To begin class, my students will complete the following online practice as a review of scatter plots. I plan for my students to spend five minutes completing the exercise. Then, we will spend five minutes reviewing what they learned. In particular, I will ask students to explain their reasoning. Much of our work today will require careful explanations.
It is usually the case that some of my students draw a blank when they try to recall this material. But, I want to communicate to them the fact that this content is review. Learning the vocabulary and using the different representations correctly are my expectations for what my students can do independently. I provide students with graph paper or an individual white board with a graph on it as a resource to use as they work on these two Warmup_Problems.
The price of gas compared to the price of milk is an example of correlation between variables when there is no causation.
In my course the students have learned how to write the equation of a line. We have also discussed the selection of two "good points" for determining a best fit line. During the discussion of this Warm Up, I also introduce the terms interpolate and extrapolate. Since we will be using scatter plots to make predictions, I want my students to begin learning how to talk about their predictions using precise language.
URL for Online Practice: http://www.regentsprep.org/regents/math/algebra/ad4/PracPlot.htm
(Last accessed 3-21-15)
Vocabulary early on in Algebra I can be essential to students for when they see r, r squared, and residuals in later math classes. Even though this is more of an introduction to these concepts, it is imperative that teachers try to use the correct vocabulary.
Teachers need to have a good grasp on these concepts before presenting it to students. This online blog shows some good examples for teachers to look at and work through. These concepts have been known to be difficult for teachers and students.
Teachers need to collaborate with AP Stat teachers as well for a stronger understanding. They will not teach as much as the AP Stat teachers on these concepts in Algebra I, but their presentations will be more clear and precise with a deeper understanding.
After reviewing the Warm Up with the students, I provide students with a Guided Practice. The purpose of this lesson is to take the students beyond the basics of Scatter Plots that were introduced in 8th grade. I use this Guided Practice to help students build their understanding of the following:
Once they are familiar with these terms, my students will be better able to explain their thinking when they interpret scatter plots of bivariate data. For today, students do not calculate by hand. Our efforts focus on developing intuitions and vocabulary for interpreting values taken from a graphing calculator. I want my students to be able to interpret these values meaningfully.
The Guided Practice includes two sets of bivariate data. I will work through one of the data sets with the students. Then, my students will work with the other data set on their own. Inmy demonstration I focus on the correlation coefficient (r) and the determination coefficient (r-squared). As we work the problem I encourage my students to contribute by asking them to explain concept from the Warmup: pattern of correlation, line of best fit, interpolation, extrapolation. I plan to introduce the following terms:
After creating a scatter plot and a line of best fit for the first data set, I will ask my students to draw a line that best represents the data in the second set. I expect that they will correctly draw a line in the direction of the points. I will remind them to try to keep about the same amount of points above and below the line. Then I will pose the question, "Why do we try to keep about the same amount of points above and below the line?" We will discuss this for a few minutes before I share the following video resources:
1. Correlation Coefficient (r)
URL: https://www.youtube.com/watch?v=ugd4k3dC_8Y
URL: https://www.youtube.com/watch?v=jEEJNz0RK4Q
3. Coefficient of Determination (r squared)
URL: http://www.statisticshowto.com/what-is-a-coefficient-of-determination/
After watching the videos I expect my students to understand that a strong correlation means that the r-value is close to one or negative one. In addition, the r-squared value helps to explain the percentage of the data points that fall on the regression line. Thus, this value is better if it is close to zero.
The final activity on the Guided Practice worksheet asks students to complete their approximations of r and r-squared and then compare it to the calculator. I will use a calculator to demonstrate using the speed to braking distance on dry pavement data. Then, I will have students complete the same process with the wet pavement data.
With about 10 minutes remaining in the period, I will hand my students an Exit Slip. The Exit Slip asks students to explain their results of the Wet Pavement Data set from the Guided Practice. I first want students to state the differences in their approximations for r and r-squared compared to the calculator. I ask them to explain the differences between the approximate and calculated values in their own words.
I also want my students argue whether or not a linear regression was an appropriate technique for modeling the data. Does the resulting model represent the data well. Here, I am expecting students to explain that the correlation coefficient r, should be close to positive one or negative one for a strong linear fit.
Finally, I would like students to square their r-value consider how much error results from using this model to represent the given data. At the end of the lesson, I want to remind students that the closer the r-squared statistic is to zero, the better the fit for the Least Regression Line.