Performance Task: Mini-Problems

Transcription

1 Performance Task: Mini-Problems In this task, students will create a table and develop an equation. STANDARDS ADDRESSED IN THIS TASK: Use functions to model relationships between quantities. MCC8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. MCC8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Investigate patterns of association in bivariate data. MCC8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. MCC8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. MCC8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. MCC8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. July 2014 Page 120 of 195

2 STANDARDS FOR MATHEMATICAL PRACTICE: This task uses all of the practices with emphasis on: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. BACKGROUND KNOWLEDGE: In order for students to be successful, the following skills and concepts need to be maintained: MCC5.G.1, MCC5.G.2, MCC6.EE.5, MCC6.EE.6, MCC6.EE.7, MCC7.EE.4 COMMON MISCONCEPTIONS: Some students may not pay attention to the scale on a graph, assuming that the scale units are always one. When making axes for a graph, some students may not using equal intervals to create the scale. Students often confuse a recursive rule with an explicit formula for a function. For example, after identifying that a linear function shows an increase of 2 in the values of the output for every change of 1 in the input, some students will represent the equation as y = x + 2 instead of realizing that this means y = 2x + b. When tables are constructed with increasing consecutive integers for input values, then the distinction between the recursive and explicit formulas is about whether you are reasoning vertically or horizontally in the table. Both types of reasoning and both types of formulas are important for developing proficiency with functions. Some students may infer a cause and effect between independent and dependent variables, but this is often not the case. ESSENTIAL QUESTIONS: How can I write a function to model a linear relationship? What strategies can I use to help me understand and represent real situations involving linear relationships? How can the properties of lines help me to understand graphing linear functions? What can I infer from the data? How can functions be used to model real-world situations? How does a change in one variable affect the other variable in a given situation? July 2014 Page 121 of 195

3 Which tells me more about the relationship I am investigating a table, a graph or an equation? Why? How can I analyze a scatter plot? How can I use a linear model to solve problems? How can I use bivariate data to solve problems? MATERIALS: Graph paper Straightedge Scientific or Graphing Calculator (optional) Copies of task for students GROUPING: Individual/Partner TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION: These Mini-Problems may be utilized in a variety of ways: use them together, assign a different one to each group and let them share or video their results small group, individually, take home assignments DIFFERENTIATION: Extension: Write your own mini-problem similar to the task problems. Be sure to create a situation that will lend itself nicely to the creation of a table, graph and equation in slope-intercept form. Create clear directions as to what you would like a classmate to do in order to analyze your problem situation and develop questions to go with your problem. You will be trading problems with a classmate. Intervention/Scaffolding: Help struggling students set up the tables and graphs. Prompt struggling students with guiding questions. Group students in a way that promotes cooperative learning. July 2014 Page 122 of 195

4 Mini-Problems 1. Susan and Shumaq work at Philips Arena selling programs to the Hawks games. They get paid $25 per game plus $0.50 for each program they sell. a. Make a table showing the pay they can expect for any game as a function of the number of programs they sell. Include values for 0 to 100 programs in steps of 5. Solutions Programs Sold Pay Earned $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $75.00 b. Write an equation relating programs sold S and pay P for a game. P =.50(S) + 25 July 2014 Page 123 of 195

5 c. Graph the relation between programs sold and pay (for 0 to 100 programs). Since pay depends on the number of programs sole, pay is the dependent (y-axis) variable. The number of programs sold is the independent variable (x-axis). Solutions d. How do the numbers from the pay rule of $25 plus $0.50 per program sold relate to the patterns in the table, the graph, and the equation? Lowest amount paid is $25 then is increases in increments of $2.50. July 2014 Page 124 of 195

6 2. Amani and Greg work for a small production company creating music videos. Amani is a producer with several years of experience, so she is paid $20 per hour. This is Greg s first job in this line of work, so he earns only $8.50 per hour. a. Make a table showing how Amani s and Greg s earnings will grow over the summer as a function of hours worked. Solutions Hours Worked Amani s Earnings Greg s Earnings 0 $0 $0 5 $100 $ $200 $85 15 $300 $ $400 $ $500 $ $600 $ $700 $ $800 $ $900 $ $1000 $ $1100 $ $1200 $ $1300 $ $1400 $ $1500 $ $1600 $ $1700 $ $1800 $ $1900 $ $2000 $850 b. Write equations relating hours worked H and pay earned for each worker. Use A for Amani s earnings and G for Greg s earnings. Equation for Amani s Earnings Equation for Greg s Earnings A = 20(H) G = 8.50(H) July 2014 Page 125 of 195

7 c. Sketch graphs of the two pay rules on the same coordinate system. 3. Laine had a babysitting service the summer before she left for Georgia Tech. She earned $1200. While she did spend some of it on new college clothes, she saved quite a bit to have some spending money while away at college. The following table shows her bank account balance over a 9-week period in the fall semester. Week Number Money Left a. Make a scatter plot of Laine s bank balance data. Draw a line that models the trend in that plot. July 2014 Page 126 of 195

8 Solutions b. Write an equation for the linear model. Use W for weeks and M for money left. Remember that money left depends on the week number. Answers may vary but should be in the neighborhood of M =840 24(W) c. Predict Laine s bank balance after 10, 15, and 20 weeks. Balance after: Week #10 Week #15 Week #20 M = (10) M = (15) M = (20) M = M = M = M = 600 M = 480 M = 360 d. Explain how the patterns in the table, graph, and equation are related to each other. What do they say about Laine s spending habits? The pattern indicates that Laine s money supply is dwindling. It appears to be decreasing by about $24 a day. July 2014 Page 127 of 195

9 4. Eric is an umpire for a softball league. The job pays $960 for the season. The league plays 3 nights a week for 8 weeks. Eric is already aware that he will need a substitute a few of those nights. He will have to pay the substitute from his salary. Solutions a. What should Eric pay a substitute for one night? 3 nights per week for 8 weeks = 24 total nights to work 960/24 = $40 per night b. Use the letters N for nights a substitute works, S for pay to the substitute, and E for Eric s total earnings. Write an equation relating N and S, beginning S = S = 40 (N) Write an equation relating N and E, beginning E = E = (N) c. Create graphs of the equations in part b. Graph of S = 40 (N) July 2014 Page 128 of 195

10 Graph of E = (N) d. How do the equations and the patterns of the graphs show Eric s earning prospects from two views? From Eric s point of view which is represented by the equation E = (N), his earnings are decreasing by a rate of $40 every night that he cannot work. From the substitute s point which is represented by the equations S = 40(N), the earnings are increasing every night he/she gets to go to work. July 2014 Page 129 of 195

11 SE TASK: Mini-Problems 1. Susan and Shumaq work at Philips Arena selling programs to the Hawks games. They get paid $25 per game plus $0.50 for each program they sell. a. Make a table showing the pay they can expect for any game as a function of the number of programs they sell. Include values for 0 to 100 programs in steps of 5. b. Write an equation relating programs sold S and pay P for a game. c. Graph the relation between programs sold and pay (for 0 to 100 programs). Since pay depends on the number of programs sole, pay is the dependent (y-axis) variable. The number of programs sold is the independent variable (x-axis). d. How do the numbers from the pay rule of $25 plus $0.50 per program sold relate to the patterns in the table, the graph, and the equation? 2. Amani and Greg work for a small production company creating music videos. Amani is a producer with several years of experience, so she is paid $20 per hour. This is Greg s first job in this line of work, so he earns only $8.50 per hour. a. Make a table showing how Amani s and Greg s earnings will grow over the summer as a function of hours worked. b. Write equations relating hours worked H and pay earned for each worker. Use A for Amani s earnings and G for Greg s earnings. c. Sketch graphs of the two pay rules on the same coordinate system. 3. Laine had a babysitting service the summer before she left for Georgia Tech. She earned $1200. While she did spend some of it on new college clothes, she saved quite a bit to have some spending money while away at college. The following table shows her bank account balance over a 9-week period in the fall semester. Week Number Money Left Problem continues on next page. July 2014 Page 130 of 195

12 Problem continued from previous page. a. Make a scatter plot of Laine s bank balance data. Draw a line that models the trend in that plot. b. Write an equation for the linear model. Use W for weeks and M for money left. Remember that money left depends on the week number. c. Predict Laine s bank balance after 10, 15, and 20 weeks. d. Explain how the patterns in the table, graph, and equation are related to each other. What do they say about Laine s spending habits? 4. Eric is an umpire for a softball league. The job pays $960 for the season. The league plays 3 nights a week for 8 weeks. Eric is already aware that he will need a substitute a few of those nights. He will have to pay the substitute from his salary. a. What should Eric pay a substitute for one night? b. Use the letters N for nights a substitute works, S for pay to the substitute, and E for Eric s total earnings. Write an equation relating N and S, beginning S = Write an equation relating N and E, beginning E = c. Create graphs of the equations in part b d. How do the equations and the patterns of the graphs show Eric s earning prospects from two views? July 2014 Page 131 of 195