The mathematical purpose of this activity is for students to describe how two variables are related, and to determine whether or not there is a causal relationship. Students should begin to recognize that while some variables may be related, one does not cause the other to change. At this point, the mathematics of scatter plot analysis cannot lead to a conclusion of whether there is a causal relationship. The relationship must be thought through carefully before a conclusion is made about whether the related variables in a situation have a causal relationship. As students analyze the relationships, they are modeling with mathematics (MP4).

The goal is for students to develop an understanding of what it means for a relationship between two variables to be causal, a term that will be introduced more formally in the next activity.

Select several students to share their reasoning for the relationship between the variables. For each pair of variables, ask students what might have caused the variables to be linked. In some cases (the first two here), one of the variables causes a change in the other. In other cases (the last two here), an additional variable or situation is the cause of the change.

Tell students that, since most people are used to seeing independent variables on the -axis and dependent variables on the -axis, the convention is to put the variable that caused the change on the -axis and the other variable on the -axis. The book club scatter plot should probably have the axes switched to meet the convention.

Explain that to truly determine causality, a careful experiment should be arranged to ensure other factors are not the source of change. Without those additional experiments, a more precise description would be to say that the amount of rain is an influential variable on jacket wearing.

Here are some questions for discussion.

• “Why does an increase in precipitation cause an increase in the percentage of people wearing rain jackets?” (When it rains, people usually wear coats to stay dry.)
• “An increase in the time it takes to read a book does not cause the number of pages in the book to increase. Does an increase in the number of pages in a book cause the time it takes to read a book to increase? Explain your reasoning.” (Yes, if you switch the axes on the graph, you can see that relationship. It is causal because it takes longer to read more pages.)
• “The relationship shown in the graph of the height of a pine tree and the number of tree rings appears to be strong. Does an increase in height cause an increase in the number of tree rings?” (It is not the height that causes the increase in rings, but probably the age of the tree that causes the increase. Age is positively related to height, so that likely explains the relationship seen between the two variables.)

The mathematical purpose of this activity is for students to practice thinking about the idea of a causal relationship. They think of situations in which the relationship is causal and those in which it is not.

This activity is a good chance to connect mathematical learning with learning happening in other subjects. Collaborate with other subject teachers to collect categories or quantities that could be related either causally or not.

Students may still wonder how two variables can be correlated without having a causal relationship. It may help to provide an example, such as sales of ice cream or sales of sunburn remedies. Ask students why these variables might be related and whether increasing one would cause the other variable to increase. Ask students to think of something that might cause two distinct outcomes to result.

The goal of this discussion is for students to gain a deeper understanding of what it means for variables to have a causal relationship.

Here are some questions for discussion.

• “Which pair of variables was the most difficult for you to describe? Explain your reasoning.” (I had a hard time describing a pair of variables that were strongly related but where a third factor might be the cause. It was difficult because I kept coming up with causal relationships.)
• “For question 1, how did you convince yourself or your group that one variable causes a change in the other?” (To convince them, I gave an example of using the variables in context. My two variables were snail weight and shell volume. When a snail has more weight, it increases in volume, which means it needs a bigger shell.)