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Unit 6, Lesson 8

Using the Correlation Coefficient

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Integrated Math 1

8.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

The mathematical purpose of this activity is for students to match bivariate data with its context. Students should think about whether they expect a strong correlation and whether the relationship has a positive or negative correlation. Monitor for students who discuss linear relationships or variability in the data.

Launch

Arrange students in groups of 2 to 4. For the Warm-up, provide access to the scatter plots and contexts.

Activity

None

Match the variables to the scatter plot you think they best fit. Be prepared to explain your reasoning.

	-variable	-variable
1.	low temperature in Celsius for Denver, CO, on a given day	boxes of cereal in stock at a grocery store in Miami, FL, on a given day
2.	number of free throws shot in a game	basketball team score in a game
3.	measured student height in feet	measured student height in inches
4.	number of minutes spent in a waiting room	hospital satisfaction rating given by patient

Student Response

to access Student Response.

Building on Student Thinking

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 -variable may change as the -variable increases.

Activity Synthesis

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 models 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, so that led me to choose scatter plot C. Second, I knew that the third set of variables 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.)

8.2

Activity

Instructional Routines

Aspects of Mathematical Modeling Fit It MLR8: Discussion Supports

Materials

To Gather

Graphing technology

Activity Narrative

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. A strong relationship is a relationship between two numerical variables that is represented very well by a linear model. A weak relationship is not represented very well by a linear model.

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).

8.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

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.

Launch

Arrange students in groups of 2. Give students an example of the type of response expected by telling them, “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).”

Representation: Access for Perception. Provide students with a physical copy of the example of the type of response from the Launch. Encourage students to highlight or annotate to anchor understanding. Check for understanding by inviting students to rephrase in their own words. Consider keeping the display of the example visible throughout the activity.
Supports accessibility for: Language, Memory

Lesson Synthesis

Here are some questions for discussion.

“What does it mean for two variables to have a weak, positive relationship?” (When one of the variables increases, the other variable also tends to increase, but since the linear relationship is weak, the data does not follow a linear path.)
“If the -value for a line of best fit in a scatter plot is 0.8, what would you expect the data represented by the scatter plot to look like?” (I would expect the scatter plot to show the data generally increasing from left to right and the data to look somewhat like a line, but not perfectly like a line.)
“What does the correlation coefficient tell you about the relationship between two variables?” (It tells me if the two variables are positively or negatively related. It also tells me how strong the relationship between the two variables is.)

to access Cool-down.

Student Lesson Summary

The value for the correlation coefficient can be used to determine the strength of the relationship between the two variables represented in the data.

In general, when the variables increase together, we can say they have a positive relationship. If an increase in one variable’s data tends to be paired with a decrease in the other variable’s data, the variables have a negative relationship. When the data is tightly clustered around the best fit line, we say there is a strong relationship. When the data is loosely spread around the best fit line, we say there is a weak relationship.

A correlation coefficient with a value near 1 suggests a strong, positive relationship between the variables. This means that most of the data tends to be tightly clustered around a line and that when one of the variables increases in value, the other does as well. The number of schools in a community and the population of the community is an example of variables that have a strong, positive correlation. When there is a large population, there is usually a large number of schools, and small communities tend to have fewer schools, so the correlation is positive. These variables are closely tied together, so the correlation is strong.

Similarly, a correlation coefficient near -1 suggests a strong, negative relationship between the variables. Again, most of the data tend to be tightly clustered around a line, but now, when one value increases, the other decreases. The variables time since leaving home and the distance left to reach school have a strong, negative correlation. As the travel time increases, the distance to school tends to decrease, so this is a negative correlation. The variables are again closely, linearly related, so this is a strong correlation.

Weaker correlations mean there may be other reasons the data is variable other than the connection between the two variables. For example, variables number of pets and number of siblings have a weak correlation. There may be some relationship, but there are many factors that account for the variability in the number of pets other than the number of siblings.

The context of the situation should be considered when determining whether the correlation value is strong or weak. In physics, where things are measured with precise instruments, a correlation coefficient of 0.8 may not be considered strong. In social sciences, where data is collected through surveys, a correlation coefficient of 0.8 may be very strong.

Launch

Show students how to find a correlation coefficient using technology. The digital version of this activity includes instructions for finding the correlation coefficient using Desmos. If they will be using technology other than Desmos (available in Math Tools), alternate instructions may need to be prepared. Here is the data from scatter plot A in the Warm-up to use in the demonstration.

height in feet ()	height in inches ()
5.5	66
5.25	63
5	60
5.5	66
6	72
5.8	70
5.9	71
6.25	75
5.4	65
5.1	61
4.9	59
5.7	68
5.75	69
5.9	71
5.5	66

The equation of the line of best fit is , and the correlation coefficient is 0.999. Since there is a direct conversion between the units, we expect there to be a strong, positive relationship between the variables.

Although it is not perfect (, not 1) due to rounding, the best-fit line models the data very well, so there is a strong relationship between height in feet and height in inches. Since an increase in one of the values is paired with an increase in the other, there is a positive relationship.

Give students 10 minutes of work time to answer the questions.

Activity Synthesis

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 relationship and negative relationship to describe whether one variable increases or decreases as the other variable increases. Also define strong relationship and weak relationship as a description of whether a linear model can represent the data well or not.

Give some guidelines as to when to call a relationship strong or not. For example, when , there is a strong, linear relationship, when , the relationship is weak, and when , 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, 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 , and it is positive because time, , tends to increase as distance, , 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 .” (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 someone to run a mile and their speed. As their speed increases, the time it takes decreases.)

MLR8 Discussion Supports. Invite students to repeat their reasoning using mathematical language: “Can you say that again but using the word 'strong' or 'weak'?”
Advances: Conversing, Representing

Priya takes note of the distance the car is driven and the time it takes to get to the destination for many trips.

distance (mi) ()	travel time (min) ()
2	4
5	7
10	11
10	15
12	16
15	22
20	23
25	25
26	28
30	36
32	35
40	37
50	51
65	70
78	72

Distance is one factor that influences the travel time of Priya’s car trips. What are some other factors?
Which of these factors (including distance) most likely has the most consistent influence for all the car trips? Explain your reasoning.
Use technology to create a scatter plot of the data and add the best fit line to the graph.
What do the slope and -intercept for the line of best fit mean in this situation?
Use technology to find the correlation coefficient for this data. Based on the value, how would you describe the strength of the linear relationship?
How long do you think it will take Priya to make a trip of 90 miles if the linear relationship continues? If she drives 90 miles, do you think the prediction you made will be close to the actual value? Explain your reasoning.

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Lesson 8

Using the Correlation Coefficient

Warm-up

Standards Alignment

Building On

Addressing

S-ID.6

Building Toward

S-ID.7

S-ID.8

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Launch

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

S-ID.6.a

S-ID.7

S-ID.8

Building Toward

S-ID.9

Standards Alignment

Building On

Addressing

S-ID.8

Building Toward

S-ID.9

Activity

Student Response

Building on Student Thinking