At the Crossroads

DEPENDENT AND INDEPENDENT TEACHER EDITION List of Activities for this Unit:

ACTIVITY STRAND DESCRIPTION 1. Function Determine whether variables are AS dependent or independent 2. Exploring Dependent & Independent AS Determine which quantities are dependent and independent 3. Dependent Stories AS Write a verbal description and an equation using points from the table. 4. Describe the Walk AS Write an equation and relate it to the original problem 5. Wandering Around AS Write an equation using points from a table AS Determine if a given scatterplot is 6. Guess My Function a function 7. Mathematical Definition of Functions AS Determine if given situations represent a function 8. Coordinate Points AS Determine if given situations represent a function 9. What’s the Situation? AS Determine which situations represent functions 10. Valentine’s Day Idea AS Use systems of equations to determine which flower shop offers the better deal 11. New Rose Offers AS A variation of “Valentines Day Idea” using new values 12. Using Graphs for New Rose Offer AS Use graphs of equations to determine which flower shop offers the better deal and under what conditions 13. The Birthday Gift AS Write and graph a function, then describe how the line changes when values change 14. Spending Money AS Write and graph a function, then describe how the line changes when values change 15. Money, Money, Money AS Write and graph a function 16. At the Crossroads (let’s get fit) & MC Use linear equations to determine items which fitness club offers the better deal (More 2-equations cost comparison and multiple choice problems)

Teacher: Ch 7 “Dependent and Independent” 1 At the Intersection COE Connections

Motion Detector Graph paper MATERIALS Graphing Calculators Rulers Colored pencils

Payday Warm-Ups Tall Paul (in Segmented Extras Folder)

Vocabulary: Mathematics and ELL

Axis x-axis, y-axis Intersection Cause Manipulated Variable Coordinate Membership (Health Club) Coordinate Plane Motion Detector Coordinate Points Predict, Prediction Dependent Variable Relevant Effect Responding Variable Equation Sketch Function Table Graph Taxi Independent Variable Viewing Window (calculator) Interpret

Length of the Unit: 450 minutes; an additional 250 minutes for optional activities

Teacher: Ch 7 “Dependent and Independent” 2 Essential Questions:

 What is meant by the terms “dependent variable” and “independent variable”?  Science uses “responding” and “manipulated” instead of “dependent and “independent”. How can we use all terms without confusion?  How do the dependent variable and independent variable relate to graphing?  What is meant by the term “function”?  How do you know when points on a graph represent a function?  How is an algebraic equation developed from information provided?  What do the variables represent in an equation? How are the variables replaced in order to reach a solution?  How information is converted to coordinates and placed on a coordinate grid?  How are coordinates written and then turned into a symbolic representation?  What is an informative title for a graph?  What are appropriate intervals for the x- and y-axes?  What do the intercepts of a graph represent in the context of a given situation?  What does the slope of a graph represent in the context of this scenario?  How can a graph be used to make predictions?  What is meant when two lines in a graph intersect?  How can a conclusion be supported using mathematical information and calculations?  How can I use a calculator to assist in the solving of a problem through graphing?  How do I choose a window for the calculator when graphing?  What does the window do to help me when graphing?  How is a conclusion supported by data?  How is computation used to give an answer within a given context?  What information is presented in a box and whisker plot?  What information can be easily extracted from a box and whisker plot?  What evidence from graphs or plots can be used to support/justify your conclusion?

Lesson Overview:

 Before allowing the students the opportunity to start the activity 16: access their prior knowledge with regards to joining a health club. Why would a person join one club vs. another? Is cost the only factor? Spending dollars vs. saving dollars? Ask the students if any of them belong to a health club. What do they like about the health club? Discuss why more and more companies, apartment complexes, hotels and colleges have good fitness centers. Why do these places include fitness centers? What is the benefit of including a fitness center in these places?  There are several ways that a function can be written as a statement. Be sure to address a variety of possibilities. These include but are not limited to using terms such as; depends, determines, hinges on, establishes, decides, causes and effects.  Important concepts are identifying the two quantities involved and determining the dependent and independent relationships. Many times, parts of this statement can be interchanged. For example, if discussing gas in a car and the amount of money you have, either statement of a function is accurate but the dependency changes:  The amount of money I have after I put gas in my car depends on how much gas I put in my car.  How much gas I put in my car depends on how much money I have.

Teacher: Ch 7 “Dependent and Independent” 3  Discuss with students that sometimes the situation above exists; however, the question statement will help decide the dependency. Don’t let this become an issue that distracts students from the concept of independent and dependent quantities.  In Activity 1, many of these statements could be turned around and the dependent statement could become the independent. Students may point out that the number of hours they work depends on how much money they need to make. The idea here is to associate independent and dependent quantities with specific statements and words such as depends, determines, and function. As the statements are written, the dependency is established. Allow and encourage students to discuss how changing the dependent or independent statement affects the other.  Explain what moving toward/away from motion detector means.  Activities 4-6 may need additional scaffolding regarding slope and y = mx + b dependent upon the needs of the students.  How can you support a conclusion that you make? What evidence from graphs can be used to support/justify your conclusion?  A good warm-up could be Payday or Tall Paul.  When reading a graph, where are the x- and y-intercepts? What is the relationship between the intercepts and the context of any problem?  Activities 13, 14 and 15 are graphing calculator-based units that are included in this overall lesson, but are viewed as optional.  Use resources from your building.

Performance Expectations:

4.4.D Graph and identify points in the first quadrant of the coordinate plane using ordered pairs. 5.2.H Solve single- and multi-step word problems involving addition and subtraction of whole numbers, fractions (including mixed numbers), and decimals, and verify the solutions. 5.4.B Write a rule to describe the relationship between two sets of data that are linearly related. 5.4.D Graph ordered pairs in the coordinate plane for two sets of data related by a linear rule and draw the line they determine. 6.1.H Solve single- and multi-step word problems involving operations with fractions and decimals and verify the solutions. 6.2.A Write a mathematical expression or equation with variables to represent information in a table or given situation. 6.2.B Draw a first-quadrant graph in the coordinate plane to represent information in a table or given situation. 7.1.G Solve single- and multi-step word problems involving rational numbers and verify the solutions. 7.5.A Graph ordered pairs of rational numbers and determine the coordinates of a given point in the coordinate plane. 8.1.A Solve one-variable linear equations. 8.1.C Represent a linear function with a verbal description, table, graph, or symbolic expression, and make connections among these representations. 8.1.D Determine the slope and y-intercept of a linear function described by a symbolic expression, table, or graph. 8.1.E Interpret the slope and y-intercept of the graph of a linear function representing a contextual situation. 8.1.F Solve single- and multi-step word problems involving linear functions and verify the solutions.

Teacher: Ch 7 “Dependent and Independent” 4 8.1.G Determine and justify whether a given verbal description, table, graph, or symbolic expression represents a linear relationship. A1.1.A Select and justify functions and equations to model and solve problems. A1.1.B Solve problems that can be represented by linear functions, equations, and inequalities. A1.1.C Solve problems that can be represented by a system of two linear equations or inequalities. A1.3.A Determine whether a relationship is a function and identify the domain, range, roots, and independent and dependent variables. A1.4.B Write and graph an equation for a line given the slope and the y-intercept, the slope and a point on the line, or two points on the line, and translate between forms of linear equations. A1.4.C Identify and interpret the slope and intercepts of a linear function, including equations for parallel and perpendicular lines. A1.4.D Write and solve systems of two linear equations and inequalities in two variables. A1.6.D Find the equation of a linear function that best fits bivariate data that are linearly related, interpret the slope and y-intercept of the line, and use the equation to make predictions. G.6.F Solve problems involving measurement conversions within and between systems, including those involving derived units, and analyze solutions in terms of reasonableness of solutions and appropriate units.

Teacher: Ch 7 “Dependent and Independent” 5 Performance Expectations and Aligned Problems Chapter 7 1- 2- 3- 4- 5- 6- 7- 8- 9- 10- 11- 12- 13- 14- 15- 16- Func Explor Depend Descr Wander Guess Mathe Coor What’s Valen New Using The Spen Mo At “Dependent tion ing ent ibe ing My matical dinate the Sit tine’s Day Rose Graphs Birth ding ney the and Depen Stories the Around Fun Def Points uation Idea Offers for New day Mon , Cro dent Walk ction inition ? Rose Gift ey Mo ssro Independent & of Fun Offer ney ads ” Indepe ctions , (let’ ndent Mo s Subsections: ney get fit) & MC ite ms Problems 3, 1, 2, 1, 2, 1, Supporting: 1 1 3 8 3, 5 5, 6 3, 4 3, 4 3 PE 4.4.D 12 Problems Supporting: 3, PE 5.2.H ≈ 1- 1, 5, 4, 1-4 1-4 2 3 1, 6 4, 14 6 6 6.1.H ≈ 5 7.1.G ≈ 8.1.F Problems Supporting: 1, 1- PE 5.4.B ≈ 1-4 1-4 2 3 2 2 4 12- 14 6.2.A ≈ 15 A1.4.B Problems 3, 5, 1, 6, Supporting: 1-4 1-4 3 8 3, 5 12 3 12 PE 5.4.D Problems 3, 5, 1, 6, Supporting: 1-4 1-4 3 8 3, 5 12 3 12 PE 6.2.B Problems 3, 1, 6, Supporting: 1-4 1-4 1 1 3 8 3, 5 5,12 3 12 PE 7.5.A Problems 2, 1, 5, 1, 5, 4, Supporting: 2, 9 3 5, 6 6 6 PE 8.1.A 10 1, Problems 2, 2, 1, 1- 1, 6, 6, Supporting: 1-4 1-4 2-7 3 7- 5- 3, 14 8 12- PE 8.1.C 11 11 4 15 4, Problems 2, 2, 1- 1, 2, 1, 6, Supporting: 1-4 1-4 2-7 3 7- 6- 14 8 3 12- PE 8.1.D 11 12 15 2, 3, 4, Problems 2, 3, 1, 2, 5, 1, 6, Supporting: 1-4 1-4 2-7 3 5- 8 7- 3 12- PE 8.1.E 12 12 15

Teacher: Ch 7 “Dependent and Independent” 6 Chapter 7 1- 2- 3- 4- 5- 6- 7- 8- 9- 10- 11- 12- 13- 14- 15- 16- Fun Explor Depend Descr Wande Guess Mathe Coor What’s Valen New Using The Spen Mo At “Dependent ction ing ent ibe ring My matical dinate the Sit tine’s Day Rose Graph Birth ding ney the and Depen Stories the Aroun Fun Def Points uation Idea Offe s for day Mon , Cro dent Walk d ction inition ? rs New Gift ey Mo ssro Independent” & of Fun Rose ney ads Subsections: Indepe ctions Offer , (let’ ndent Mo s ney get fit) & MC ite ms Problems 1, 1, 4, Supporting: 2, 5, 1-4 1-4 2 3 8 2, 5 3, 6, PE 8.1.G 12 4 12- 15 Problems 1, 1, 4, Supporting: 2, 5, 1-4 1-4 2 3 8 2, 5 3, 6, PE A1.1.A 12 4 12- 15 Problems 2- 2, 9, 5, 6, 4, Supporting: 3 2-7 5, 6 5, 10 12 6 PE A1.1.B 7-9 Problems 1, Supporting: 2, 3, 5, 7, 3 4, 6-9 8, 9 8 PE A1.1.C ≈ 6 A1.4.D Problems Supporting: All All Al All All All 4 4 2 PE A1.3.A Problems 1, 1, 5, 2, 4, Supporting: 1, 2, 1, 1-4 1-4 2, 3-7 3 7- 6- 6, PE A1.4.C 8 3 11 12 12- 15 Problems 1, Supporting: 2, 2, 1, 1- 1, 6, 6, PE A1.6.D 1-4 1-4 2-7 3 7- 5- 3, 14 8 12- 11 11 4 Weakly 15 aligned Problems Supporting: PE G.6.F 1, 2, 2, 1, All linear 1- 1, 6, 6, 1-4 1-4 2-7 3 7- 5- 3, 14 8 12- eqns in 11 11 4 context are 15 conversion problems

Assessment: Use the multiple choice and short answer items from Algebraic Sense and Number Sense that are included in the CD. They can be used as formative and/or summative assessments attached to this lesson or later when the students are being given an overall summative assessment.

Teacher: Ch 7 “Dependent and Independent” 7 Activity 1: Function

I. Read each numbered statement. Determine the two variables (in words) in each situation and identify each as independent or dependent.

a. Independent (Manipulated) variable – Not subject to control by others, self-governing. The independent variable is usually associated with the horizontal, or x axis. A common example of something not subject to the control of others, would be time, in the form of minutes, hours, days, etc. The independent variable is also referred to as the “input”. Another way of thinking of the independent variable is it is “free” and not dependent on some other action. For instance if I drop a ball off a building, the time required for the ball to hit the ground is “free” and is not in the least affected by the height of the building, me, the ball, or anything else. Thus in this case time is independent of the height of the ball (distance above the ground). b. Dependent (Responding) variable – Relying on or subject to something else for support. The dependent variable is usually associated with the vertical, or y axis. It’s sometimes easier to figure out the dependent variable first, because we can see when something is relying on something else for support. Such as in our example above, time is not dependent on height, but the ball’s distance above the ground is dependent on the time it was dropped.

1. How fast the grass grows depends on how much rain we get. (Example solution already given to student.)

a. Independent variable how much rain______

b. Dependent variable – how fast the grass grows______

2. The number of problems missed on a test determines your grade on the test.

a. Independent variable -___Number of problems missed.______

b. Dependent variable - ____Your grade on the test.______

3. How long I talk on my cell phone depends on the number of minutes on my calling plan.

a. Independent variable -___Number of minutes on my calling plan.______

b. Dependent variable - ____How long I talk on my cell phone.______

4. The amount of money I make is a function of the number of hours I work.

a. Independent variable -___Number of hours I work.______

b. Dependent variable - ____How much money I make.______

Teacher: Ch 7 “Dependent and Independent” 8 5. The number of cakes sold in a bake sale determines the amount of money made.

a. Independent variable -___How many cakes are sold.______

b. Dependent variable - ___Amount of money that is made.______

6. The winner of the football game depends on which side scored the most points.

a. Independent variable -____Which side scored the most point.______

b. Dependent variable - _____Winner of the football game.______

7. The amount of memory on a CD determines how many songs I can download to it.

a. Independent variable -_____Amount of memory on a CD.______

b. Dependent variable - ______How many songs I can download.______

II. The American Heritage Dictionary defines function as something closely related to another thing and dependent upon it for its existence, value, or significance. The words function and depends can be used interchangeably. Examples of Function – Write sentences that represent a function. Determine the independent and dependent parts. Include several different words such as depends, determines, and ‘is a function of’ in your statements.

1. How much money I make depends on the number of hours I work a. I control the hours I work – independent variable b. The money depends on the hours – dependent variable

2. ______Answers will vary.______

a. Independent variable -______Answers will vary b. Dependent variable - ______

3. ______

a. Independent variable -______

b. Dependent variable - ______4. ______

b. Dependent variable - ______

Teacher: Ch 7 “Dependent and Independent” 9 5. ______

6. ______

III. Identify the cause and effect variables from the dependent functions you created in numbers 2-6.

Cause Effect

Answers will vary

Teacher: Ch 7 “Dependent and Independent” 10 Activity 2: Exploring Dependent and Independent

1. Mathematics is considered a universal language. People from all parts of the world read and interpret mathematical statements the same way. In order for this to happen, universally accepted mathematical terms, words, and symbols were established and defined. We call these “conventions”. Order of operations is one of these conventions. Worldwide, all people agree that problems will be simplified using the order of operations you have learned. Another convention is the agreement that the x-axis will represent the independent (manipulated) variable and the y-axis will represent the dependent (responding) variable. Applying this agreement, determine the independent and dependent quantities of each statement and label the axes on the graph.

a. My grade in Algebra depends on how long I study.

b. The longer I procrastinate, the less amount of work I accomplish.

c. The number of CDs I buy depends on how much money ($’s) I have.

d. The cost of the taxi ride is a function of the number of miles I ride.

Teacher: Ch 7 “Dependent and Independent” 11 Activity 3: Dependent Stories

1. Use the axes’ labels to write a dependent (responding) statement.

Example: Statement: How far I walk is dependent on how long I

walk.______

______

Statement: ___Total cost in dollars will depend on the

number of minutes talked.______

______

Statement: ___The profit in dollars will depend on how

many tickets are sold.______

______

Teacher: Ch 7 “Dependent and Independent” 12 Activity 4: Describe the Walk

1. Sketch and label the graph and units in the table. Write a description of the walk or run and an equation. Include the units of measure in the table that you decide upon.

Table Time (Hrs) Distance (Mi) 0 15 1 21 2 27

a. Verbal Description: ____I started my run 15 miles from home. At a speed of 6 miles per

hour, I ran for 2 hours ending 27 miles from home. ______

______Answers will vary.______

b. Equation: ___D = 6t + 15 ; where D = distance and t = time.______

Table Time (hrs) Distance (mi) )

3 i 21 m (

4 e 20 c n a t s

5 i 19 D

a. Verbal Description: ____Must include starting at (3,21) and ending at (5,19). Must include

units in table and graph.______Answers will vary.______Time (hrs) b. Equation: _____ D = -1t + 24 ; where D = distance and t = time.______

Teacher: Ch 7 “Dependent and Independent” 13 3. Sketch and label the graph and units in the table. Write a description of the walk or run and an equation. Include the units of measure in the table that you decide upon.

Table Time (hrs) Distance (mi) ) 0 1 i m (

3 7 c n a t s 6 13 i D

Time (hrs) a. Verbal Description: ___Must include starting at (0,1) and ending at (6,13). Must include

units in table and graph.______

b. Equation: _____D = 2t+1 ; where D = distance and t = time.______

Table Time (hrs) Distance (mi) ) i m

6 a 17 t s i 10 D 29

Time (hrs)

a. Verbal Description: _____Must include starting at (2,5) and ending at (10,29). Must

include units in table and graph._____Answers will vary.______

b. Equation: ______D = 3t – 1 ; where D = distance and t = time.______

Teacher: Ch 7 “Dependent and Independent” 14 Activity 5: Wandering Around

1. Label the table and graph. Fill in the table, sketch the graph and write an equation for the situation.

a. Ryan was walking away from the motion detector at 2 feet per second. You missed where he started, but you know that he was at the 9 foot mark after walking for 3 seconds.

Equation (equation of the line): ____D = 2t + 3 ; where D = distance and t = time_____

b. How does the equation relate to the original problem? ___It describes Ryan’s movement

away from the motion detector and allows prediction.______

2. Madeline was walking toward the motion detector at 3 feet per second. You missed where she started, but you know that she was at the 9 foot mark after walking for 2 seconds.

Table Time (Sec) Distance (ft) ) t

0 15 f (

1 12 n a t s i

2 9 D 3 6

Time (sec)

a. Equation (equation of the line): ______D = 15 – 3t ; where D = distance and t = time___

b. How does the equation relate to the original problem? ___It describes Madeline’s

movement away from the motion detector and allows prediction.______

Teacher: Ch 7 “Dependent and Independent” 15 3. Robyn started 1 foot from the motion detector. You looked up and she was at 5 feet at the 2nd second.

a. Equation (equation of the line): ____D = 4t + 1 ; where D = distance and t = time_____

b. How does the equation relate to the original problem? ____It describes Robyn’s movement

away from the motion detector and allows prediction.______

4. You looked up and Chet was walking toward the motion detector. He was at the 6 foot mark at the 1st second and the 1 foot mark at the 2nd second

a. Equation (equation of the line): _D = -5t + 11 ; where D = distance and t = time._____

b. How does the equation relate to the original problem? ____It describes Chet’s movement

toward the motion detector and allows prediction.______

Teacher: Ch 7 “Dependent and Independent” 16 Activity 6: Guess My Function!

Write an equation that represents the relationship between x and y in each table:

1. x y 2. x y 0 11 0 1.3 1 16 1 2.5 2 21 2 3.7

1. ______y = 5x + 11______

2. ______y = 1.2x + 1.3______

3. ______y = -2x + 4______

4. ______y = -7x + 25______

5. ______y = 4x + 16______

6. ______y = 3x – 4.9______

Teacher: Ch 7 “Dependent and Independent” 17 7. ______y = -x + 24______

8. ______y = -5x + 41______

9. ______y = 6x + 11______

10. ______y = -x + 15______

11. ______y = 2x + 4 ______

12. ______y = -½x + 25 ______

13. ______y = 2x + 6______

14. ______y = - ¼x + 8.05 ______

Teacher: Ch 7 “Dependent and Independent” 18 Activity 7: Mathematical Definitions of Functions

1. Look at the scatter plots. List the coordinate points, connect the points from left to right, and decide if the data represents a function based on the definition of a function. The definition of a function is for each independent value there is only one dependent value.

Coordinates: ___(2,4) ; (3,6) ; (4,8)______Coordinates: ____(-7,10) ; (-5,8) ; (-2,4)______(5,10) ; (5,12) ; (7,14)______(3,6) ; (5,12) ; (7,14) ; (9,18)______Yes or No ___No______Yes or No ____Yes_____

Coordinates: ____(-7,10) ; (-5,8) ; (-2,4)__ Coordinates: ____(-2,-1) ; (0,3) ; (1,5)______(-2,12) ; (3,6) ; (7,14) ; (9,18)______(3,9) ; (5,13)______

Yes or No ___No______Yes or No __Yes_____

Teacher: Ch 7 “Dependent and Independent” 19 Coordinates: ___(-5,3) ; (-3,5) ; (-2,7)______Coordinates: ___(-5,3) ; (-5,4) ; (-5,5)_____

____(-1,8) ; (0,9) ; (1,8) ; (2,7) ; (3,5) ; (5,3)_ ___(-2,7) ; (-1,8) ; (0,9) ; (1,8) ; (2,7) ; (3,5)__

Yes or No ____Yes______Yes or No ____No____

2. What do the coordinates tell you about whether the data represents a function? Make a conclusion about what a graph looks like when it follows the definition of a function. Support your conclusion using an example or other evidence from this lesson. ______Answers will vary______

______- some x-coordinate will have 2 or more y-values, then not a function_____

______

Teacher: Ch 7 “Dependent and Independent” 20 Activity 8: Coordinate Points

1. Using each of the lists of coordinate points: create a table of values and a scatter plot for each group of coordinates. Explain which groups of coordinate points follow the definition of a function. a. (-1, -2); (3, 3); (-3, 6); (0, -3); (5, 1)

X Y -1 -2 3 3 -3 6 0 -3 5 1

Does this group of coordinate points represent a function? ___Yes_____

Why? _____The same x-coordinate does not have more than 1 y-value.______

b. (-4, 9); (-2, 5); (1, -1); (-2, 7); (4, -3)

X Y -4 9 -2 5 1 -1 -2 7 4 -3

Does this group of coordinate points follow the definition of a function? ___No______

Why? _____The x-coordinate of -2 has more than 1 y-value. Therefore, it does not

follow the definition of a function.______

Teacher: Ch 7 “Dependent and Independent” 21 c. (-1, -5); (2, -5); (4, -5); (-5, 2); (-3, 3)

X Y -1 -5 2 -5 4 -5 -5 2 -3 3

Does this group of coordinate points follow the definition of a function? ____Yes______

Why? ______No x-coordinate has more than 1 y-value.______

d. (-3, 5); (-1, -1); (1, -1); (2, -7); (-3, -5)

X Y -3 5 -1 -1 1 -1 2 -7 -3 -5

Does this group of coordinate points follow the definition of a function? ___No______

Why? ______The x-coordinate of -3 has more than 1 y-value. Therefore, it does not

follow the definition of a function.______

______

Teacher: Ch 7 “Dependent and Independent” 22 Activity 9: What’s the Situation?

1. Read each of the following situations and determine whether it represents a function. Justify your answer.

Example: The ages of the students in this classroom matched to their shoe size. ______No, this is not a functional situation. Different aged students may have the same shoe size.______

a. The numbers on a telephone depend on the letters on the phone. ______Yes, this is a

functional situation. No letter has more than one number assigned to it.______

b. The letters on the telephone depend on the numbers on the phone. ____No, this is not a

functional situation. Some numbers on a phone have more than one letter.______

c. The number of days in a month depends on the month. __Yes,______this is a functional situation. No x-coordinate has more than 1 y-value.______

d. The cost of renting a car per day if there is a charge of $0.50 per mile. ___No, this is not a

functional situation because a one-day rental could have more than one cost.______

e. The number of minutes used on a cell phone in January depends on the students in this class.

_Yes,_this is a functional situation. Each statement is unique. Therefore, no x-coordinate

has more than 1 y-value.______

Teacher: Ch 7 “Dependent and Independent” 23 Activity 10: Valentine’s Day Idea

The school’s drill team has contacted several flower distributors and has narrowed the choice to two companies.

Option 1: Roses-R-Red has offered to sell its roses for a fixed down payment of $20.00 and an additional charge of $0.75 per stem.

Option 2: The Flower Power has offered to sell its roses for a fixed down payment of $60.00 and an additional charge of $0.50 per stem.

1. Predict which offer is the better deal. ______Answers will vary.______

Explain why you made your choice: ______Answers will vary.______

2. From the description of the two offers, complete the chart to find an algebraic rule that will determine the cost of n roses. Rose Offers

Process Process Cost at Cost at Number of Roses Column Column Roses-R-Red Flower Power (Roses-R-Red) (Flower Power) 10 20 + 10(0.75) $27.50 60 + 10(0.50) $65.00 20 20 + 20(0.75) $35.00 60 + 20(0.50) $70.00 40 20 + 40(0.75) $50.00 60 + 40(0.50) $80.00 60 20 + 60(0.75) $65.00 60 + 60(0.50) $90.00 120 20 + 120(0.75) $110.00 60 + 120(0.50) $120.00 150 20 + 150(0.75) $132.50 60 + 150(0.50) $135.00 210 20 + 210(0.75) $177.50 60 + 210(0.50) $165.00 240 20 + 240(0.75) $200.00 60 + 240(0.50) $180.00 1,000 20 + 1000(0.75) $770.00 60 + 1000(0.50) $560.00 n 20 + 0.75n 60 + 0.50n

a. Write a rule for the cost of roses at Roses-R-Red using words and symbols.

Teacher: Ch 7 “Dependent and Independent” 24 1) Sentence using words: ____$20 fee + 0.75 per rose______

2) Sentence using symbols: __ C = 20 + 0.75 n______

b. Write a rule for the cost of roses at Flower Power using words and symbols.

1) Sentence using words: ____$60 fee + 0.50 per rose______

2) Sentence using symbols: ___ C = 60 + 0.50 n______

c. What patterns do you observe from the table of values? ____ Answers will vary , but all should

mention increasing costs.______

d. What happens to the cost of the roses as the number of roses purchased increases? ___The__

_cost increases as the number of roses purchased increases.______

e. What would a graph of this relationship look like? ____ The graph will be linear with a

positive slope.______

f. How many roses can you buy from Roses-R-Red for $65.00? ______60______Support your answer using words, numbers and/or diagrams.

Looked in table or 65 = 20 + 0.75n 45 = 0.75n n = 60

g. How many roses can you buy from Flower Power for $65.00? _____10______Support your answer using words, numbers and/or diagrams.

Looked in table or 65 = 60 + 0.50n 5 = 0.50n n = 10

h. What is the point where the two flower dealers charge the same amount? n = 160___

i. Write an equation that represents the point where the two flower shops charge the same

amount. ______20 + 0.75 n = 60 + 0.50 n______

Teacher: Ch 7 “Dependent and Independent” 25 3. Graph the two functions on the same coordinate grid and label.

______Cost of Roses at Rose-R-Red & Flower Power______

4. What effect does the $0.75 per stem cost have on the graph of the Roses-R-Red function?

______That is the slope – the cost per rose. The rate of change.______

5. What effect does the $20.00 have on the graph? _____The $20.00 is the y-intercept. In this case,

the starting point for the graph.______

6. What effect does the $0.50 per stem cost have on the graph of the Flower Power function?

_____That is the slope – the cost per rose. The rate of change.______

Teacher: Ch 7 “Dependent and Independent” 26 7. What effect does the $60.00 have on the graph? ______The $60.00 is the y-intercept. In this

case, the starting point for the graph.______

8. What is the significance of the point of intersection? ____The point of intersection is the point

where it doesn’t matter from whom you purchase roses because he cost will be the same.______

9. What is the point of intersection of the two graphs? ____At the coordinate (160, 140).______Show the work that supports your answer.

20 + 0.75n = 60 + 0.50n 20 + 0.75(160) = C -20 -20 20+ 120 = C

0.75n = 70 + 0.50n 140 = C -0.50n -0.50n

0.25n = 40 Purchase 160 roses will cost n = 160 $140.00 from both places.

10. Which flower dealer offers the better deal and when? ___At 160 roses.______

- Roses-R-Red is the better deal if you purchase less than 160 roses. - They are equal at 160 roses. - Flower Power is the better deal if you purchase more than 160 roses.

Teacher: Ch 7 “Dependent and Independent” 27 Activity 11: New Rose Offers

To entice potential new customers, Roses-R-Red decides to eliminate the $20.00 fixed charge. According to its new offer, the drill team pays only for the roses they buy. When the Flower Power learned about the new offer by its competitor, it immediately entered the price war by reducing its $60.00 fixed charge to a $40.00 fixed charge.

1. How do the new offers compare to the original offers? ___Answers will vary.______

2. Make a prediction. Will it change which company has the better offer? ___Answers will vary.__ Support your prediction: ______Answers will vary.______

3. From the description of each of the two new deals, complete the table and write new algebraic rules that will determine the cost of n roses.

New Rose Offers a. Process Process Number of Roses Cost at Cost at Column Column Roses-R-Red Flower Power (Roses-R-Red) (Flower Power) 10 10(0.75) $7.50 40 + 10(0.50) $45.00

20 20(0.75) $15.00 40 + 20(0.50) $50.00

40 40(0.75) $30.00 40 + 40(0.50) $60.00

60 60(0.75) $45.00 40 + 60(0.50) $70.00

120 120(0.75) $90.00 40 + 120(0.50) $100.00

150 150(0.75) $112.50 40 + 150(0.50) $115.00

210 210(0.75) $157.50 40 + 210(0.50) $145.00

240 240(0.75) $180.00 40 + 240(0.50) $160.00

1,000 1,000(0.75) $750.00 40 + 1,000(0.50) $540.00

n n(0.75) 40 + n(0.50)

b. What patterns do you observe in the new table of values? __The cost for roses has decreased

significantly.______

Teacher: Ch 7 “Dependent and Independent” 28 c. Compare the costs in this table to the costs in the first table. What changes do you observe?

___The y-value for Rose-R-Red and Flower Power have all decreased by $20. When graphing,

Rose-R-red y-intercept will be 20 less than the original. Same situation for Flower Power.____

d. How many roses can you buy from Roses-R-Red for $65.00? ______About 87 roses_____ Support your answer using words, numbers and/or diagrams. 65 .75 C = 0.75n = n .75 .75 65 = 0.75n

e. How many roses can you buy from Flower Power for $65.00? ______50______Support your answer using words, numbers and/or diagrams. 25 0.50n C = 40 + 0.50n = 0.50 0.50n 65 = 40 + 0.50n -40 -40 n = 50

f. Write an equation that represents the point where the two flower shops charge the same amount. ______0.75 n = 40 + 0.50 n______

g. At what point will the two flower dealers charge the same amount? _n = 160_____

What is the charge? ____$120_____

0.25n 40 0.75n = 40 + 0.50n = -0.50n -0.50n 0.25 0.25

160  0.75 = $120 n =160

h. Which company offers the better deal? - From table

- Roses-R-Red is the better deal if you purchase less than 160 roses. - They are equal at 160 roses. - Flower Power is the better deal if you purchase more than 160 roses.

Teacher: Ch 7 “Dependent and Independent” 29 Activity 12: Using Graphs for New Rose Offers

1. Graph both the original offer, y = 0.75x + 20, and the new offer, y = 0.75x, together.

a. What effect does subtracting $20.00 from the old rule have on the new graph of the Roses-

R-Red function? ____The graph starts at 0 instead of at 20. The lines are parallel.______

2. Turn off the above two graphs. Graph both the original offer, y = 0.5x + 60, and the new offer, y = 0.5x + 40, together.

a. What effect does subtracting $20.00 from the old rule have on the new graph of the Flower

Power function? ____The graph starts at 40 instead of 60. The lines are parallel.______

3. What are the coordinates of the point where the original two functions intersect? __(160,140)__

4. What are the coordinates of the point where two new functions intersect? ___(160,120)______

5. What is the significance of this point of intersection? ___It is the point where both companies

charge the same amount for the same number of roses purchased. 160 roses for $140.00 for original

and 160 roses for $120.00. ______

6. What affect did the new pricing structure have on the point of intersection? Why? __None. The

point of intersection doesn’t change for the number of roses because each company chopped their

price by the same amount of -$20. Only the cost changes.______

7. Which flower dealer now offers the better deal? ____There is not a change in who offers the better

deal. The original and new price structure centers the same results.______

- Roses-R-Red is better when purchasing less than 160 roses. - The cost is the same for both companies when 160 roses are purchased. - Flower Power is the better deal when purchasing more than 160 roses.

Teacher: Ch 7 “Dependent and Independent” 30 8. Sketch the graphs of all four functions, label the graphs and label the relevant points of intersection. Check your graphs on your graphing calculator. You may have to adjust your window settings to see all your graphs. Cost of Purchasing Roses from Roses-R-Red and Flower Power ) $ (

t s o C

Number of Roses Purchased

Teacher: Ch 7 “Dependent and Independent” 31 Activity 13: The Birthday Gift

1. Susan’s grandmother gave her $25.00 for her birthday. Instead of spending the money, she decided to start a savings program by depositing the $25.00 in a non-interest paying savings club account in the bank. Each week, Susan plans to deposit an additional $2.50 into the account. Make a table of values for the situation.

Time (Weeks) Process Amount Saved (A) 0 $25 $25.00

1 $2.50 + $25.00 $27.50

2 $2.50 + $27.50 $30.00

3 $2.50 + $30.00 $32.50

4 $2.50 + $32.50 $35.00

5 $2.50 + $35.00 $37.50

2. Write a function rule for the amount of money Susan will have after t weeks and define the variables. ______A = 25. + 2.50 t ; where A = amount saved and t = number of weeks______

3. Find a viewing window for the problem situation and sketch your graph.

4. Justify your window choices.

Teacher: Ch 7 “Dependent and Independent” 32 5. Graph the function rule by hand on the coordinate grid.

5. How much money will Susan have after 7 weeks? ___$42.50______Support your answer using words, numbers and/or diagrams.

A = 25 + 2.50t A = 25 + 2.50(7) A = 25 + 17.50 A = $42.50

6. Susan wants to buy a school ring. When will she have enough money to buy the $159.99 ring?

______54 weeks______Support your answer using words, numbers and/or diagrams.

159.99 = 25 + 2.50t -25 -25

134.99 2.50t = 2.50 2.50

t = 53.996

7. How would the line for the graph change if Susan deposits only $15.00 of the $25.00 given to her

Teacher: Ch 7 “Dependent and Independent” 33 by her grandmother? ____Susan’s starting value will be lower. She will “start” on the y-axis at 15

instead of 25. Parallel to the original line.______

8. How would the line for the graph change if Susan deposits the $25.00 from her grandmother plus

another $15.00 she already had? ____Susan’s starting value will be higher. She will “start” on the y-

axis at 40 instead of 25. Parallel to the original line.______

9. How would the line change if Susan deposits the $25.00 from her grandmother, but decides she

can only save $2.00 per week? ____Susan’s rate of saving has decreased, therefore the amount of

money will not grow as fast. The line will be less steep than the original. The new line is not parallel

to the original.______

10. What changes in the situation resulted in a change in the steepness of the line? __When the rate

of savings changes, the slope of the line changes.______

11. What changes in the situation resulted in a change in the starting point of the line? _When the

starting value in Susan’s savings plan changed, the starting point, or y-intercept of the line,

changed.______

12. Write the coordinates of the point where the original line intersects the y-axis. __(0,25)_____

This point is called the y-intercept. What do these coordinates represent in the context of this

situation? ____This point represents the starting value of Susan’s savings plan.______

Teacher: Ch 7 “Dependent and Independent” 34 Activity 14: Spending Money

1. Anton worked all summer and saved $1,090.00 and plans to spend the money during the school year. He has budgeted spending $30.00 per week. Make a table of values for the situation.

Time (Weeks) Process Amount Saved (A) 0 $1,090.00 $1,090.00

1 $1,090.00 – 1week($30.00/week) $1,060.00

2 $1,090 – 2week($30/week) $1,030.00

3 $1,090 – 3week($30/week) $1,000.00

4 $1,090 – 4week($30/week) $970.00

5 $1,090 – 5week($30/week) $940.00

2. Write a function rule for the amount of money Anton will have after t weeks. Define the variables. ______A = 1,090 – 30 t ; when t = number of weeks and A = total left in savings.______

3. Find a viewing window for the problem situation and sketch a graph.

4. Justify your window choices. ______Answers will vary.______

______

Teacher: Ch 7 “Dependent and Independent” 35 5. Graph the function rule by hand on the coordinate grid.

5. How much money will Anton have after 11 weeks? ___$760.00______Support your answer using words, numbers and/or diagrams.

A = 1090 – 30t A = 1090 – 30(11) A = 1090 – 330 A = 760

6. When will Anton run out of money? _____37 weeks______

Support your answer using words, numbers and/or diagrams.

0 = 1090 – 30t After 36 weeks, Anton will only have -1090 -1090 $10.00 left to spend.

-1090 - 30t = -30 - 30

t = 36.333 t 37

Teacher: Ch 7 “Dependent and Independent” 36 7. How would the line for the graph change if Anton had initially saved $1,300 during the summer? _____Anton’s starting value will be higher, so the line will “start” on the y-axis at 1,300 instead of

1,090. The sloped stayed the same, so the new line would be parallel to the original.______

8. How would the line for the graph change if Anton spends $200.00 on school clothes and started

the school year with $890.00? ___Anton’s starting value will lower, so the line will “start” on the y-

axis at 890 instead of 1090. The y-intercept changed. The slop stayed the same, so the new line

would be parallel to the original.______

9. How would the line change if Anton starts with $1,090.00, but decides he will only spend $25.00

per week? ___Anton’s rate of spending changed, so now the money will not deplete so fast. The line

will be less steep than the original.______

10. What change in the situation resulted in a change in the steepness of the line? __How much

Anton spent per month.______

11. What change in the situation resulted in a change in the starting point of the line? __How much

Anton decided to put in the bank.______

12. Graph the three new situations (problems 7, 8, and 9) on the coordinate grid.

Teacher: Ch 7 “Dependent and Independent” 37 Activity 15: Money, Money, Money

1. Bill has $1,090.00 and he will spend $30.00 per week. Susan has $25.00 and will save $2.50 per week. Determine a viewing window that includes both situations and sketch both graphs.

2. Justify your window choices.

______Answers will vary.___Note: Susans’s graph not shown-check graph below. ______

3. Graph both function rules by hand on the coordinate grid. ) $ (

e c n a l a B

s ’ g n i v a S

Time (weeks)

Teacher: Ch 7 “Dependent and Independent” 38 4. When will Bill and Susan have the same amount of money in their accounts? ___Never______Support your answer using words, numbers and/or diagrams.

After 32 weeks, Bill has $130 and Susan has $105.00. During the 33rd week withdrawal by Bill, he will drop to $100 and Susan will increase to $107.50. So they never have the same amount in their accounts at the same time.

1090 – 30t = 25 + 2.50t – 2.50t – 2.50t

1090 – 32.50t = 25 -1090 -1090

-32.50t = -1065 t = 32.77

a. Write an equation that would allow you to determine the answer.

______1090 – 30 t = 25 + 2.50 t______Define the variable:

t = the number of weeks

6. How much money will Susan have when Bill is out of money? ______Support your answer using words, numbers and/or diagrams.

After 36 weeks, Bill is effectively out of money, as he only has $10 left.

So, after 36 weeks, Susan will have

A = 25 + 2.50(36) A = 25 + 90 A = $115

Once Bill spends the $10, Susan will have added $2.50 in that week to now have $117.50

Teacher: Ch 7 “Dependent and Independent” 39 Activity 16: At The Crossroads

FitnessPLUS Bodyworks

Initiation Fee $50 Initiation Fee $225 Monthly Fee $60 Monthly Fee $35

1. Candra wants to join a health club. Write two equations to compare the costs of belonging to each health club (in slope-intercept form). Let C represent the total cost of membership. Let N represent the number of months of membership.

Total Cost = Monthly fee times N + Initiation fee

FitnessPLUS ______C = 60(N) + 50______

Bodyworks ______C = 35(N) + 225______

2. Calculate the costs for 5 and 10 month membership for each of these two health clubs.

Cost of membership FitnessPLUS Bodyworks 50 + 5(60) = $350 225 + 5(35) = 400 5 Months 50 + 10(60) = $650 225 + 10(35) = $575 10 Months

3. Which health club membership will cost less if Candra joins for:

5 months? ___ Fitness Plus______For 10 months? ____ Bodyworks______

4. Without graphing, identify the slope and the y-intercept (vertical intercept) of the graph of each cost equation. Explain the meaning of each number in terms of the total cost of belonging to each health club.

FitnessPLUS: _____slope: $60/month, each month it costs $60 to belong.______

______y-intercept: $50, initial cost for zero months.______

Bodyworks: ______slope: $ 30/month, each month it costs $35 to belong.______

______y-intercept: $225, initial cost for zero months.______

Teacher: Ch 7 “Dependent and Independent” 40 5. Find the cost of a 7-month and 8-month membership for each health club.

FitnessPLUS Bodyworks

7-month membership: __60(7) + 50 = $470 __35(7) + 225 = $470_

8-month membership: __60(8) + 50 = $530 __35(8) + 225 = 505___

What do you notice? ______I notice at 7 months the cost is the same, $470. After 7 months the

Bodyworks is less expensive.______

6. Graph each equation on the same graph. Label the axes, title the graph and use appropriate scales.

Cost to Belong to Fitness Plus and Bodyworks Health Clubs

700 675 650 625 600 575 550 525

$ 475 Fitness Plus (

s 425 Bodyworks

o 400 375 C 350 y

h 300 t 275 n 250 o 225 M 200 175 150 125 100 75 50 25 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Months

7. Explain what the point where the two lines intersect means in the context of this scenario.

_ The intersection point is when the cost of belonging to either health club is the same, at 7 months.

Teacher: Ch 7 “Dependent and Independent” 41 8. Which health club costs less to belong to for one year? Use information from the graph to support your answer. ____After 7 months, Bodyworks costs less to belong. Therefore, after one year Bodyworks will cost

less.______

9. Candra plans on belonging to a health club for many years. Make a recommendation to Candra regarding which club to join from a purely financial perspective, but also communicate to her other criteria she should use before making her decision. ____Bodyworks will be cheaper in the long run, but you must also look at:______

_____- Equipment______

_____- Instruction______

_____- Proximity to home or work______

_____- Where your workout buddies belong______

10. Of the people in attendance at a recent baseball game,

 one-third had grandstand tickets,  one-fourth had bleacher tickets, and  the remaining 11,250 people in attendance had other tickets.

Which was the total number of people in attendance at the game?

 A. 27,000  B. 20,000  C. 16,000  D. 18,000

11. Many sociologists predicted in 1990 that the trend shown on this graph will continue for many years.

Which is the median age a man would marry for the first time in the year 2000?

 A. Younger than 26.0  B. Between 26.0 and 26.9  C. Between 27.0 and 28.0  D. Older than 28.0

Teacher: Ch 7 “Dependent and Independent” 42 12. A painter uses 2 gallons of paint to cover a 400 square-foot area.

Which line on the graph best represents the total number of gallons of paint he will need to paint x square feet?

 A. l

 B. m

 C. n

 D. q

13. The annual salary for a teacher is $34,000, plus an additional $500 per year for each year of experience.

Which equation represents y, the annual salary for a teacher with x years of experience?

500  A. y = + 34,000 x

500  B. y = - + 34,000 x

 C. y= -500 x + 34,000

 D. y=500 x + 34,000

Teacher: Ch 7 “Dependent and Independent” 43 14. The table shows data collected from a timed practice period for a bicyclist riding at a constant rate.

Which equation represents the linear relationship between y, the total number of minutes ridden, and x, the total number of miles ridden?

485  A. y=10 x - 2

1 97  B. y= x + 10 10

1  C. y= x + 25 10

10  D. y= x 3

15. Which situation could be represented by the equation y = 3x + 5?

 A. A bricklayer lays 5 bricks in 3 minutes. What is the total number of bricks he can lay in x minutes?

 B. An express package costs $5.00 plus $3.00 per pound to ship. What is the total cost of shipping x pounds?

 C. A jogger is 3 miles from home, jogging toward home at 5 miles per hour. What is the total number of minutes it will take her to get home?

 D. A math teacher gave some problems worth 5 points and a bonus problem worth 3 points. What was the total value of the problems?

Teacher: Ch 7 “Dependent and Independent” 44