluate

1) 2 (x + 1)

2) 5 (b + 4)

3) 9( 3 + 9)

4) 2 ( x + y + 6)

5) 3 ( n + 5)

6) 8 (a + 9)

7) 5 ( x + z)

Use the Distributive property to solve each of the following!

8) 5 x 25 9) 2 x 42 10) 7 x 63

Translating Expressions

Expressions are mathematical phrases (sentences)

Numerical Expression Algebraic Expressions

*numbers and operation signs only

(all of your order of operation problems)

2 x (7-4) + 10

*numbers, operation signs and variables

14 + n nn-1

Key Vocabulary:

Variable- ______________________________________________________________

Coefficient- ___________________________________________________________

x + 5

x – 5

5x

5

x

Remember….

When translating an algebraic expression using variables:

DO NOT USE THE MULTIPLICARION SIGN!!

Why? Which of the following is the variable? 4 x f

We now write 4f when translating four times f. When the 4 is next to the (f) it means

multiplication!

More Than / Less Than

___________________

From

___________________

Translating Expressions

Identify the variable and coefficient in each expression below.

1) 9 – 8n 2) 3k + 5 3) 22 – p + 6r

Write each phrase as an algebraic expression.

4) 7 less than m

5) The quotient of 3 and

y

6) The total of 5 and c

7) The difference of 6

and r

8) N divided by 2

9) The product of k and

9

10) The Jets won 5 more games than they lost. Let “L” represent the amount of games the Jets

lost. Write an algebraic expression to show the number of games they won.

11) Lucas has 2 times as much money as Hannah. Let “m” represent the amount of money

Hannah has. Write an algebraic expression to show the amount of money that Lucas has.

12) Kara is 4 years older than Donna. Let “a” represent Donna’s age. Write an algebraic

expression to show ow old Kara is.

Evaluating Expressions

2 Types of Expressions

Numerical Expression Algebraic Expression

Numbers and operations only!

Ex: 16 + 12 ÷ 4

Operations, numbers and variables!

Ex: n+ 12 ÷ 4

Steps to Evaluating Expressions!

1) Write the original algebraic expression

2) Substitute (replace) the variable with the correct number.

3) Use order of operations to solve.

Substitution = replace the variable with the value given!

Evaluate each of the following when n = 3 and x = 4.

1) n + 4

2) 6n

3) nx

4) 9

n

Evaluate each of the following when Evaluate if a = 2 and b = 3

5) ab 6) 10 – 1 7) 9a + b

Evaluating Expressions

8)

9)

10)

n 3 5 8 12 25

2 + 3n

k 2(k – 4)

6

9

12

x 3( x – 4) = x2

5

6

7

8

9

Combining Like Terms

“Like Terms” can be thought of as a family of people!

Families always come together!

Vocabulary

Variable- a symbol that represents an unknown number

Coefficient- the number in front of the variables(s) in a term

Term- when addition or subtraction signs separate an algebraic expression in to parts, each part

is a term.

Like Terms- terms that have the same variable raised to the same power but, can have different

coefficients

Constant- number all by itself

Combine- put something together (SIMPLIFY!)

Identify the variables, coefficients, constants, terms and like terms.

1) 9x + 3 – 4x 2) 3b + 5n + 2p – 11

Variables- Variables-

Coefficients- Coefficients-

Constants- Constants-

Terms- Terms-

Like Terms- Like Terms

Combining Like Terms

Determine whether each of the following are like or unlike terms.

1) Apple, Orange

2) Broccoli, Asparagus

3) 10x, 9

4) 12x, 12y

5) 10x, 2x

6) 2 apples, 3 apples

7) 8a, 2a

8) 19c, 23c

9) 25, 98f

10) 3 fire engines, 9 fire engines

11) 5t, 9t

12) 34x, 27x, 12x

13) 5 quarters, 9 dimes

14) 12w, 25w

15) W, a, t, r

16) 2a, 5a, 7a

Rules to Combine Like Terms!

1) Use shape (or color) to determine like term families

2) Add the coefficients of the like terms together

3) Bring the variables along

Examples:

1) 13b + 2b 2) 8a + 5b 3) 9m – 4m

4) 2x + 4 + 3x 5) 8a + 5a + 9 -1 6) 3x – 2x + 2y

Combining Like Terms

Try these on your own: Simplify the expressions

1) 5x + 3x 2) 8y + 4y 3) 6w + 3w + w

4) 2x2+ x2+ 8 5) 3x + 2x + y + 3y 6) 7 + 6x + 2 + 3x

7) x + 2x + 2y + 3y + 8 9) 9x + 8y + 4 + 2x + y – 2 10) 0.5x – 0.25x – 2

11) 7y – 0.2x + 0.4x – 0.9y 12) 8 + 0.7y – 8 13) 0.23z – 10 + 0.32

14) 1

2𝑥 −

1

3𝑥 − 8 15)

1

4−

1

6𝑧 + 5 +

9

2𝑥 16)

3

4𝑑 +

2

3𝑥 −

1

8𝑑 +

2

6𝑥

17) 51

2𝑥 − 10 + 9

5

8 𝑦 18) 10

1

2𝑤 − 9

1

3𝑘 − 8𝑤 19) 5k – 0.34g + 1g – 7

Challenge: 3x + 4y + 8w - 20 + 3ab + 8x + 2ab + 4y + 5w – 10

Distributing Expressions

Distributive Property- is used to multiply a single term and two or more terms inside a set

of parentheses.

Is 6(13) and 6(10 + 3) the same?

Equivalent expressions- expressions that have the same value.

The expressions 3( 2 + 4) and (3 x 2) + ( 3 x 4) are equivalent expressions because they have

the same value, 18

What does it mean to distribute?

Example: We want to distribute cookies to both Frosty and Santa so, that each of them will

get cookies!

( + ) ( ) + ( )

Distributing Expressions

Use the distributive property to write an equivalent expression. Then, use it to evaluate the

expression.

1) 2( 3 + 9)

2) 2( 6 + 4)

3) 5(5 + 3)

4) 7(9 + 11)

Rewrite using the distributive property in terms of x. (With variables)

5) 5 (2x + 5)

6) 3 ( 2 + 10b)

7) 7 ( 2x – 6)

8) 9 (4w + 12)

9) 8 ( 8x + 9)

10) 3(5t + 9 )

11) 4 ( 7t + 8 + z)

12) 2( 8y + 9x – 1)

13) 6(x + y + 5)

14) 8(4x + 2x – 2)

Factoring Numerical Expressions

Vocabulary

Greatest Common Factor- (GCF) The greatest common factor is the largest common factor of

two or more given numbers

Numerical Expressions- a number phrase without an equal sign (numbers and operation signs

only)

Equivalent Expressions- two or more expressions that have the same value.

**You can use the greatest common factor (GCF) and the Distributive Property to factor

numerical expressions.

Factoring- reverse Distributive Property

Example 1: 27 + 39

Step 1: Find the GCF of 27 and 39.

Step 2: Divide 27 and 39 by the GCF.

Step 3: Rewrite the expression as the Distributive Property

Example 2: Find an equivalent expression for 15 + 45

Factoring Numerical Expressions

Practice:

1) 24 + 40 2) 18 + 30 3) 21 + 42

4) 27 + 54 5) 60 + 12 6) 36 + 54

7) 35 + 45 8) 28 + 49 9) 16 – 12

10) 9 + 21 11) 80 – 56 12) 36 + 30

13) 63 + 9 + 81 14) 12 + 24 + 26 15) 4 + 66 + 88

16) 49 – 28 17) 102 – 10 + 2 18) 45 + 60

Factoring Numerical & Algebraic Expressions

Vocabulary

Greatest Common Factor- (GCF) The greatest common factor is the largest common factor of

two or more given numbers

Numerical Expressions- a number phrase without an equal sign (numbers and operation signs

only)

Algebraic Expressions- a combination of variables, numbers and at least one operation. ** NO

equal sign

Equivalent Expressions- two or more expressions that have the same value.

Example 1: Factoring Algebraic Expressions.

Factor the following algebraic expressions using the distributive property and the GCF

27x – 39

Step 1: Find the GCF

Step 2: Divide the terms by the GCF

Step 3: Bring down all the variables

Example 2:

Factor the following algebraic expressions using the distributive property and the GCF

36x + 120

Step 1: Find the GCF

Step 2: Divide the terms by the GCF

Step 3: Bring down all the variables

Factoring Numerical & Algebraic Expressions

Practice:

1. 15m + 35n

2. 40 k – 25 c

3. 6x + 16

4. 12b + 24

5. 20x + 25

6. 6y + 3

7. 7x – 56

8. 18x + 42y

9. 8x + 12y + 20

10. 48x – 32r – 8

11. 8x – 5y

12. 8c – 2

13. 3b + 27

14. 2x + 10 + 8y + 6b

15. 18x + 24b + 36a + 54

16. 20 + 35x + 15b + 40a + 90t

17. Bonus: a + 22b + 44c + 98d + 16t + 88m + 100