I began this lesson with what I thought would be a review of correlation from previous courses, in fact I was concerned this lesson might be too basic to be included in Algebra II because of the review. What I found was that my students had only a very basic understanding of correlation as a way of describing the relationship between two sets of data. They had never used a calculator to find correlation, nor did they really understand what negative, positive or no correlation meant in terms of the actual data. We talked as a class about the meaning of correlation and they became more engaged when I challenged them to show how strong (or weak) a specific correlation was. One of the examples I use is the rates different people pay for car insurance based on correlations between age or gender and number of accidents. I explained that there are actually a lot of other factors involved, but that ultimately the variation in rates reflects a prediction about the chance of an accident.
I believe this truly is an Algebra II lesson if your students do the calculations by hand and your class discusses not only the process of finding correlation but also the value of its application.
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.
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.
The most surprising thing about this part of the lesson was how poorly my students followed the directions for taking measurements. I actually tried the directions with two other students during a free period to trouble-shoot any confusing sections so I thought I had them well-written, but every team had to make at least one correction because they hadn't done the measurement correctly. My observation about this is that it isn't the directions per se that are the problem, it's that most of the students only read the headings and didn't bother with the specifics about how to make the measurement! I'm not sure how to "fix" this for the future other than building in a stronger accountability and student ownership piece early in the year and continuing to reinforce these practices.
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):
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.
Although we had to extend this lesson into a second class period, the questions remained the same. The most difficult part for my students was in making the connection between which measurement was the best predictor and using that data to create an equation to generalize about height. A few students were stumped at where to begin so I asked them questions like "what kind of function does your data look like when it's graphed?" and "Can you fit a straight line or curve through your graph?" The interesting questions came from a team who were adamant that there was no function to fit all the data. I agreed with them and then redirected them to creating an equation that "best fit" the data. They remained unconvinced that such an equation had any value and/or asked how they could tell which one was best. This continuing discussion helped me to understand the areas my students struggle with conceptually.