Unit 6, Lesson 7
The Correlation Coefficient
Integrated Math 1
7.1
Warm-up
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four scatter plots displaying data with linear and nonlinear trends. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the scatter plots for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their response with their group and then together find as many sets of three as they can.
Activity
NoneWhich three go together? Why do they go together?
AGraph of a scatter plot, origin O. Horizontal axis, distance (miles), scale 0 to 10, by 2’s. Vertical axis, cost (dollars), scale 0 to 22, by 2’s. The data has a visible pattern rising from near (point 5 comma 4) to (9 point 5 comma 21).
BGraph of a scatter plot, origin O, with grid. Horizontal axis, height (millimeters), scale 0 to 10, by 2’s. Vertical axis, weight (milligrams), scale 0 to 22, by 2’s. The data has no visible pattern.
CScatterplot, origin O. Horizontal, from 0 to 10, by 2’s, labeled precipitation, centimeters. Vertical, from 0 to 22, by 2’s, labeled water used for irrigation, thousands of gallons. 25 points represent the data, trend linearly downward and to the right. First data point approximately begins at 0 comma 10 and last is approximately at 7 comma 3 point 5.
DScatterplot, origin O. Horizontal, from 0 to 10, by 2’s, labeled temperature, degrees Celsius. Vertical, from 0 to 22, by 2’s, labeled number of phytoplankton, tens of thousands. 25 points that represent the data, trend slightly upward and to the right first and then very steep upward curve and to the right. First data point approximately begins at 0 comma 2 point 5 and last is approximately at 9 point 5 comma 21.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as "linear," "nonlinear," and "random." Also, press students on unsubstantiated claims.
Student Lesson Summary
While residuals can help pick the best-line to fit the data among all lines, we still need a way to determine the strength of a linear relationship. Scatter plots of data that are close to the best-fit line are better modeled by the line than are scatter plots of data that are farther from the line.
The correlation coefficient is a convenient number that can be used to describe the strength and direction of a linear relationship. Usually represented by the letter , the correlation coefficient can take values from -1 to 1. The sign of the correlation coefficient is the same as the sign of the slope for the best-fit line. The closer the correlation coefficient is to 0, the weaker the linear relationship. The closer the correlation coefficient is to 1 or -1, the better a linear model fits the data.
While it is possible to try to fit a linear model to any data, we should always look at the scatter plot to see if there is a possible linear trend. The correlation coefficient and residuals can also help determine whether the linear model makes sense to use to estimate the situation. In some cases, another type of function might be a better fit for the data, or the two variables we are examining may be uncorrelated, and we should look for connections using other variables.