AT THE CROSSROADS

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					                            DEPENDENT AND INDEPENDENT

Enduring Understanding: Develop a better understanding that using systems of equations in
graphical form is appropriate to determine a solution. Develop a better understanding of how to use
variables to write linear equations. Develop a better understanding of how to write an equation and
then substitute values in the equation to answer a question. Develop a better understanding of
identifying or writing a rule to describe a linear function. Develop a better understanding of drawing
conclusions and supporting them.

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?

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

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

Dependent and Independent                                                                            1
        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.
       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.

Materials: Motion Detector
           Graph paper
           Graphing Calculators
           Rulers
           Colored pencils

EALRs/GLEs:

1.5.1-- Apply knowledge of patterns or sequences to represent linear functions
1.5.2-- Determine an equation or rule for a linear function represented in a pattern, table, graph, or
model.
1.5.4-- Use variables to write expressions, linear equations and inequalities that represent situations
involving rational numbers, whole number powers, and square roots.
1.5.6-- Apply properties to solve multi-step equations and systems of equations.
3.2.1-- Draw and support conclusions, using inductive or deductive reasoning.

Item Specifications: AS01; AS02; AS03; SR04

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.


Dependent and Independent                                                                                 2
                                        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.

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 -_____________________________________________________________

b. Dependent variable - ______________________________________________________________

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

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________

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

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________

5. The number of cakes sold in a bake sale determines the amount of money made.

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________



Dependent and Independent                                                                                   3
6. The winner of the football game depends on which side scored the most points.

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________

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

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________

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. _______________________________________________________________________________

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________


3. _______________________________________________________________________________

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________
4. _______________________________________________________________________________

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________

5. _______________________________________________________________________________

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________




Dependent and Independent                                                                         4
6. _______________________________________________________________________________

a. Independent variable -_____________________________________________________________

b. Dependent variable - ______________________________________________________________


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


                       Cause                                             Effect




Dependent and Independent                                                                           5
                    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
           I have.




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




Dependent and Independent                                                                         6
                            Activity 3: Dependent (Responding) 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

                                            _______________________________________




a.

                                            Statement: ___________________________________

                                            _____________________________________________

                                            _____________________________________________

                                            _____________________________________________

                                            _____________________________________________




b.

                                            Statement: ___________________________________

                                            ____________________________________________

                                            ____________________________________________

                                            ____________________________________________

                                            ____________________________________________




Dependent and Independent                                                                          7
                                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.




        a. Verbal Description: ________________________________________________________

        ___________________________________________________________________________

        ___________________________________________________________________________

        b. Equation: ____________________________________________________________

2. 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.




        a. Verbal Description: ________________________________________________________

        ___________________________________________________________________________

        ___________________________________________________________________________

        b. Equation: ____________________________________________________________




Dependent and Independent                                                                             8
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.




        a. Verbal Description: ________________________________________________________

        ___________________________________________________________________________

        ___________________________________________________________________________

        b. Equation: ____________________________________________________________


4. 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.




        a. Verbal Description: ________________________________________________________

        ___________________________________________________________________________

        ___________________________________________________________________________

        b. Equation: ____________________________________________________________




Dependent and Independent                                                                             9
                                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): ____________________________________________

        b. How does the equation relate to the original problem? _________________________
        ___________________________________________________________________________

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.




        a. Equation (equation of the line): __________________________________________

        b. How does the equation relate to the original problem? _________________________
        ___________________________________________________________________________




Dependent and Independent                                                                         10
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): __________________________________________

        b. How does the equation relate to the original problem? _________________________
        ___________________________________________________________________________




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


                       Table
            Time (Sec)      Distance (ft)




        a. Equation (equation of the line): __________________________________________

        b. How does the equation relate to the original problem? _________________________
        ___________________________________________________________________________




Dependent and Independent                                                                          11
                               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. _______________________________________________________________________________

2. _______________________________________________________________________________




3. _______________________________________________________________________________

4. _______________________________________________________________________________




5. _______________________________________________________________________________

6. _______________________________________________________________________________




Dependent and Independent                                                                 12
7. _______________________________________________________________________________

8. _______________________________________________________________________________




9. _______________________________________________________________________________

10. ______________________________________________________________________________




11. ______________________________________________________________________________

12. ______________________________________________________________________________




13. ______________________________________________________________________________

14. ______________________________________________________________________________




Dependent and Independent                                                       13
                     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: _________________________                Coordinates: ________________________
_____________________________________                 ____________________________________
        Yes or No _______________                               Yes or No ______________




Coordinates: _________________________                Coordinates: _______________________
_____________________________________                 ___________________________________

        Yes or No _______________                           Yes or No ______________


Dependent and Independent                                                                              14
Coordinates: _________________________             Coordinates: _________________________

_____________________________________              _____________________________________

        Yes or No _______________                           Yes or 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.
_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________




Dependent and Independent                                                                        15
                                     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




        Does this group of coordinate points represent a function? _____________

        Why? ____________________________________________________________________

        __________________________________________________________________________

        __________________________________________________________________________

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

                    X                       Y




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

        Why? __________________________________________________________________

        ________________________________________________________________________

        ______________________________________________________________




Dependent and Independent                                                                           16
        c. (-1, -5); (2, -5); (4, -5); (-5, 2); (-3, 3)

                    X                       Y




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

        Why? ____________________________________________________________________

        __________________________________________________________________________

        __________________________________________________________________________

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

                    X                       Y




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

        Why? ____________________________________________________________________

        __________________________________________________________________________

        __________________________________________________________________________




Dependent and Independent                                                                         17
                              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. ____________________
_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

    b. The letters on the telephone depend on the numbers on the phone. ___________________
_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

    c. The number of days in a month depends on the month. ____________________________
_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

    d. The cost of renting a car if there is a charge of $0.50 per mile. ______________________
_________________________________________________________________________________

_________________________________________________________________________________

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

_________________________________________________________________________________

_________________________________________________________________________________

Dependent and Independent                                                                             18
                             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. _______________________________________________

Explain why you made your choice: __________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________



Dependent and Independent                                                                          19
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

Number of Roses           Process              Cost at               Process              Cost at
                          Column             Roses-R-Red             Column            Flower Power
                       (Roses-R-Red)                             (Flower Power)
                        20 + 10(0.75)           $27.50            60 + 10(0.50)            $65.00
        10

        20

        40

        60

       120

       150

       210

       240

      1,000

         n

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

        1) Sentence using words: _____________________________________________________

_________________________________________________________________________________

        2) Sentence using symbols: ___________________________________________________

_________________________________________________________________________________

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

        1) Sentence using words: _____________________________________________________

_________________________________________________________________________________

        2) Sentence using symbols: ___________________________________________________

_________________________________________________________________________________




Dependent and Independent                                                                           20
    c. What patterns do you observe from the table of values? _______________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

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

_________________________________________________________________________________

_________________________________________________________________________________

    e. What would a graph of this relationship look like? ___________________________________

_________________________________________________________________________________

_________________________________________________________________________________

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




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




    h. What is the point where the two flower dealers charge the same amount? ___________

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

    amount. _________________________________________________________________




Dependent and Independent                                                                       21
3. Graph the two functions on the same coordinate grid an label.

                 ________________________________________________________




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

_________________________________________________________________________________

_________________________________________________________________________________


5. What effect does the $20.00 have on the graph? _______________________________________

_________________________________________________________________________________




Dependent and Independent                                                                    22
6. What effect does the $0.50 per stem cost have on the graph of the Flower Power function?

_________________________________________________________________________________

_________________________________________________________________________________


7. What effect does the $60.00 have on the graph? _______________________________________

_________________________________________________________________________________


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

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


9. What is the point of intersection of the two graphs? _____________________________________
Show the work that supports your answer.




10. Which flower dealer offers the better deal and when? ___________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________




Dependent and Independent                                                                     23
                                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?___________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

2. Make a prediction. Will it change which company has the better offer?______________________
Support your prediction: ___________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________




Dependent and Independent                                                                             24
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(0.75)            $7.50            40 + 10(0.50)            $45.00
         10

         20

         40

         60

         120

         150

         210

         240

         300

         n

    b. What patterns do you observe in the new table of values? _____________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

    c. Compare the costs in this table to the costs in the first table. What changes do you observe?

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________



Dependent and Independent                                                                              25
    d. How many roses can you buy from Roses-R-Red for $65.00? __________________________
    Support your answer using words, numbers and/or diagrams.




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




    f. Write an equation that represents the point where the two flower shops charge the same
    amount.
    ________________________________________________________________________

    g. At what point will the two flower dealers charge the same amount? ___________

    What is the charge? __________________




    h. Which company offers the better deal? ____________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________




Dependent and Independent                                                                       26
                      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? _____________________________________________________________

        ___________________________________________________________________________

        ___________________________________________________________________________

        ___________________________________________________________________________


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? _____________________________________________________________

        ___________________________________________________________________________

        ___________________________________________________________________________

        ___________________________________________________________________________


3. What are the coordinates of the point where the original two functions intersect? _____________


4. What are the coordinates of the point where two new functions intersect? ___________________


5. What is the significance of this point of intersection? ____________________________________

_________________________________________________________________________________

_________________________________________________________________________________

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

_________________________________________________________________________________

_________________________________________________________________________________




Dependent and Independent                                                                            27
7. Which flower dealer now offers the better deal? ________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________



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.

            ______________________________________________________




Dependent and Independent                                                                         28
                               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)

                                                                               $25.00




2. Write a function rule for the amount of money Susan will have after t weeks and define the
variables.
_________________________________________________________________________________

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




4. Justify your window choices.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

Dependent and Independent                                                                        29
5. Graph the function rule by hand on the coordinate grid.

                        _____________________________________________




5. How much money will Susan have after 7 weeks? ____________________________________
Support your answer using words, numbers and/or diagrams.




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

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




Dependent and Independent                                                                       30
7. How would the line for the graph change if Susan deposits only $15.00 of the $25.00 given to her

by her grandmother? ________________________________________________________________

_________________________________________________________________________________


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? _______________________________________________________

_________________________________________________________________________________


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? ________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


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

_________________________________________________________________________________


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

_________________________________________________________________________________

_________________________________________________________________________________


12. Write the coordinates of the point where the original line intersects the y-axis. _______________

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

situation? _________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________




Dependent and Independent                                                                          31
                                  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)

                                                                            $1,090.00




2. Write a function rule for the amount of money Anton will have after t weeks. Define the
variables.
_________________________________________________________________________________

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




4. Justify your window choices.
_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________



Dependent and Independent                                                                       32
5. Graph the function rule by hand on the coordinate grid.

                        _____________________________________________




5. How much money will Anton have after 11 weeks? ____________________________________
Support your answer using words, numbers and/or diagrams.




6. When will Anton run out of money?

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




Dependent and Independent                                                            33
7. How would the line for the graph change if Anton had initially saved $1,300 during the summer?
_________________________________________________________________________________

_________________________________________________________________________________

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? ___________________________________________________
_________________________________________________________________________________


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

per week? ________________________________________________________________________
_________________________________________________________________________________


10. What change in the situation resulted in a change in the steepness of the line? ______________
_________________________________________________________________________________


11. What change in the situation resulted in a change in the starting point of the line? __________
_________________________________________________________________________________

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

                        _____________________________________________




Dependent and Independent                                                                          34
                            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.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

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

                 ________________________________________________________




Dependent and Independent                                                                       35
4. When will Bill and Susan have the same amount of money in their accounts? _________________
Support your answer using words, numbers and/or diagrams.




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

        ___________________________________________________________________________
        Define the variable:




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




Dependent and Independent                                                                   36
                                         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             ________________________________________________________

Bodyworks               ________________________________________________________

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




                                          Cost of membership
                                             FitnessPLUS                       Bodyworks

             5 Months

             10 Months

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

        5 months? ______________________             For 10 months? ________________________


4. Without graphing, identify the slope and the 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: ____________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

Dependent and Independent                                                                            37
Bodyworks: _____________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________



5. Find the cost of a 7-month and 8-month membership for each health club.

                                FitnessPLUS                            Bodyworks

7-month membership:         __________________                  ____________________



8-month membership:         __________________                  ____________________




What do you notice? ______________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________




Dependent and Independent                                                              38
6. Graph each equation on the same graph. Label the axes, title the graph and use appropriate scales.

                __________________________________________________________




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

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________



Dependent and Independent                                                                           39
8. Which health club costs less to belong to for one year? Use information from the graph to support
your answer.
_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

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.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


Dependent and Independent                                                                             40
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




Dependent and Independent                                                                           41
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




Dependent and Independent                                                                              42
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?




Dependent and Independent                                                                              43

				
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