Lesson 7

This warm-up prompts students to compare four scatter plots displaying data with linear and nonlinear trends. It gives students a reason to use language precisely (MP6) and gives you the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another. To allow all students to access the activity, each item has one obvious reason it does not belong. Encourage students to move past the obvious reasons and find reasons based on mathematical properties.

Arrange students in groups of 2–4. Display the scatter plots for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reasoning as to why a particular item does not belong and together, find at least one reason each item doesn’t belong.

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 asking 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 linear, nonlinear, and random. Also, press students on unsubstantiated claims.

In this activity, students are given cards displaying scatter plots of data that can be fit by linear models with varying accuracy. Cards show data that is random, poorly fit by a linear model, well fit by a linear model, and data that is better fit by another type of function, such as quadratic or exponential. Students should begin to recognize these differences and the connection to the correlation coefficient.

A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Monitor for different ways groups choose to categorize the scatter plots, but especially for categories that distinguish between plots that would be modeled well with a linear function and those that would not.

Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the scatter plots, they should work with their partner to come up with categories. Distribute one copy of the blackline master to each group.

Select groups of students to share their categories and how they sorted their equations. You can choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between plots that would be modeled well with a linear function and those that would not. Attend to the language that students use to describe their categories and equations, giving them opportunities to describe their equations more precisely. Highlight the use of terms like linear model, fit, and non-linear.

Display the scatter plots with the best fit lines and \(r\) values.

A: \(y = \text{-}0.83x-1.85, r = \text{-}0.61\)

B: \(y = \text{-}0.07x + 5.63, r = \text{-}0.06\)

C: \(y = \text{-}0.31x+5.76, r = \text{-}0.88\)

D: \(y = \text{-}0.8x+6, r = \text{-}1\)

E: \(y=0.69x+4.49, r = 0.46\)

F: \(y = 1.43x + 2.86\), \(r = 0.94\)

G: \(y= 3x + 2, r = 1\)

H: \(y = \text{-}0.02x+1.64, r = \text{-}0.13\)

I: \(y = 14.55x-36.18, r = 0.95\)

J: \(y = \text{-}0.62x + 6.52, r = \text{-}0.96\)

Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Among things students should notice are:

1. the sign of \(r\) is the same as the sign of the slope of the best fit line
2. the values for \(r\) seem to go from -1 to 1
3. the closer \(r\) is to 1 or -1, the stronger the linear relationship between the variables
4. the closer \(r\) is to 0, the weaker the linear relationship between the variables

Note that the sign of the correlation coefficient matches the sign of the slope of the best fit line, but the value for \(r\) is not otherwise related to the slope. If \(r = 0.8\), the best fit line will have a positive slope, but whether the slope is 0.2 or 2,000 is not clear without examining the data.

In this activity, students gain a better understanding of correlation coefficients by taking turns with a partner to match scatter plots and correlation coefficients. Students trade roles, explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

Tell students that the \(r\) value is called a correlation coefficient. A correlation coefficient is one way to measure the strength of a linear relationship. Tell students that:

1. the sign of \(r\) is the same as the sign of the slope of the best fit line
2. the values for \(r\) go from -1 to 1, inclusive
3. the closer \(r\) is to 1 or -1, the stronger the linear relationship between the variables
4. the closer \(r\) is to 0, the weaker the linear relationship between the variables

Arrange students in groups of 2. Tell students that for each scatter plot, one partner finds the associated correlation coefficient and explains why they think it goes with that scatter plot. The other partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next scatter plot, the students swap roles. If necessary, demonstrate this protocol before students start working.

Students may struggle with starting to match the scatter plots with a correlation coefficient. Guide students by asking them about the sign of the correlation coefficients. Ask them to sort the cards into groups that make sense and use that to connect to the correlation coefficient values. Ask them: “How does the sign of the correlation coefficient relate to the linear model?”

The purpose of this discussion is for students to understand that the correlation coefficient is a formal way to quantify the strength of a linear relationship between variables, and that the sign of the correlation coefficient tells you whether or not the variables show a positive or negative association.

Here are some questions for discussion.

• “What does the sign of the correlation coefficient tell you about the data?” (If it is negative, then \(y\) tends to decrease as \(x\) increases. If it is positive, then \(y\) tends to increase as \(x\) increases.)
• “What does it mean to have a correlation coefficient of 1 or -1?” (It means that the data is perfectly linear and is fit exactly by a linear function.)