In this activity, students are given cards displaying scatter plots of data that can be fit by linear models with varying accuracy. Cards show data that is random, poorly fit by a linear model, well fit by a linear model, and better fit by another type of function, such as quadratic or exponential. Students should begin to recognize these differences and the connection to the correlation coefficient.

Students sort different scatter plots during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Monitor for different ways groups choose to categorize the scatter plots, but especially for categories that distinguish between plots that would be modeled well with a linear function and those that would not. 

Select groups of students to share their categories and how they sorted their scatter plots. Discuss as many different types of categories as time allows, but ensure that one set of categories distinguishes between plots that would be modeled well with a linear function and those that would not. Attend to the language that students use to describe their categories and scatter plots, giving them opportunities to describe their scatter plots more precisely. Highlight the use of terms like "linear model," "fit," and "nonlinear."

Display the scatter plots with the best-fit lines and -values.

Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice, then follow with a whole-class discussion.

Among things students should notice are:

• The sign of is the same as the sign of the slope of the best-fit line.
• The values for seem to go from -1 to 1.
• The closer is to 1 or -1, the stronger the linear relationship between the variables.
• The closer is to 0, the weaker the linear relationship between the variables.

Note that the sign of the correlation coefficient matches the sign of the slope of the best-fit line, but the value for is not otherwise related to the slope. If , the best-fit line will have a positive slope, but whether the slope is 0.2 or 2,000 is not clear without examining the data.

Math Community

Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm '' in action today and it really helped me/my group because .”

In this activity, students gain a better understanding of correlation coefficients by taking turns with a partner to match scatter plots and correlation coefficients. Students trade roles, explaining their thinking and listening, which provides opportunities for them to explain their reasoning and critique the reasoning of others (MP3).

Students may struggle with starting to match the scatter plots with a correlation coefficient. Guide students by asking them about the sign of the correlation coefficients. Ask them to sort the cards into groups that make sense and use those to make a connection to the correlation coefficient values. Ask them: “How does the sign of the correlation coefficient relate to the linear model?”

The purpose of this discussion is for students to understand that the correlation coefficient is a formal way to quantify the strength of a linear relationship between variables and that the sign of the correlation coefficient tells us whether or not the variables show a positive or negative association.

Here are some questions for discussion.

• “What does the sign of the correlation coefficient tell you about the data?” (If it is negative, then tends to decrease as increases. If it is positive, then tends to increase as increases.)
• “What does it mean to have a correlation coefficient of 1 or -1?” (It means that the data is perfectly linear and is fit exactly by a linear function.)