Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 7, Part C:
Modeling Linear Relationships

In This Part: How Square Can You Be? | Analyzing the Differences | Using a Scatter Plot

A scatter plot is also useful in investigating the nature of the relationship between height and arm span. Here is the scatter plot of the 24 heights and arm spans:

Consider these people from the data table:

Person #

Arm Span

Height

Height - Arm Span

 1 156 162 6 6 161 162 1 9 170 170 0 14 177 176 -1 19 188 182 -6

The scatter plot below shows the five points for these people together with a graph of the line Height = Arm Span. We draw such lines to explore potential models for describing relationship between two variables, such as height and arm span.

Problem C7

 a. Why is the point for Person 1 above the line Height = Arm Span? b. Why is the point for Person 9 on the line Height = Arm Span? c. Why is the point for Person 19 below the line Height = Arm Span? d. Why is it helpful to draw the line where Height = Arm Span? How does this line help us analyze differences?

 Problem C8 The points for Person 1 and Person 6 are both above the line. Why is the point for Person 1 farther away from the line? Person 1 has an arm span of 156 cm and a height of 162 cm, which is the coordinate point (156, 162) in the scatter plot. A hypothetical person represented by the coordinate point (156, 156) would be on the line Height = Arm Span, as shown below: The vertical distance from a point to the line is the absolute value of the difference in y-coordinates from the first point and the point on the line directly above (or below) that point. In this case, Person 1's point (156, 162) is six above the line's point (156, 156). Therefore, the vertical distance from Person 1's point to the line is 6. Put another way, the vertical distance from (156,162) to the line Height = Arm Span is the magnitude (or absolute value) of the difference between the height and the arm span. The vertical distance is: |Height - Arm Span| = |162 - 156| = |6| = 6. In a similar way, the vertical distance from the point for Person 6 (which is [161,162]) to the line Height = Arm Span is: |Height - Arm Span| = |162 - 161| = |1| = 1.

 Problem C9 The points for Persons 14 and 19 are both below the line Height = Arm Span. Determine the vertical distance from each of their points to the line. The following scatter plot shows four points corresponding to four new people and the graph of the line Height = Arm Span:

Problem C10

Consider the four points corresponding to Persons 2, 4, 7, and 23. Use the scatter plot to determine the following:

 a. Which of the four people have heights greater than their arm spans? b. Which of the four people have heights that are less than their arm spans? c. Which of the four has the greatest difference between height and arm span? d. Which of the four has the smallest difference between height and arm span?

 Answer questions (c) and (d) by comparing those points to the line Height = Arm Span.   Close Tip Answer questions (c) and (d) by comparing those points to the line Height = Arm Span.

 Here is the scatter plot of all 24 people and the graph of the line Height = Arm Span:

Problem C11