Course 1 Unit 6

Expressions

Name: ___________________

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Lesson 6-1: Powers and Exponentsbase – _________________________________________________________________

exponent – ____________________________________________________________

Example:

powers – _____________________________________________________________

Example:

perfect squares – ______________________________________________________

Example 1Write 6 x 6 x 6 x 6 using an exponent.

_____ is the base and it is multiplied _____times.So _____ is the exponent.

= ______Example 2Write 4 x 4 x 4 using an exponent.

_____ is the base and it is multiplied _____times.So _____ is the exponent.

= ______

Got it? Write each product using an exponent.1) 7 x 7 x 7 x 7 2) 9 x 9 x 9 x 9 x 9 x 9 x 9

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Example 3Write 52 as a product of the same factor. Then find the value.

52 = ______ x ______

= _______

Example 4Write 1.53as a product of the same factor. Then find the value.1.53= ______x ______x ______Multiply:

Add the decimal back in with 3 places to the right of the decimal.= ________

Example 5

Write ( 12)3

as a product of the same factor. Then find the value.

( 12)3

= ______ x ______ x _______

= _______

Got it? Write each power as a product of the same factor. Then find the value.

3) 105 4) 2.12 5)( 14)2

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Example 6The zoo has an aquarium that holds around 74 gallons of water. About how many gallons of water does the aquarium hold? 74 = ______ x _______ x _______ x _______Multiply:

So, the aquarium holds about ___________ gallons of water.

Got it? 6) Michigan has more than 104 inland lakes. Find the value of 104 .

Guided Practice:Write each product using exponents.

1. 8 x 8 x 8 = ___________ 2. 1 x 1 x 1 x 1 x 1 = ___________

Write each power as a product of the same factor. Then find the value.

3. ( 17)3

= ______________ 4. 1.42=¿¿

5. Coal mines have shafts that can be as much as 73 feet deep. About how many feet deep into Earth’s c rust are these shafts?

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Lesson 6-2 Numerical Expressionsnumerical expression – _________________________________________________________________________________________________________________________

Example: _____________________

order of operations – ___________________________________________________________________________________________________________________________

PEMDAS or _____________________________________________________________1) Simplify anything inside _______________________2) Evaluate the _______________________3) ___________________ and ____________________ in order from left to right.4) _____________________ and ___________________ in order from left to right.

Example 1Find the value of the numerical expression. 10 – 2 + 8Use an ice cream cone!1. There are no parentheses or exponents.2. There are no multiplication or division symbols.3. Add and subtract from left to right.

Example 2Find the value of the numerical expression. 4 + 3 x 5Use an ice cream cone!1. There are no parentheses or exponents.

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2. Multiply first.

3. Then add:Got it? Evaluate each expression. Use an ice cream cone!1) 10 + 2 x 15 2) 16 ÷ 2 x 4

Example 3Find the value of the numerical expression.

20 ÷ 4 + 17 x (9 – 6)Use an ice cream cone!1. Solve inside the parentheses first.

20 ÷ 4 + 17 x (9 – 6)

2. There are no exponents.3. Divide and multiply left to right.

4. Add and subtract left to right.

Example 43 x 62 + 4Use an ice cream cone!1. There are no parentheses.2. Simplify the exponent first.

3 x 62 + 4= 3 x 36 + 4

3. Multiply.36 x 3 = 108

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4. Add.108 + 4 = 112

Example 55 + (82-2) x 2Use an ice cream cone!1. Because the exponent is inside the parentheses, simplify the exponent first.

5 + (82-2) x 2= 5 + (64 – 2) x 2

2. Then solve (64-2).

3. Multiply.5 + 62 x 2

4. Add.

Got it? Evaluate each expression. Use an ice cream cone!3) 25 x (5 – 2) ÷ 5 – 12

4) 24 ÷ (23 + 4)

5) 22 ÷ 2 x 32

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Example 6Write an expression for the total cost of 5 candies, 2 candles, and 4 lip balms. Find the total cost.

5 x $5 + 2 x $7 + 4 x $2= 25 + 2 x 7 + 4 x 2= 25 + 14 + 4 x 2= 25 + 14 + 8

= _______Got it? 6) Alexis and 3 friends are at the mall. Each person buys a pretzel for $4, sauce for $1, and a drink for $2. Write an expression for the total and find the total cost.

Guided Practice: 1. 9 + 3 – 5 2. (26 + 5) x 2 – 15

3. 52 + 8 ÷ 2 4. 27 ÷ (3 + 6) x 5 – 12

5. Tickets to a play cost $10 for members and $24 for nonmembers. Write an expression to find the total cost for 4 nonmembers tickets and 2 member tickets. Then find the total cost.

Lesson 6-3 Algebra: Variables and Expressions

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algebra – ______________________________________________________________

variable – _____________________________________________________________

algebraic expressions –_________________________________________________

What is the difference between a numeric expression and an algebraic expression?

Numeric expression ONLY has _______________ and operations. Algebraic expression has _______________ AND _______________ AND

operations.

Variables and MultiplicationThe letter x is often used as a variable. To avoid confusion, we show ______________________________in other

ways:5 ∙ x 5(x) 5x

Example 1Evaluate 16 + b if b = 25.

Replace b with 25 and solve.16 + b

= 16 + ______= _______

Example 2:Evaluate x – y if x = 64 and y = 27.

Replace x with 64 and y with 27x – y

= ______ – ______~ 9 ~

= _______

Example 3

Evaluate 6x if x = 12.

6x

= 6 ∙ 12= 61 ∙

12

= 62= ______

Got it? Evaluate each expression if:

a = 6, b = 4, and c = 13.

1) a + 8 2) a – b 3) 9c

Example 4Evaluate 5t + 4 if t = 3.

5t + 4= 5(3) + 4

= ______ + 4= ______

Example 5

Evaluate 4x2 if x = 184x2

= 4 ∙ ( 18)2

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= 4 ∙ (18 ∙ 18)

= 4 ∙ ¿¿

= 41 ∙ 164= ¿¿

Example 6

Evaluate 10a + 7 if a = 15.

10a + 7

= 10(15)+ 7

= 101 ∙ 15 + 7

= ¿¿ + 7= 2 + 7= ______

Got it?

Evaluate each expression if d = 12 and e = 13.

4) 2d – 5 5) 50 – 3d 6) 9e2

Example 7Emily is wrapping a gift for her brother’s birthday. The box has side lengths that are 12 foot. Use the expression 6s2, where s represents the length of a side, to find the surface area of the box she is wrapping. Write your answer in square feet.

6s2 = 6 ∙ (12 ¿¿2

= 6 ∙ (12 ∙ 12)

= 6 ∙ ¿¿

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= 62 ∙ ¿¿

= ¿¿ or _______ square feet.

Got it? 7) The amount of money that remains from a 20-dollar bill after Malina buys 4 party favors for p dollars each is 20 – 4p. Find the amount remaining if each favor cost $3.Guided Practice:

Evaluate each expression if m = 4, z = 9, and r = 16 .

1. 3 + m 2. z – m 3. 12r

4. 4m – 2 5. 60r – 4 6. 3 r2

7. The amount of money that remains from a 20-dollar bill after Malina buys 4 party favor for p dollars each is 20 – 4p. Find the amount remaining if each favor cost $3.

Lesson 6-4 Algebra: Write Expressionsdefining a variable – ___________________________________________________

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Example 1Write the phrase as an algebraic expression.

Eight dollars more than Ryan earned

Example 2Write the phrase as an algebraic expression.

Ten dollars less than the original price

Example 3Write the phrase as an algebraic expression.

Four times the number of gallons

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Got it? Write an expression for each situation. 1) four points fewer than the Bulls scored

b = bulls score expression: _________

2) 12 times the number of feetf = feet expression: _________

3) the total cost of a shirt and an $8 pair of sockss = shirt expression: _________

Example 4Write the phrase “5 less than 3 times the number of points” as an algebraic expression.

Got it? 4) Write the phrase “$3 more than four times the cost of a pretzel” as an algebraic expression.

c = ____________ expression: ___________

Example 5

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Terry bought a magazine for $5, and 2 bottles of nail polish. Write an expression to represent the total amount she spent. Then find the total amount if each bottle of nail polish cost $3.

Step 1: 2 bottles of nail polish plus $5Step 2: let n = nail polish

Step 3: ___________To solve, insert 3 for n.

2(_____) + 5= ______ + 5

= $______Got it? 5) Shoko bought a box of popcorn for $3.50 and three medium drinks. Define a variable and write an expression to represent the total amount she spent. Then find the total amount if one drink costs $1.50.

Guided Practice:Define a variable and write each phrase as an algebraic expression. 1. four times more money than Elliotm = __________________ expression: ______________________

2. half as many pages as Georgeb = __________________ expression: ______________________

3. the width of a box that is 4 inches less than the lengthL = __________________ expression: ______________________

4. the cost of 5 CDs and a $12 DVDC = __________________ expression: ______________________

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Lesson 6-5: Algebra: Propertiesproperties – ____________________________________________________________

equivalent expressions – _______________________________________________

• Example: _____________________

• Non example: ______________________

Example 1Determine whether the two expressions are equivalent. If so, tell what property is applied. If not, explain why.

15 + (5 + 8) and (15 + 5) + 8They are equivalent by the

____________________________________________. 15 + (5 + 8) = (15 + 5) + 8

Example 2Determine whether the two expressions are equivalent. If so, tell what property is applied. If not, explain why.

(20 – 12) – 3 and 20 – (12 – 3)

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The expressions are not equivalent because the _________________________

is not true for ___________________________.(20 – 12) – 3 ≠ 20 – (12 – 3)

Example 3Determine whether the two expressions are equivalent. If so, tell what property is applied. If not, explain why.

34 + 0 and 34The expressions are equivalent by the

__________________________________.34 + 0 = 34

Example 4:Determine whether the two expressions are equivalent. If so, tell what property is applied. If not, explain why.

25 ÷ 5 and 5 ÷ 25The expressions are not equivalent because the

_________________________ does not apply to __________________________.

25 ÷ 5 ≠ 5 ÷ 25Got it? Determine whether the two expressions are equivalent. If so, tell what property is applied. If not, explain why.1) 5 x (6 x 3) and (5 x 6) x 3

2) 27 ÷ 3 and 3 ÷ 27

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3) (7 – 2) – 1 and 7 – (2 – 1)

4) 12 • 1 and 12

Example 5In a recent season, the Kansas Jayhawks had 15 guards, 4 forwards, and 3 centers on their roster. Write two equivalent expressions using the Associative Property that can be used to find the total number of players on their roster.

15 + (4 + 3) = (15 + 4) + 3

Got it? 5) Drew earned $7 babysitting and $12 cleaning out the garage. Write two equivalent expressions using the Commutative Property that can be used to find the total amount he earned.

Example 6

The area of a triangle can be found using the expression 12bh, where b is base and h is height. Find the area of the triangle shown.

12bh = 12(15)(12)

Because of the Commutative Property, switch the 15 and 12.12(12)(15)

6(15) = ________ square feet

Got it?

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6) Victoria earned $6 an hour while working 11 hours over the weekend. She put 13 of what she earned in a savings account. Find out how much she put into the account.

Guided Practice:Determine whether the two expressions are equivalent. If so, tell what property is applied. If not, explain why.1. (35 + 17) + 43 = 35 + (17 + 43)

2. 59 x 1 = 59

3. What two operations do not work for the Commutative and Associative Property?

4. What two operations only work for the Commutative and Associative Property?

Lesson 6-6: The Distributive Property Part 1Distributive Property – _________________________________________________________________________________________________________________________Example: 3(4 + 2) = 12 + 6

a(b + c) = ab + ac Example 1

Find 9 x 413 by using the Distributive Property.

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Think of the problem as 9(4 + 13)

Use “swoops”.

9(4 + 13)

______ + ______ = _______

Got it? Find each product using the Distributive Property.

1) 5 x 235 2) 12 x 214 3) 2 x 3.6

Example 2Use the Distributive Property to rewrite the expression 2(x + 3).

2(x + 3)=2(x) + 3(2)

= ______ + _______Got it? Use the Distributive Property to rewrite each expression.4) 8(x + 3) 5) 5(9 + x) 6) 2( x + 3)

Example 3Francesca is making jewelry for four friends. Each pair of earrings uses 4.5 centimeters of wire and each bracelet uses 13 centimeters. Write two equivalent expressions and then find how much total wire is needed.

4(4.5) + 4(13) = 4(4.5 + 13)18 + 52 = 704(17.5) = 70

So, she needs _______ centimeters of wire.

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Guided Practice:1. Find 9 x 8 23 using the distributive property. Show work.

Use the Distributive Property to simplify.2. 3(x + 1) 3. 5(x + 8) 4. 4(x + 6)

Lesson 6 – 6 Distributive Property Part 2factoring the expression – _____________________________________________________________________________________________________________________Example: 10 + 20

= 10(1 + 2) Think: common (leftovers)

Example 4Factor 12 + 8.Make a tree diagram of 12 and 8.

(What do they have in common?)Write what they have in common out in front…

4( + )

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Write what is leftover in parentheses…4(3 + 2)

Got it? Factor each expression.8) 9 + 21

9) 14 + 28

10) 80 + 56

Example 5Factor 3x + 15.Make a tree diagram of 3x and 15.

What do they have in common?3( + )

What is leftover?3(x + 5)

Got it? Factor each expression.

11) 16 + 4x

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12) 7x + 42

13) 36x + 30

Guided Practice: Factor each expression. 1. 25 + 60 2. 4x + 40

3. What’s the difference between distributing and factoring?

Lesson 6-7 Equivalent Expressions Part 1term – _________________________________________________________________

Ex: 3x + 7 + x

coefficient – ___________________________________________________________

Ex: 4n

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constant – _____________________________________________________________

Ex: 3x + 7 + x

like terms – ____________________________________________________________

Ex: 4n + 2n + 3y

Example 1Simplify the expression 4(6x).

4 · 6 · x______ · x

______

Simplify the expression x + x + 3.x + x + 3______ + 3

Got it? Simplify each expression.1) (3 · x) · 11 2) x + x + x 3) 7x + 8 + x

Example 2Three friends will pay $x each for admission to the museum plus $1 each to view the mummy exhibit. A fourth friend will pay admission but will not view the mummy exhibit. Write and simplify an expression that represents the total cost.

The expression 3(x + 1) + x represents the total cost.To simplify use the distributive property:

______ + ______ + x

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_______ + 3

Got it? 4)Write and simplify an expression for the total cost of six friends to go to the museum if only four friends view the mummy exhibit.

Remember that admission is $x and the mummy exhibit is $1.

5) Simplify the expression (14y + x) + 22y.

6) 3x + 9y + 2x

Guided PracticeSimplify each expression.1. 5(6x) 2. 3x + x – 9

3. Mikayla bought five skirts at $x each. Three of the five skirts came with a matching top for an additional $9 each. Write and simplify an expression that represents the total cost of her purchase.

Lesson 6-7 Equivalent Expressions Part 2Example 4Simplify 4(2x + y) using the Distributive Property.

4(2x) + 4(y)

= ______ + ______

Got it? 7) Simplify 7(3x + y).

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Example 5Factor 27x + 18y.Step 1: Make a tree diagram.

Step 2: What do they have in common? 9( + )

Step 3: What are the leftovers? 9(______ + ______)

Example 6Factor each expression. 10x + 15yStep 1: Make a tree diagram.

Step 2: What do they have in common? 5( + )

=Step 3: What are the leftovers? 5(______ + ______)

Got it? 8) Factor 12x + 8y 9) Factor 45x + 72y

Example 7The farmer’s market sells fruit baskets. Each fruit basket has 3 apples and 1 pear. Use a to represent apples and p to represent pears. Write and simplify an expression that represents the total of each type of fruit in 5 baskets.

Each basket is represented by the expression _________________.

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Five baskets would be ______________.Use the distributive property to rewrite.

5(______) + 5(_______)

_______ + ________Got it? 10) The free gift bag from Claire Cosmetics includes 5 bottles of nail polish and 2 tubes of lip gloss. Use p to represent each bottle of nail polish and g to represent each tube of lip gloss. Write and simplify an expression that represents the total of each type of gift in 8 free gift bags.

Guided Practice:Simplify the expression.1. 2x + 5y + 7x 2. 4(2x + 5y)

3. Twenty-six students will attend the field trip to the career center. The cost to attend the career center is $x. Fifteen of these students will pay an additional $3 to attend the automotive class. Which expression represents the total cost?

a. 15(x + 3) + 26x b. 26(x + 3)

c. 26(x + 3) + 15x d. 15(x + 3) + 11x

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