Lesson 1 of 8
Objective: SWBAT represent bivariate data using a scatter plot and determine a line of best fit that models the data.
As my students enter class today I hand each a Just Curious Entrance Slip. I will ask students to complete the slip. We will use a Think-Pair-Share protocol for today's Launch:
- Complete the Entrance Slip individually
- Discussing your response with a shoulder partner
- Be prepared to share with the whole class
For the prompts on this slip, I usually get responses like:
1. Study Time/Higher grades; yes; strongly
2. Cell phone time/height; No
3. Computer memory/internet speed; yes; strongly
4. Quarterback practice/team performance; yes; somewhat
These prompts usually stir-up debate, which I welcome (see my Facilitating Discussion while Managing Debate reflection). We are preparing to discuss correlation, so I want students to begin thinking about the idea that opinions about the meaning of data can differ. And, that it is important to work together to come to agreement about the ways that consensus will be arrived at when conclusions are necessary. For example, a student may argue that "Studying more doesn't always lead to higher test grades." I may respond by asking if this is generally what happens. I may say, "How can we use what we are learning in this course to convince someone one way or the other?"
During the discussion I am looking for an opportunity to introduce the term Correlation. I like to do so by saying something like, "Oh, you are thinking that there is correlation between the amount of studying and the test grades..." I want my students to consider this as an idea that they are using, even if they do not know the name for it yet. In my experience, the class opinion is usually that there is a relatively strong correlation between study time and test grades.
At the end of today's Launch I will ask my students to paste their Entrance Slip into their notebooks. Then, I will ask them to write a definition of the term correlation beneath it. I'll say something like, "I gave you a name for an idea that you were using, so I think it is okay for you to write this definition in your own words. Acceptable answers include:
- When two events are related
- When one event has an affect on the other
Finally, I will say, "In my opinion, there is no correlation between the two variables given in #2 on the Entrance Slip." I'll pause for a minute to let students think about this statement. Then, I'll say, "Take another look at your definition of correlation. Is there anything else that you want to add before we move on?"
Activity 1: Exploring
At the start of this activity I introduce my students to an easy to use virtual manipulative:
I begin stating that we are going to plot data points and look for correlation on a graph. Then, I ask students to scroll down to the second set of directions just above the second grid titled Line of Best Fit.
At this point I give students a few minutes to play with the interactive grid as I walk around observing them. Most of my students are quick to get the hang of this tool. As I circulate I make sure all of my students know what to do when creating a scatter plot that represents weak or strong correlation.
Next, I ask my students to read the brief explanations of the terms outlier and best fit line. Some of the questions I ask as I monitor around are:
- Which of the points on your graph may be an outlier?
- Why is this a strong/weak correlation?
- Why is the best fit line there....and not here?
- What does this positive/negative correlating line remind you of?
Once my students appear to have the hang of using this Applet, I ask the class to show me how they are feeling using a visual cue. Sometimes I use a rubric like How do you feel to give students a sense of how to respond. In this lesson the students are learning about the technology and the mathematical concepts, so an unstructured response may not tell me what I really need to know.
Once I am satisfied that students are ready for a more formal presentation I will move on to a brief presentation on Scatter Plots. This presentation is meant to be relatively interactive. I will call on students to explain their thinking after they have a minute to look at the scatter plot on each slide. The objective of this conversation is to prepare students to successfully complete our next activity.
Activity 2: Challenge
For this activity my students pair up and complete a challenge several times, trying to come up with a strategy to complete the challenge successfully. The process of developing a successful strategy should help students to develop a more intuitive understanding of correlation. For this activity, the pairing can be random. The interactive grid from Activity 1 is used for the challenge.
Here's the Challenge:
- Plot exactly 5 data points to create a scatter plot that has a strong positive correlation.
- Using as few points as possible, add more points to your plot until it now shows a strong negative correlation. Think carefully as add new points!
- After steps 1 and 2, discuss what strategies can be used to complete both challenges with as close to 5 points as possible. Then, play again using one of these strategies.
- Be ready to explain your best strategy: Why does it work? Demonstrate on the screen for all to see.
After my students have explored the challenge, I will ask a student volunteer to come up to the teacher's computer and show the class their best strategy starting from a blank grid.
Here's a video of a strategy: Challenge Scatterplot
The student plotted 5 points close together showing a strong positive correlation on one end, then one point on the far lower end. A winner of course, changing the from strong positive to strong negative with a single point. But one very bright student raised his hand and said that this wasn't very reliable and that this may be an outlier. I asked him to tell me more and he said we would need more data to know if it were truly a strong negative trend. He came up to the board and plotted points continuing the positive trend of the 5 clustered points, changing the trend to positive again This demonstrated how the far off point could very well be an outlier. Nonetheless, we had a winner.
Here is a simulation of what he did:Student demo
I always ask the student to explain what he/she is going to do to change the correlation from strong positive to strong negative using the least number of points. This usually leads to some students claiming that they have a better strategy. I am psyched when this happens, because I know that we are going to have a rich discussion of correlation. I allow the class to enjoy the excitement of competing strategies. But, I make sure my students leave with a clear understanding of strong and weak correlation.
A cool and interesting to implement extension exercise is to ask students to go to google images and type in:
real world scatter plot correlations
I ask my students to search and to find two examples of scatter plots relating variables that interest them. "If possible," I say, "find one with positive correlation and one with negative correlation." I like to collect screen shots of the plots that my students choose and organize them into a powerpoint presentation. (Sometimes I give students this task as an extra credit assignment).