Lesson 9
Causal Relationships
Lesson Narrative
The mathematical purpose of this lesson is to understand that the relationship between variables can be, but is not always, a causal relationship. A causal relationship is one in which a change in one of the variables directly causes a change in the other variable. The work of this lesson connects to previous work because students interpreted the relationship between two variables using the correlation coefficient. The work of this lesson connects to upcoming work because students will analyze bivariate data and draw conclusions from their analysis.
When students determine that there is a causal relationship, they are attending to precision (MP6), because they are refining their language to be more precise. When students analyze the relationship between two variables to determine whether they are causal or not, they are modeling with mathematics (MP4).
Learning Goals
Teacher Facing
 Describe (orally and in writing) the strength and sign of the relationship between two variables.
 Investigate the relationship between two variables to analyze whether or not the relationship is causal.
Student Facing
 Let’s get a closer look at related variables.
Learning Targets
Student Facing
 I can look for connections between two variables to analyze whether or not there is a causal relationship.
CCSS Standards
Glossary Entries

causal relationship
A relationship is one in which a change in one of the variables causes a change in the other variable.

correlation coefficient
A number between 1 and 1 that describes the strength and direction of a linear association between two numerical variables. The sign of the correlation coefficient is the same as the sign of the slope of the best fit line. The closer the correlation coefficient is to 0, the weaker the linear relationship. When the correlation coefficient is closer to 1 or 1, the linear model fits the data better.
The first figure shows a correlation coefficient which is close to 1, the second a correlation coefficient which is positive but closer to 0, and the third a correlation coefficient which is close to 1.

negative relationship
A relationship between two numerical variables is negative if an increase in the data for one variable tends to be paired with a decrease in the data for the other variable.

positive relationship
A relationship between two numerical variables is positive if an increase in the data for one variable tends to be paired with an increase in the data for the other variable.

strong relationship
A relationship between two numerical variables is strong if the data is tightly clustered around the best fit line.

weak relationship
A relationship between two numerical variables is weak if the data is loosely spread around the best fit line.
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