9.1: Which One Doesn’t Belong: Correlations (5 minutes)
This warm-up prompts students to compare four data representations. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Arrange students in groups of 2–4. Display the pairs of variables for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.
Which one doesn’t belong?
A. the number of pictures painted and the amount of paint left in the paint can
B. amount of ice cream eaten the previous summer and number of movies seen this summer
C. distance run and number of water breaks during the run
D. number of people who contracted a genetic disease and presence of the gene that raises risk for the disease
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as positive and negative correlation. Also, press students on unsubstantiated claims.
9.2: Correlation Relationships (15 minutes)
The purpose of this activity is for students to begin to recognize that correlated variables can have different relationships. Students examine two pairs of variables that are similarly correlated, but with only one of the pairs having a causal relationship. This activity prepares students to recognize the difference between correlation and causal relationships in the associated Algebra 1 lesson.
Arrange students in groups of 2.
For each pair of graphs, the linear model fits the data about the same. What do you notice about the variables? How might the variables be related?
1a. the number of cows in some states and the number of chickens in those same states
1b. the number of cows in some states and the number of farms in those same states
2a. the worth of a person’s house and the worth of that same person’s car
2b. the worth of a person's car and their income
The purpose of this discussion is for students to consider the type of relationship between variables in a real situation. Select students to share their answers to the questions.
- "Are the number of cows and the number of chickens correlated? Explain how you know." (Yes. They are correlated because there is a trend in the data that is reasonably fit by a linear model.)
- "Do you think a scatter plot for the number of farms and the number of chickens would also be correlated? What about the year and average price of houses? Explain your reasoning." (Both cases should be correlated. It makes sense in terms of the situation that more farms should likely mean more chickens and as time passes, the price of homes rise. It also makes sense in terms of the correlation since farms are correlated with cows and cows are correlated with chickens.)
9.3: It Takes Two (20 minutes)
The mathematical purpose of this activity is for students to extend their thinking about relationships between variables to understand that they are not unique to causal relationships. Students identify variables that may be involved in a situation, then determine whether they are related and whether one causes the other to change. This activity prepares for the supported Algebra 1 lessons when students compare relationships and create their own situation to represent correlations and causal relationships. Students attend to precision (MP6) as they use precise language to describe the difference between correlations and causal relationships.
Mai is training for the upcoming track season by running 8 laps around the school track each morning before school. She records her time to complete the 8 laps and notices that she is finishing faster and faster as time goes on. She also notices that she feels better in the morning and her grades in her first class are improving as her times improve.
- In addition to the 2 listed, what other variables are changing in this situation?
- time to complete 8 laps
- number of mornings Mai has run 8 laps
- Select 3 pairs of variables from the list. For each pair determine if they are related, then decide whether you think one variable causes the other to change. Explain your reasoning.
The purpose of this discussion is for students to share their reasoning about correlation and causal relationships. Although students may not use the term causal relationship at this point, highlight any differences that arise when discussing their answers to each question. Select students to share their responses. Here are sample questions to promote class discussion:
- "Why is it important to consider variables that might be related to a situation?" (Some variables may be related, but it might be a third thing that causes the changes.)
- "What are some ways that variables A and B could be related?" (A could cause B to change, B could cause A to change, or another variable, C, could cause both A and B to change.)
- "Why might it be important to think about the variables to determine the relationship rather than just looking at a scatter plot of data?" (Some pairs of variables may be more directly related or even have one variable cause the other to change while others may be indirectly related.)
- "If two variables are correlated, will changing one of the variables change the other?" (Not necessarily. For example, from the last activity, having more cows will not make a farm get more chickens even though they are correlated.)