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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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