SCATTER PLOTS AND TREND LINES

Transcription

1 6 SCATTER PLOTS AND TREND LINES INSTRUCTIONAL ACTIVITY Lesson 2 LEARNING GOAL Students will use a scatter plot to identify outliers and clusters, estimate a linear model, and use the linear model to make predictions. The critical outcome of this activity is for students to identify outliers on a scatter plot, informally draw and assess the accuracy of a trend line, and use the trend line to make predictions. Additionally, students should be able to identify the y-intercept and slope of a trend line and to describe what they mean in terms of the data. PRIMARY ACTIVITY Students will use the scatter plot from LESSON 1 to identify outliers and clusters of data. Then, given a foot length (or height), students will predict height (or foot length), first visually, using the scatter plot alone, then with a trend line (determined informally). OTHER VOCABULARY Students will need to know the meaning of Slope y-intercept Trend line Outlier Data cluster MATERIALS Scatter plot from LESSON 1

2 7 IMPLEMENTATION This lesson begins with the data collected in LESSON 1. Create at least one outlier for discussion in this lesson when recording the results from the previous lesson. One example could be if your class has a very young student who was accelerated by several years or a student who has long feet but is significantly shorter than other students with the same size feet. Require students to add the outlier to their scatter plot from LESSON 1 (do not tell students it is an outlier). Ask students what they notice about the point. Students should have experience identifying outliers in work with one-variable data. Define an outlier in bivariate data as a data value that substantially deviates or is distant from an overall pattern in the data set. Lead students to identify any other outliers from the scatter plot they created in LESSON 1 it is possible that the only outlier will be the one you included. Ask each student to think of a data point that would have been an outlier if it was part of the data. Have students share with a partner and confirm each other s outlier, then share their outliers with the rest of the class. Determine if the student can EXPLAIN OUTLIERS: Do any of the points look out of place? What is different about the points you identified? What should you consider when creating a data point that would be considered an outlier? How did you determine if your partner s data point was an outlier? Determine if the student can EXPLAIN CLUSTERING OF DATA IN A SCATTER PLOT: Where are most of your data points? Why is the data clustered in this portion of the scatter plot?

3 8 After a discussion of outliers, students will start predicting values within the data. Provide your students either your foot length or height and have students predict the other measure using their scatter plot alone. (The scatter plot should appear roughly linear, with a positive slope, so students should be able to make a reasonable prediction.) Ask students to share their predictions and describe how they arrived at that number. Make the measurement and share with students to determine how close their estimates are. It is likely that the teacher s foot length and height fall within or close to the student data. After discussing a point within the data, discuss a point that falls outside or beyond the class data. For example, choose a basketball player (e.g., Kevin Durant height: 206 cm, foot length: 31.8 cm) to add to the scatter plot to require students to estimate a value beyond the data. Ask students to share their predictions for the measurement not given and ask how they arrived at that number. Some students may, without a prompt to do so, draw a line to help estimate a value that is well beyond the majority of the data. Lead all students to the idea of drawing a trend line through data in a scatter plot to help them estimate values that are not in their data set. Determine if the student can USE GRAPHS TO READ BETWEEN THE DATA: How did you estimate the unknown measurement using the scatter plot? How does the structure of the scatter plot help you estimate? Determine if the student can USE GRAPHS TO READ BEYOND THE DATA: How did you estimate an unknown measurement outside of the values plotted on the scatter plot? What strategies can help you make a more accurate estimate?

4 9 Determine if the student can EXPLAIN A LINEAR OR NONLINEAR ASSOCIATION OF DATA DISPLAYED IN A SCATTER PLOT: Do the points on the scatter plot seem to follow a pattern? Does the pattern appear to be straight (linear) or curved? Can you think of a relationship between two quantities that might not be linear? Does the pattern have a positive or negative relationship? Can you think of a relationship that might be negative? Define a trend line as a line that best defines or expresses the trend of the data, approximates the pattern in the data, and is centrally located within the points. Direct students to draw a trend line on their data if they haven t already. If they drew a line while predicting values earlier in the lesson, require students to assess whether the line fits the criteria for a trend line and adjust the line if needed. Elicit student thinking: How does the name trend line relate to the way the line is drawn on a scatter plot? Determine if the student can REPRESENT A TREND LINE FOR A SCATTER PLOT: How do you know the line you drew fits the data? (Student should provide reasoning that includes the criteria of the trend line.) How many points are within one centimeter (above or below) the trend line? Two centimeters? Five centimeters? Once all students have their trend line drawn, students will explore the meaning of the y-intercept in terms of foot length and height.

5 10 If the x-axis is constructed with a break, Focus students on the trend line. Ask students if it would be accurate to continue the trend line through the y-axis. Alternatively, if the trend line is already drawn through the y-axis, ask students if this is an accurate y-intercept. Through discussion, students should come to understand that this point is not accurate because of the break in the graph. Ask students what the y-intercept would mean in terms of the problem situation. Depending on how the axes are oriented, this is either someone s height who has a foot length of 0 cm or someone s foot length who has a height of 0 cm. Ask students to explain whether this is a reasonable data value for this situation. Discuss where the values on the scatter plot begin to make sense. Students should discuss topics such as possible heights and foot lengths when babies are born and identify where these points would be located on the scatter plot. If the x-axis is constructed without a break, Focus students on the point where their trend line crosses the y-axis, and ask them what this point is called. Ask students what this point means in terms of the problem situation. Depending how the axes are oriented, this is either someone s height who has a foot length of 0 cm or someone s foot length who has a height of 0 cm. Ask students to explain whether this is a reasonable data value for this situation. Discuss where the values on the scatter plot begin to make sense. Students should discuss topics such as possible heights and foot lengths when babies are born and identify where these points would be located on the scatter plot. Elicit student thinking: Where is the scatter plot realistic?

6 11 Determine if the student can EXPLAIN AN INTERCEPT IN THE CONTEXT OF A PROBLEM OR DATA SET: Are you able to find the y-intercept of the trend line? If so, what is the y- intercept? What does the y-intercept mean in terms of the variables in the problem situation? Is this a realistic combination of foot length and height? After discussing the y-intercept, shift the conversation toward the meaning of the slope of the trend line. Focus students on how the trend line is changing. Ask whether the trend line is increasing or decreasing and what this means in terms of the problem situation. Discuss more specifically how much the line increases over each unit on the x-axis. Students should be able to answer quantitatively and describe the meaning in terms of foot length and height (e.g., As foot length increases by 1 cm, height increases by about cm. ). Determine if the student can EXPLAIN THE SLOPE IN THE CONTEXT OF A PROBLEM OR DATA SET: What is the approximate slope of the trend line? What does the slope mean in terms of the variables in the problem situation? At the end of this activity, Require students to sketch a scatter plot with data points that suggest a linear trend. Ask students to label the axes with possible variables (e.g., hours studied and test grade) and draw a trend line. Students should then answer the following three questions about their scatter plot and trend line: How do you know the trend line you drew is appropriate for the data? Describe the meaning of the y-intercept of the trend line in terms of your data. Describe the meaning of the slope of the trend line in terms of your data.