Lesson 9

The mathematical purpose of this activity is for students to describe the relationship between variables using mathematical terminology, such as strong or weak relationship and positive or negative relationship. These terms were defined in the previous lesson. Students must reason abstractly and quantitatively (MP2) to determine the type of relationship.

Arrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

To help students understand some of the context, explain that, for many modern cars, it is recommended that the oil be changed every 5,000 miles driven or every 5 months.

The purpose of this discussion is for students to discuss the strength of relationships in preparation for having students distinguish between causal relationships and statistical relationships.

Encourage the use of the terms strong or weak relationship and positive or negative relationship in the discussion.

The discussion should focus on the reasoning for the last 2 problems and how they are similar and different. Here are some questions for discussion.

• “How is the relationship between the car price and number of oil changes similar to the relationship between the car price and number of miles driven?” (They both have a strong, negative relationship.)
• “How is the relationship between the car price and number of oil changes different from the relationship between the car price and number of miles driven?” (The number of miles the car has been driven seems more directly related to the price than the number of oil changes. It probably has a stronger negative relationship than the number of oil changes.)

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 some variables may be related, but one does not cause the other to change. At this point, the mathematics of scatter plot analysis cannot decide whether there is a causal relationship. The relationship must be thought through carefully to decide, based on the situation, whether the related variables have a causal relationship. As students analyze the relationships, they are modeling with mathematics (MP4).

Arrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

The goal is for students to develop an understanding of what it means for a relationship between two variables to be a causal relationship.

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 \(x\)-axis and dependent variables on the \(y\)-axis, the convention is to put the causal variable on the \(x\)-axis and the other variable on the \(y\)-axis when one of the variables does cause the other to change. The book club scatter plot should probably have the axes switched to meet the convention.

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 in the height and test score graphs appears to be strong. Does an increase in height cause an increase in test scores?” (It is not the height that causes the increase in test score, but probably the age or grade level of the students might be causing the increase. Age and grade level are 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 using the term causal relationship and think of situations to which it might apply and where relationships are not causal.

Tell students that “people often use the phrase ‘correlation, not causation’ or ‘association, not causation’ to refer to these situations in which there is a relationship, but it is not a causal relationship. A causal relationship means that a change in one of the variables actually causes a change in the other variable ”

Arrange students in groups of 2–4. Give students several minutes to record answers to the questions individually and then ask students to share their answers with each other and determine if the answer is correct.

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 and 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, then it increases in volume, which means it needs a bigger shell.)