The Correlation Coefficient

5 min

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.

Student Task

Which three go together? Why do they go together?

A

<p>Graph of a scatter plot, origin O. distance (miles) and cost (dollars).</p>
Graph 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).

B

<p>Graph of a scatter plot, origin O, with grid. Height (millimeters) and weight (milligrams).</p>
Graph 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.

C

<p>A scatterplot.</p>
Scatterplot, 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.

D

<p>A scatterplot.</p>
Scatterplot, 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.

Sample Response

Sample responses:

A, B, and C go together because the xx-axes are all measured in units of length.

A, B, and D go together because the trends in the data appear to be increasing.

A, C, and D go together because it seems they will be fit well by a model function.

B, C, and D go together because they are not fit perfectly by a linear model.

Activity Synthesis (Teacher Notes)

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.

Standards
Addressing
  • HSS-ID.B.6·Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
  • S-ID.6·Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
  • S-ID.6·Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
  • S-ID.6·Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
Building Toward
  • HSS-ID.C.8·Compute (using technology) and interpret the correlation coefficient of a linear fit.
  • S-ID.8·Compute (using technology) and interpret the correlation coefficient of a linear fit.
  • S-ID.8·Compute (using technology) and interpret the correlation coefficient of a linear fit.
  • S-ID.8·Compute (using technology) and interpret the correlation coefficient of a linear fit.

20 min

10 min