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Domain: Statistics and Probability (SP) Cluster: Investigate patterns of association in bivariate data. Standard: 8.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. Standards for Mathematical Practice (MP): MP.2. Reason abstractly and quantitatively. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. Connections: This Cluster is connected to the grade 8 Critical Area of Focus #1, Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations. Explanations and Examples 8.SP.1 Bivariate data refers to two variable data, one to be graphed on the x-axis and the other on the y-axis. Students represent measurement (numerical) data on a scatter plot, recognizing patterns of association. These patterns may be linear (positive, negative or no association) or non-linear Students build on their previous knowledge of scatter plots examine relationships between variables. They analyze scatterplots to determine positive and negative associations, the degree of association, and type of association. Students examine outliers to determine if data points are valid or represent a recording or measurement error. Students can use tools such as those at the National Center for Educational Statistics to create a graph or generate data sets. (http://nces.ed.gov/nceskids/createagraph/default.aspx) Examples: Data for 10 students‘ Math and Science scores are provided in the chart. Describe the association between the Math and Science scores. Student Math Science 1 64 68 2 50 70 3 85 83 4 34 33 5 56 60 6 24 27 7 72 74 8 63 63 9 42 40 10 93 96 Data for 10 students‘ Math scores and the distance they live from school are provided in the table below. Describe the association between the Math scores and the distance they live from school. Student Math score Dist from school (miles) 1 2 3 4 5 6 7 8 9 10 64 50 85 34 56 24 72 63 42 93 0.5 1.8 1 2.3 3.4 0.2 2.5 1.6 0.8 2.5 49 Data from a local fast food restaurant is provided showing the number of staff members and the average time for filling an order are provided in the table below. Describe the association between the number of staff and the average time for filling an order. Number of staff Average time to fill order (seconds) 3 180 4 138 5 120 6 108 7 96 8 84 The chart below lists the life expectancy in years for people in the United States every five years from 1970 to 2005. What would you expect the life expectancy of a person in the United States to be in 2010, 2015, and 2020 based upon this data? Explain how you determined your values. Date 1970 1975 1980 1985 1990 1995 2000 2005 Life Expectancy (in years) 70.8 72.6 73.7 74.7 75.4 75.8 76.8 77.4 Instructional Strategies Building on the study of statistics using univariate data in Grades 6 and 7, students are now ready to study bivariate data. Students will extend their descriptions and understanding of variation to the graphical displays of bivariate data. Scatter plots are the most common form of displaying bivariate data in Grade 8. Provide scatter plots and have students practice informally finding the line of best fit. Students should create and interpret scatter plots, focusing on outliers, positive or negative association, linearity or curvature. By changing the data slightly, students can have a rich discussion about the effects of the change on the graph. Have students use a graphing calculator to determine a linear regression and discuss how this relates to the graph. Students should informally draw a line of best fit for a scatter plot and informally measure the strength of fit. Discussion should include ―What does it mean to be above the line, below the line?‖ The study of the line of best fit ties directly to the algebraic study of slope and intercept. Students should interpret the slope and intercept of the line of best fit in the context of the data. Then students can make predictions based on the line of best fit. Common Misconceptions: Students may believe: Bivariate data is only displayed in scatter plots. 8.SP.4 in this cluster provides the opportunity to display bivariate, categorical data in a table. In general, students think there is only one correct answer in mathematics. Students may mistakenly think their lines of best fit for the same set of data will be exactly the same. Because students are informally drawing lines of best fit, the lines will vary slightly. To obtain the exact line of best fit, students would use technology to find the line of regression. 50 8.SP.1 Domain: Statistics and Probability Cluster: Investigate patterns of association in bivariate data. Standard: 8.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. Standards for Mathematical Practice (MP): MP.2. Reason abstractly and quantitatively. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. Connections: See 8.SP.1. Explanations and Examples 8.SP.2 Students understand that a straight line can represent a scatter plot with linear association. The most appropriate linear model is the line that comes closest to most data points. The use of linear regression is not expected. Examples: The capacity of the fuel tank in a car is 13.5 gallons. The table below shows the number of miles traveled and how many gallons of gas are left in the tank. Describe the relationship between the variables. If the data is linear, determine a line of best fit. Do you think the line represents a good fit for the data set? Why or why not? What is the average fuel efficiency of the car in miles per gallon? Miles 0 Traveled Gallons 0 Used 75 120 160 250 300 2.3 4.5 5.7 9.7 10.7 Instructional Strategies See 8.SP.1. 51 8.SP.2 Domain: Statistics and Probability Cluster: Investigate patterns of association in bivariate data. Standard: 8.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. Standards for Mathematical Practice (MP): MP.2. Reason abstractly and quantitatively. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. Connections: See 8.SP.1. Explanations and Examples 8.SP.3 Linear models can be represented with a linear equation. Students interpret the slope and yintercept of the line in the context of the problem. Examples: 1. Given data from students‘ math scores and absences, make a scatterplot. 2. Draw a line of best fit, paying attention to the closeness of the data points on either side of the line. 3. From the line of best fit, determine an approximate linear equation that models the given data (about y = 25 x 95 ) 4. Students should recognize that 95 represents the y intercept and 3 25 represents 3 the slope of the line. 52 5. Students can use this linear model to solve problems. For example, through substitution, they can use the equation to determine that a student with 4 absences should expect to receive a math score of about 62. They can then compare this value to their line. Instructional Strategies See 8.SP.1. 53 8.SP.3 Domain: Statistics and Probability Cluster: Investigate patterns of association in bivariate data. Standard: 8.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. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Standards for Mathematical Practice (MP): MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. Connections: See 8.SP.1. Explanations and Examples 8.SP.4 Students recognize that categorical data can also be described numerically through the use of a two-way table. A two-way table is a table that shows categorical data classified in two different ways. The frequency of the occurrences are used to identify possible associations between the variables. For example, a survey was conducted to determine if boys eat breakfast more often than girls. The following table shows the results: Male Female Eat breakfast on a 190 110 regular basis Do not eat breakfast 130 165 on a regular basis Students can use the information from the table to compare the probabilities of males eating breakfast (190 of the 320 males à 59%) and females eating breakfast (110 of the 375 females à29%) to answer the question. From this data, it can be determined that males do eat breakfast more regularly than females. Example: The table illustrates the results when 100 students were asked the survey questions: Do you have a curfew? and Do you have assigned chores? Is there evidence that those who have a curfew also tend to have chores? Solution: Of the students who answered that they had a curfew, 40 had chores and 10 did not. Of the students who answered they did not have a curfew, 10 had chores and 40 did not. From this sample, there appears to be a positive correlation between having a curfew and having chores. 54 8.SP.4