Lesson 8

The mathematical purpose of this activity is for students to match bivariate data with its context. Students should think about whether they might expect a strong correlation or not as well as whether the relationship has a positive or negative correlation. Monitor for students who discuss linear relationships or variability in the data.

Arrange students in groups of two to four. Provide access to the scatter plots and contexts for the warm-up.

Students may struggle with matching the pairs of variables with a scatter plot. Encourage students to think about how related the variables are and how the \(y\) variable might change as the \(x\) variable increases.

The goal of this discussion is for students to discuss how the characteristics of the scatter plots allowed them to determine the context. For each context, select groups to share their match and reasoning. Select groups who used linear model and variability in their small group discussions.

Here are some questions for discussion.

• “How did the concept of linear relationships help you to make a match?” (For the height in inches and the height in feet, I knew that the data would be very linear. I chose scatter plot A because it was the only one that appeared linear. I did wonder why scatter plot A was not perfectly linear. Maybe it had to do with rounding or measurement error.)
• “How did you use the concept of linearity to help you to make a match?” (First, I expected the temperature and cereal context relationship to be totally random rather than linear at all, so that fit with scatter plot C. Second, I knew that problem 3 matched with scatter plot A, because I knew it should be almost perfectly linear. For a given height in feet, there is only one height in inches.)
• “Which matches were the most difficult? What helped you figure them out?” (It was hardest to find the matches for scatter plots B and D because they were pretty scattered. I noticed that the data in scatter plot D displayed a negative relationship, so I looked for a context that would have a negative relationship. It makes sense that customer satisfaction decreases as wait time increases.)

The mathematical purpose of this activity is for students to use technology to find a correlation coefficient and use it to interpret the strength of the linear relationship in context. In addition, students interpret the coefficients of the equation of the line of best fit and use the equation to make predictions. As students examine the situation and describe variables that could influence the results, they are modeling with mathematics (MP4).

Students may misunderstand how to interpret negative correlation coefficients. Ask students how the sign of the correlation coefficient is related to the linear model for the situation. Students may benefit from drawing an example scatter plot representing the situation.

The purpose of this discussion is for students to understand that the correlation coefficient quantifies the strength of a linear relationship.

Introduce the terms positive and negative relationship. Give some guidelines as to when to call a relationship strong or not. For example: When \(|r| \geq 0.8\), there is a strong, linear relationship, when \(|r| \leq 0.5\) the relationship is weak, and when \(0.5 < |r| < 0.8\), it is moderately strong. Although these are good guidelines, they should not be treated as a rule. Context is also important when determining whether to call a relationship strong or weak.

In general, \(r\) is related to how much of an improvement a linear model is over just using the mean as an estimate for the data.

Here are some questions for discussion.

• “How would you describe the relationship between distance and time for Priya’s trips? Explain your reasoning.” (It is a strong positive relationship. It is strong because \(r=0.99\), and it is positive because time, \(y\), tends to increase as distance, \(x\), increases.)
• Refer students to scatter plot D from the warm-up. “How would you describe the relationship between wait time and customer satisfaction? The correlation coefficient is \(r=\text{-}0.52\).” (It is a negative relationship. It is a moderately strong relationship, since the linear model is an improvement over merely using the mean for the data, but it is not a great improvement.)

• “What is an example of data with a strong, negative relationship?” (An example of a strong negative relationship could be between the time it takes you to run a mile and your speed. As your speed increases, the time decreases.)

The mathematical purpose of this activity is to use the correlation coefficient to describe the relationship between two variables. Students examine a pair of variables and a correlation coefficient to describe the relationship between the variables as strong or weak and as positive or negative. Students must reason abstractly and quantitatively (MP2) when they interpret the situation to describe the relationship.

Arrange students in groups of 2. Give students an example of the type of response expected by telling students: “The cost of a package of light bulbs and the number of light bulbs in the package has a correlation coefficient value near 1. This means that these variables have a very strong, positive relationship. In other words, the price of the package is very closely related to the number of light bulbs in the package (strong relationship) and when one of the variables goes up, the other variable tends to go up, too (positive relationship).”

The purpose of this discussion is for students to interpret the data based on the relationship between the two variables that they determined using the correlation coefficient.

Ask:

• “Do these correlations make sense based on what you understand about these variables?” (Yes, mostly. Walking more, for example, usually means you take more steps.)
• “Based on the information here, does a greater fat intake cause cancer?” (Not necessarily. There seems to be a link, but there are many other factors about countries with greater fat intake that might play a part.) Causal relationships is the focus of the next lesson, so it is okay for students to struggle with this question now.
• “What does it mean for the relationship between the score on a science exam and the number of words written to be weak and positive?” (It means that, in general, the exam score tended to increase as the number of words written increased, but the relationship between these two variables was weak. This makes sense, because if you wrote very few words, you probably would not get a good score, but writing lots of words does not necessarily guarantee you a good score.)