For Warm Up today, I have provided students a story and asked them to create a graph that matches it. Although there are many correct answers, I am looking to make sure students are paying attention to the pertinent details of time and variance of slope. Because this is our first attempt at this type of activity, I will use it to probe student understanding through questioning.
When the timer sounds, I select a volunteer (whom I have already pre-selected while 'taking roll' during warm-up) and ask them to sketch their graph on the smartboard. I read the story aloud and ask the class to confirm if the graph, indeed, matches the story. I then ask the probing questions like, "How did you decide how steep to make the first part of your graph? What about the second part?" I then ask for another volunteer who has a different graph. Typically, if one does volunteer, the graph is either a steeper version of the first, or it reveals a serious misconception (like no line for the place in the story where Jacob stops and waits). Either way, I use the sample as a teachable moment and move on to the day's main lesson.
Before beginning today's activity, I want to take a few minutes to introduce key vocabulary related to correlation since that is the main feature of the lesson objective. I begin by showing a scatter plot that has a fairly strong positive correlation. I ask students to offer suggestions about why this would be considered a positive correlation. Typically, students use their previous knowledge of slope and volunteer that information. I then quickly move to the next slide (which on the smart board appears not to have a title--it will appear when the board is touched). I ask students what type of correlation this scatter plot represents. My question is immediately answered, so I confirm by touching the board and revealing the title. I then say, before moving to the next slide, that sometimes the data do not reveal a correlation. I then show the slide and ask students why would say that this data has no correlation. I select a volunteer to explain his/her thinking and then ask the class to agree or disagree with a thumbs up or thumbs down.
Next, I explain that in today's lesson, we are going to gather data and determine if the data, once organized, has a positive, negative, or no correlation.
As students came to class, I had them pick up the lab sheet for today's activity. This is a routine I establish the first week of school, so students know when entering class to look on the bookshelf for anything that needs to be picked up.
I ask the students to get their lab sheet and then explain that today we are going to be doing a simple task: writing the letters "O" and "K" in the boxes they see on their papers. I bring up a copy of the lab sheet on the smartboard and demonstrate. I tell students not to begin until I tell them. I go on to explain that we are going to do this five times, for differing amounts of time, and it is very important that they start and stop when I say. Otherwise, our data will not be accurate.
I then say, "Ready, set, go!" and the students begin writing. I watch my stopwatch for 5 seconds and yell, "Stop!" I then tell students to move to the next row of boxes and get ready. I repeat these procedures for 2, 7, 1, and 3.5 seconds. (These times are in the table of the labsheet.)
Then, I tell students to put their pencil in their other hand so we can gather data from our non-dominate hand. This, of course, causes about 30 seconds of reactions as students first imagine and then begin to attempt holding their pencil in their non-dominant. I explain that I recognize that this will be more difficult, but that it is really important that they do their best so we get good data. I then call, "On your mark, get set, go!" Students giggle and groan, but make the best of this first attempt. I stop them after 6 seconds. I then repeat for 3, 1, 4, and 8 seconds.
Once we finish the data collection, I explain that we are now going to create a table of our data. I demonstrate on the smart board how to count each box with a letter in it and record the number from each row into the table. I ask students to put their pencils down when the are finished. While student complete their data tables, I distribute colored pencils and clear rulers to each table.
Next, I explain that we are going to represent this data graphically. I tell them that first, we must label our graph. I ask the students to tell me what labels we should use. I then ask, "Which label is for our independent data?" I remind them that in Science they have learned about these and ask which data is dependent on the other: "Is time dependent on the number of letters you wrote or is the number of letters you wrote dependent on the time?" When students confirm the latter, I explain that today, I would like for everyone to set their graphs up the same so that we can best compare our data at the end of class. (In future lessons, I will not prescribe how to set up graphs. I want students to have this experience to build upon.)
I then go on to set up the x-axis for time, skipping a box between numbers (0-8) to better show the distribution of data. I then label the axis time.
Next, I ask students how we should label the y-axis scale. One student asks, "What's the most letters?" so I poll the class to find out. The largest number is 17, so we decide collectively that a scale of 1 would be most appropriate, since there are 20 boxes on the grid provided.
Then, I go to the smartboard to model plotting the data points. I tell students to plot their dominant hand data in pencil and their non-dominant hand in colored pencil.
After a few minutes, I stop the students and tell them I need to introduce another concept called a "line of best fit." I begin by asking students if the data they collected was linear or non-linear data. I explain that most of the time, scatter plots are non-linear, but they often times have a correlation. I then ask if they notice a correlation in their data. I explain that although they see an obvious correlation, as mathematicians we want to better represent this correlation with a line of best fit, which is basically an estimate of all the data points so the correlation can be described algebraically.
I also explain as I demonstrate with a set of data on the smartboard, that without appropriate tools like a graphing calculator, creating a line of best fit is not an exact science, but there are some basic guidelines: First, the line should be placed along the path of the data points. Second, once you've drawn your line, you should see the same or close to the same number of points above and below the line. There may even be some points touching the line. I reiterate that I am not looking for perfection, but that the clear rulers should help them, since they will be able to see where the data points fall before drawing their lines. I then circulate and provide additional guidance and affirmation as needed as this tends to be a very frustrating concept for students.
As students finish their work, I ask them to tape their graphs on our gallery wall.
Once all the students have posted their work on the gallery wall, I invite students to stand and walk to the gallery to look at the graphs. I begin asking questions:
-"What type of correlation do you see in the graphs?"
-"What does that correlation mean?"
-"Is every graph identical?" "What is different about them?"
-"Raise your hand if you are left-handed. Can you tell where his/her graph is without looking at the names?"
-"Do the two lines have the same slope?" "What does the slope represent?"
-"Would their ever be a time when the lines would have the same slope?"
Finally, I preview the following day's lesson by explaining that tomorrow, they will get another chance to gather data and determine the correlation.