Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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Name: ______________________________________

NOTE: You should be prepared for daily quizzes. Every student is expected to do every assignment for the entire unit, or else Homework Club will be assigned! (Students with 100% Homework completion at the end of the semester will be rewarded with a pizza party and 2% grade increase.) HW reminders:

If you cannot solve a problem, get help before the assignment is due. Need help? Try www.khanacademy.com or www.classzone.com Extra Help? Visit www.mathguy.us

1.1 Evaluating Expressions

Essential Question: How do you interpret and evaluate algebraic expressions that model real-world

situations?

Warm-up

Fill in the blank:

1. 6 + ___ = 17 2. ____

2= 10 3. 6 ∙ ____ = −18

4. Find the value of 1 − 6 ∙ (7 − 4) + 5²

Day Date Assignment (Due the next class meeting) Monday Tuesday

8/12/13 (A) 8/13/13 (B)

1.1 Worksheet, Pay Lab Fee $4 to bookkeeper, sign syllabus

Wednesday Thursday

8/14/13 (A) 8/15/13 (B)

1.2 Worksheet Did you pay your lab fee?

Friday Monday

8/16/13 (A) 8/19/13 (B)

1.3 Worksheet Did you pay your lab fee?

Tuesday Wednesday

8/20/13 (A) 8/21/13 (B)

1.4 Worksheet Did you pay your lab fee?

Thursday Friday

8/22/13 (A) 8/23/13 (B)

1.5 Worksheet Did you pay your lab fee?

Monday Tuesday

8/26/13 (A) 8/27/13 (B)

Unit 1 Practice Test Did you pay your lab fee?

Wednesday Thursday

8/28/13 (A) 8/29/13 (B)

Unit 1 Test

Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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Reflect #1: Did you get the same answer as your classmates on #4 in the warm-up? What rule is used when

finding the value of an expression with multiple operations?

Vocabulary

Expression

Variable Constant

Algebraic Expression

Term

Coefficient

Evaluating Expressions: When you _____________________ a value into an expression for the variable(s) in order to simplify using order of operations.

Examples: Evaluate the expressions

1) -5(x +2)² if x = -3. 2) 8h + 40 if h = −1

2.

3) 9a – 4b if a = 10 and b = -2. 4) (𝑥−ℎ)2−(𝑥−ℎ)

ℎ 𝑤ℎ𝑒𝑛 𝑥 = 2 𝑎𝑛𝑑 ℎ = −3

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Objective #1: Can you evaluate an expression? a) Evaluate the expression 𝑥(4𝑥 − 10)3 for x = 2. b.) Evaluate the expression 12.7 – 4k if k = 1.3. Examples: Complete the table by evaluating each expression (outputs) for the given values of x (inputs).

The set of input values is called the _____________________________________.

The set of output values is called the ___________________________________.

4) 2𝑥 + 5 5) 𝑥2 + 3𝑥 − 4

Input output

0

1

2

3

Domain: Domain:

Range: Range: Objective #2: Can you evaluate an expression and find the domain and range?

Complete the table by evaluating each expression (outputs) for the given values of x (inputs), then find the domain and range. a.) 3𝑥 − 2

Domain:

Range:

Input output

0

1

2

3

Input output

0

1

2

3

Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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Unit Analysis: When evaluating expressions that represent real world situations, it is important to pay attention to the units of measurement attached to the parts of the expression. For example, if p people go to

a restaurant and agree to split the $100 cost evenly, then the units of the numerator of the expression 100

𝑝 are

_________________, the units of the denominator are ________________, and the units of the value of the expression are ___________________ per ______________________. Example:

a) Lance is competing in a bike race. On the first day he rode 120 miles in 8 hours. Use the expression 𝑑

𝑡

where d is the distance traveled and t is the travel time to find his average rate of travel. What are the units for the numerator, denominator, and the value of the expression?

b) If Lance rides at the same rate on the second day, then the expression 120 + 15t gives the total distance he has traveled after the second day. How far has he traveled if he rides for 6 hours on the second day? Include units.

Reflect #2:

a) What are the terms in the expression 120 + 15t? What does each term represent in the context of the bike race?

b) What is the coefficient of the term 15t? What does it represent in the context of the problem?

1.2 Simplifying Expressions Essential Question: How can you rewrite algebraic expressions?

Warm-up

Find the value:

1. 3(8 – 13) 2. 1.3 – 2.5 3. 0.5(8 – 6)

Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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Properties of Addition and Multiplication

Distributive Property: When you ________________________ a value into parenthesis.

Ex)

Commutative Property: When you change the ______________ of an expression involving

_________________ or ___________________.

Ex)

Associative Property: When you change which items in an expression are _______________

together with the operations ________________________ or _______________________.

Ex)

Examples: Choose the best answer for each multiple choice problem involving properties.

1) Which expression below shows the associative property of addition?

A. 4 + (3 + 5) = 4 + (5 + 3)

B. 2 + (9 + 6) = (2 + 9) + 6

C. 7(3x – 1) = 21x – 7

D. 5 ∙ 3 = 3 ∙ 5

2) Which property is shown? -8(3a + 5b) = -24a – 40b A. The distributive property B. The commutative property of multiplication C. The associative property of addition D. The commutative property of addition 3) Which expression shows the commutative property of multiplication? A. 4 + (3 + 5) = (4 + 3) + 5 B. 2 + 9 = 9 + 2 C. 5(3x – 2) = 15x – 10 D. 6 ∙ 3 = 3 ∙ 6

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Objective #3: Can you identify which property is being illustrated? Identify the property being illustrated, include operation (multiplication, division, etc)

a.) (𝑎 + 𝑏) + 𝑐 = 𝑐 + (𝑎 + 𝑏) b.) 𝑎(𝑏𝑐) = (𝑎𝑏)𝑐

c.) 𝑎(𝑏 − 𝑐) = 𝑎𝑏 − 𝑎𝑐 d.) (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧) Simplify Expressions: Use properties of real numbers to _________________ any grouping symbols and

___________________ like terms.

Examples:

1. Simplify 2(3𝑥 − 4) 2. Simplify −(𝑦 + 9) − 4𝑦 + 6

3. For a picnic, you buy p packages of hamburgers at $5.99 per package and p packages of hamburger

buns for $1.49 per package. The expression 5.99p + 1.49p can be used to represent the total cost.

Simplify the expression.

4. Suppose you needed 2 more packages of hamburgers than buns. Then the expression

5.99(p + 2) + 1.49p represents the total cost. Simplify the expression.

Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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Objective #4: Can you simplify an expression? a) 7(4 − 2𝑚) b) −2(𝑚 − 4) + 3𝑚

c) 4(x + y) – 5x – y d) 10k – (k + 3)

e) Your daily workout plan involves a total of 50 minutes of running and swimming. You burn 15 calories

per minute when running and 9 calories per minute when swimming. Let r be the number of minutes you

run. The expression 15r + 9(50 – r) represents the calories burned during your workout. Simplify the

expression and identify the units for each term of your answer.

1.3 Writing Expressions Essential Question: How do you write algebraic expressions to model quantities?

Warm-up:

With a partner try to come up with as many words as you can to fill in the “Words” column below.

Think of words that mean the same thing as the operations listed.

Operation Words Examples Addition

Plus, The sum of a number and 5

Subtraction

𝑥 − 8

Multiplication

The product of a number and 2.

Division

9

𝑥

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When you see an unknown quantity, introduce a ________________________ to represent the quantity.

Example: Write an algebraic expression to model the following phrases. Simplify if possible.

1. the sum of three times a number and 4

2. 4 less than the quantity 6 times a number

3. the quotient of 3 more than Julie’s score and 5

4. the price of a meal plus a 20% tip for the meal

Reflect #4:

a) In example 4, what units are associated with the total cost? Explain.

b) What could the expression 𝑝+0.20𝑝

2 represent in the context of example 4?

c) What if the tip was 15% instead of 20% in example 4? How can you represent the total coast with a

simplified algebraic expression?

Example:

Alex buys 2 tickets to a movie. There is a 7% sales tax on the tickets and Alex receives a $3 discount on the

total order for being a student. Write an algebraic expression to represent the cost per person. Simplify the

expression and include the units.

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Objective #5: Can you write expressions?

a) 7 less than a number divided by 2 b) four times the quantity 6 and a number

b) Alex purchased a 6-hour calling card. He has used t minutes of access time. Write an algebraic

expression to represent how many minutes he has remaining.

d) Two friends went to the rodeo and each spent $12 to get in and $2 per ride. Write an expression to

represent the total cost for the friends to go to the rodeo.

Example: Write a verbal phase that could be represented by the algebraic expression.

5. 2𝑥 + 5

6. 5 ÷ (𝑛 + 3)

7. (𝑥 + 5) − 2𝑥

Objective #5: Can you write a verbal phase that could be represented by an algebraic

expression?

a. 3𝑥 − 7

b. 2𝑥+5

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Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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1.4 Writing Equations and Inequalities Essential Question: How do you represent relationships algebraically?

Warm-up

Determine whether each statement is true or false.

1. 4.1 = 4.10 2. 7 ≤ 3 3. -4 > -6

4. 2.5 ≥ 2.7 5. 2

3<

7

9

Equation: Mathematical statement that two _____________________ are equivalent.

Solution: The value or values that make the equation _______________.

Inequality: Statement that compares two __________________________ that are not strictly equal.

Symbol Meaning

<

≤

>

≥

≠

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Examples: Write an equation or inequality to model the given verbal phase.

1. The sum of 42 and a number is equal to 51.

2. The difference of 9 and the quotient of a number t and 6 is 5.

3. The product of 4 and a number is at most 45.

4. The difference of a number t and 7 is greater than 10 and less than 20.

5. Leon paid $26.50 for a shirt with a sales tax of 6% included but he doesn’t remember the price

without tax. Write an equation to represent the situation.

Objective #6: Can you write an equation or inequality to model the given verbal phase?

a. The difference of a number z and 11 is equal to 35.

b. The product of 9 and the quantity 5 more than a number is less than 6.

c. The product of 8 and a number is greater than 4 and no more than 16.

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Example: Jason can afford to spend at most $75 dollars while shopping. There is a 7% sales tax on everything he buys. Describe some costs that would be within his budget.

a) Write the inequality to model the situation.

b) Complete the table to find some costs that are within the budget.

Cost Substitute Compare Solution? $73

c) Describe the possible costs.

Objective #7: Can you model equations and inequalities?

Raylan is buying music online. The website requires a $5 subscription fee and each song costs $0.99. He has

a gift card worth $25.

a) Write an inequality to represent the number of songs he could by using only his gift card.

b) How would the problem change if he had to use all of the gift card at one time?

c) What is a possible solution to this problem? Show your work.

Objective #8: Can you decide if a value is a solution to an equation or inequality?

Is x = 4 a solution to the following? Justify your reasoning.

𝑎. ) 1.5𝑥 – 1 = 5 𝑏) 4𝑥 + 2𝑥 ≤ 21 𝑐) 7(𝑥 – 2) > 14

Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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1.5 Representing Functions Essential Question: How do you represent functions?

Warm-up

Fill in the chart below.

Chairs 1 2 3 4 5

Legs 4

Describe the relationship between the number of chairs and the number of legs.

Input (Independent Variable)

Output (Dependent Variable)

Function

Relation

Vertical Line Test Domain

Range

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Examples: Is each relation a function? Explain your reasoning. If so, identify the domain and range. 1)

x 3 6 7 -2 0 y 1 5 -9 5 -2

2)

3)

Input Output

0 2

0 3

5 4

10 5

4)

Input Output 4 8 1 2 4 8 6 12

5)

6) {(3, -2), (7, 5), (-4, -1), (6, -2)}

7) Each odd number from 3 to 9 is paired with the next greater whole number.

Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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Objective #9: Can you determine domain and range of a function?

a) b) {(-9, 5), (3, -2), (-7, 1), (8, 5)} c) d)

Functions can be described using _________________ _________________. The function f assigns the output value f(x)

to the corresponding input value x. The notation f(x) reads “f of x”. For the function in example #4 on the

previous page, f(4) = ______.

Reflect #8

a) Tell how to read the statement f(3) = 12. Then interpret what it means in terms of input and output

values.

A __________________ _____________ is an algebraic expression that defines a function. If you know an input value,

you can use the function rule to identify the output value.

x -2 -9 3 16 3 y -4 15 8 7 8

D = { }

R = { }

D = { }

R = { }

D = { }

R = { }

D = { }

R = { }

Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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Example:

The cost of buying n Snapples at $1.25 each can be represented by the function rule C(n) = 1.25n.

a) Complete the table for the given domain (input) values. Write the results as ordered pairs.

Input C(n) = 1.25n Output Ordered Pair

0

1

2

3

4

b) Graph the function by plotting the ordered pairs. Choose a beginning and end, and a scale for each

axis of the graph.

Reflect #9

a) What is the value of C(3)?

b) Use function notation to represent the cost of buying 6 Snapples. Evaluate the function for that

value and include units.

c) What is the domain and range for this function? What are reasonable values for the domain?

Unit 1: Algebraic Modeling and Unit Analysis Algebra 1

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Objective #10: Can you represent and interpret functions?

Randy wants to tile part of a floor with 25 square tiles. The tiles come in whole-number side lengths from 1

to 5 inches. If s is the side length, the area that he can cover is A(s) = 25s².

a) What is the domain of this function?

b) Make a table of values and graph the function for the given domain.

Input A(s) = 25s² Output Ordered Pair

c) What does A(4) = 400 mean in this context?