# Lesson 9

Causal Relationships

## 9.1: Used Car Relationships (5 minutes)

### Warm-up

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.

### Launch

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.

### Student Facing

Describe the strength and sign of the relationship you expect for each pair of variables. Explain your reasoning.

1. Used car price and original sale price of the car.
2. Used car price and number of cup holders in the car.
3. Used car price and number of oil changes the car has had.
4. Used car price and number of miles the car has been driven.

### Activity Synthesis

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.)

## 9.2: Cause or Effect? (15 minutes)

### Activity

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).

### Launch

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.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. After the first question invite students to suggest language or diagrams to include that will support their understanding of causal relationships. Examples to include can be phrases, such as, “increased rain causes people to wear jackets.” The two variables can be color-coded in two different colors to reveal the relationship. Both the axes and the labels can be highlighted with the color that connects it to the type of variable it addresses.
Supports accessibility for: Conceptual processing; Language

### Student Facing

Each of the scatter plots show a strong relationship. Write a sentence or two describing how you think the variables are related.

1. During the month of April, Elena keeps track of the number of inches of rain recorded for the day and the percentage of people who come to school with rain jackets.

2. A school book club has a list of 100 books for its members to read. They keep track of the number of pages in the books the members read from the list and the amount of time it took to read the book.

3. Number of tickets left for holiday parties at a venue and noise level at the party.

4. The height and score on a test of vocabulary for several children ages 6 to 13.

### Activity Synthesis

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.)

## 9.3: Find Your Cause (10 minutes)

### Activity

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.

### Launch

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.

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their writing by providing them with multiple opportunities to clarify their ideas through conversation. Give students time to meet with 2–3 partners to share their response to the last question. Students should first check to see if they agree with the experiment to show a correlation or causation. Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner: “Why do you think these are not causal?” or “How would the data show if the relationship was correlation or causal?” Next, provide students with 3–4 minutes to revise their initial draft based on feedback from their peers. This will help students understand relationships that may or may not be causal.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. Prior to directing student attention to the task, invite students to brainstorm various pairs of variables (both with correlation and without). Encourage students to free-associate and build off of each other’s ideas. When students share their work with a partner, display sentence frames to support conversation such as: “_____ reminds me of _____ because . . .” and “_____’s idea reminds me of . . .” After students have developed a list of several options, share the conditions list and then encourage them to analyze and select which one best fits each condition.
Supports accessibility for: Language; Social-emotional skills

### Student Facing

Describe a pair of variables with each condition. Explain your reasoning.

1. Two variables with a causal relationship.
2. The variables are strongly related, but a third factor might be the cause for the changes in the variables.
3. The variables are only weakly related.

### Student Facing

#### Are you ready for more?

1. Look through news articles or advertisement for claims of causation or correlation. Find 2 or 3 claims and read or watch the articles or the advertisement. Answer these questions for each of the claims.

1. What is the claim?

2. What evidence is provided for the claim?

3. Does there appear to be evidence for causation or correlation? Explain your thinking.

2. Choose the claim with the least or no evidence. Describe an experiment or other way that you could collect data to show correlation or causation.

### Anticipated Misconceptions

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.

### Activity Synthesis

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.)

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion.

• “How can you determine if there is a causal relationship between two variables? Explain your reasoning.” (To determine a causal relationship, you need to think about the context and determine if a change in one variable causes the other variable to change. One way to help determine whether there is a causal relationship is to design an experiment that controls one of the variables.)
• “Mai states that the relationship between the number of miles driven in a taxi and the price of the taxi ride is a causal relationship. Do you agree with Mai? What other information would help to further convince you one way or the other?” (I agree with her. It makes sense that the farther you go in a taxi, the more you will be charged. It might help convince me to take several taxi rides of the same length, but with different starting points or destinations and see if the cost is the same.)
• “Jada states that the relationship between the size of a pasture and the number of cows kept at various farms is a causal relationship. Do you agree with Jada? Explain your reasoning.” (I do not agree. The increase in pasture size does not cause an increase in the number of cows [nor does an increase in number of cows make the pasture larger]. The increase in land might mean there is more food for the cows, but it is the farmer who decides how many cows there are on a farm.)

## Student Lesson Summary

### Student Facing

Humans are wired to look for connections and then use those connections to learn about the world around them. One way to notice connections is by looking for a pair of variables with a relationship. In order to learn about how the variables are related, we want to control one of the variables and see if there are changes in the other variable. For example, if we notice that people who tend to eat many calories also have a higher chance of having a heart attack, we might wonder if lowering our calorie intake would improve our health.

One common mistake people tend to make using statistics is to think that all relationships between variables are causal. Scatter plots can only show a relationship between the two variables. To determine if change in one of the variables actually causes a change in the other variable, or has a causal relationship, the context must be better understood and other options ruled out.

For example, we might expect to see a strong, positive relationship between the number of snowboard rentals and sales of hot chocolate during the months of September through January. This does not mean that an increase in snowboard rentals causes people to purchase more hot chocolate. Nor does it mean that increased sales of hot chocolate cause people to rent snowboards more. More likely there is a third variable, such as colder weather, that might be causing both variables to increase at the same time.

On the other hand, sometimes there is a causal relationship. A strong, positive relationship between hot chocolate sales and small marshmallow sales may be linked, because people buying hot chocolate may want to add small marshmallows to the drink, so an increase in the sales of hot chocolate are actually causing the marshmallow sale increase.

Finding relationships with the help of the correlation coefficient is a very good way to notice that there is a connection between variables. To determine whether the relationship is causal, the next step is usually to carefully design an experiment that isolates and precisely controls only one of the variables to determine how it affects the other variable.