# Crazy Correlations

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## Objective

SWBAT evaluate the correlation between linear data sets manually and using a graphing calculator.

#### Big Idea

Is this relationship the real thing? Just how positive are you? Use a correlation coefficient to find out!

5 minutes

## Put it into Action

40 minutes

You may be thinking that this lesson belongs with Algebra I because it addresses Common Core Standard S.ID.8 "Compute (using technology) and interpret the correlation coefficient of a linear fit."  I believe that this lesson belongs in Algebra II because it requires students to explore the mathematics of correlation directly rather than just using a calculator or computer to generate a value.  This allows me to build toward an understanding of statistical concepts such as z-scores and and the student t-test.

Student Measurements
10 min: For this section your students will need meter sticks/yard sticks, rulers, a length of string at least 20 cm long, and copies of the student data chart. I have seen this kind of activity done with flexible measuring tapes, but feel that those contribute too much uncertainty to the measurements as students bend them to fit contours.  I tell my students that they will be working in teams of two for this activity.  See my strategies folder for assigning partners.  I say that they will taking body measurements and emphasize that they need to measure very accurately to get the best results. (MP6) I say that the goal of this activity is to find which measurement is the best predictor of height, that is which one has the best correlation.  I give each student a copy of the student data chart an assign each of them a number corresponding to a line on the chart.  I then allow them to choose a partner to work with in collecting the measurements because for some students this can be a bit uncomfortable.  As they begin working I walk around offering encouragement and clarification as needed. After everyone is done I ask my students to post their individual measurements in the chart I've projected on the board you can just draw it if that's easier, on the appropriate line, then copy all the class data onto their own chart.

Class Discussion 15 min: For this section you may want to have your students work with a
statistics program or graphing calculator, but it can also be done fairly well with just paper and pencil. You will also need copies of the Correlation Activity Sheet.
When all the data is posted and copied I tell my students that each team will get to compare one of the measurements to height to determine what, if any correlation exists.  I hand out copies of the Correlation Activity Sheet. I walk them through the simple example of finding the correlation coefficient on their worksheet then assign each team one measurement to use with height, asking whether they think height should be the x or y variable.  Since none of these measurements is "dependent" upon height and vice versa, this question is not as simple as it seems.  My students often suggest height should be the x variable because it's the one they'll all be working with.  I ask questions like "What are we hoping to be able to predict from our measurements?"  I keep asking questions until someone suggests that because we want to predict height, we should have it be the y variable.  I do the walkthrough because most of my students either have never seen or don't recall working with correlation as something to be calculated, they remember general trends of positive, negative or none.  By reviewing the calculations, I'm trying to reduce student frustration with the process so we can get to the comparisons...and it also reduces my frustrations about having to review the same thing with several teams instead of doing it once for the class!  (MP2)

Teamwork 15 min:
Once that's settled, I tell them they have about 15 minutes to complete their calculations. (MP1)  Some teams will need more support than others, so for those that are finished early, I suggest that they try to come up with a line-of-best-fit for their data.  Toward the end of this work period I put a table on the board for students to post their results to. When all the teams are done I ask them to post their correlations on the front board.

## Wrap it Up

10 minutes

When all the results are posted I tell my students they will be working independently for this final piece of the lesson.  I ask them to consider which of the measurements seems to be the best predictor of height and why.  Sometimes there are two very similar r-values which can lead to a good discussion about what the correlation between these two measurements might be.  I give each student a notecard and tell them to state which measurement they chose as the best predictor of height and answer the following questions (MP4, MP3):

• Why do you think the measurement is the best (support your response mathematically)
• What additional work needs to be done to create a model to predict height using your chosen measurement?

The first question gives them an opportunity to cement their understanding of the mathematics behind correlation.  The second question segues into modeling with regression equations.