For today's warm up, I have selected a scenario for students to represent graphically. This story differs from previous stories in that the character forgets his homework and has to return home, which will need to be represented as a negative slope line for that section of the graph. This may pose a challenge for students since previous warm up graphing stories have just involved changing the steepness of a positive sloped line.
For today's work time, students will be gathering data on their heights and shoe lengths in centimeters. To facilitate this data collection, I have created seven (one per table) measuring stations around the room where I have taped two meter sticks to the wall (stacked, as to measure height) and one meter stick to the floor (to measure shoe length).
I explain the day's task, which is to gather the data, plot it on a scatter plot, determine a line of best fit and then write the equation of the line that fits the data. I then model how to measure height to help any student who may lack experience measuring. I talk through explicit procedures: "Your partner will stand with his feet against the wall. You will use your hand to mark where the top of his head hits the meter stick. Remember, I have taped one meter stick on top of the other, so will this number be the one I record on the class data table?" (This is a common error, so it is important to model that students will need to add 100 to their measurement.) I then model how to record the data on the lab sheet.
I continue by explaining that after one partner is measured, they need to switch roles. Then, they need to measure their shoe length in centimeters, which I demonstrate.
Next, I reveal the large data table on the smartboard and explain that I would like each team to record their data in this table so the rest of the class can easily copy the data onto their lab sheets.
Because I have given a series of instructions, I ask a student to "recode" what I said and explain today's work to the class. I then ask for questions and then encourage students to work efficiently so we can complete the lab during class.
As students work, I circulate, providing redirection when needed. Once all the data is recorded on the smartboard, I stop students for a brief conversation about graph scales (which could not be made until we had data collected.)
I begin by asking students to look at the data collected and then ask, "How can we decide what the scale for this graph will be?" A student explained that we must look at all the data points in each column and decide how to organize it. I pondered, "So I see that everyone in the room is at least 140 centimeter tall. So does anyone know a way I could show this without counting past 140 for each data point? I saw blank stares, so I took this as a teachable moment to introduce broken scale graphs.
I showed a sample graph I had created and asked students if they noticed anything unusual about the graph. Right away, a student noticed the broken y-axis. I explained that I had used a broken scale, which means I took away the bottom section of the graph so I could more effectively represent the data, since all of us were at least 140 centimeters tall. I know there will not be any data points between 1 and 140, so I took that section of the graph away. I left my example graph for students because I did not want them to get bogged down in the process of representing the data on the graph. Rather, I wanted them to have the time to find a line of best fit and estimate a rule for that line.
I continued to circulate the room as students worked, providing guided questions as needed. I also asked several students to share their work during closure.
When the timer sounded after 30 minutes, I invited one of my pre-selected students to the smartboard to share his equation. He held up his scatter plot with its line of best fit. I asked him how he decided what his equation for the line would be and he said, "My line started at 148 so I knew that would be my y-intercept. I figured out my slope by picking some points on my line. I counted how far up and then over the next point was."
I taped his graph next to the smartboard, thanked him for contributing, and asked the next student to come and share. Her equation was slightly different than the first, so I asked her to hold up her scatter plot. I asked her how her graph varied from the first student and she said, "My line is less steep." I asked her how that related to the equation she found and she said, "It has a smaller slope than his."
I asked the third volunteer, whose graph's line was steeper than the first, and asked him the same questions. My goal in questioning was to get students to see that slight variations in the lines of best fit resulted in different equations.
I reminded students that lines of best fit, done by hand, were rarely perfect. In fact, even ones done on a calculator are estimates of the data because of rounding. I explained that this is a great example of Mathematical Practice 4: Attending to Precision, because our graphs are only as precise as we can make them without technology. Overall, the equations for the graphs were very similar and any could be used to predict the height of someone given their shoe length. I then demonstrated by taking each equation and substituting in a shoe length of 20 cm. Each result was within 5 centimeters of the other, which I explained was pretty good precision given everyone was at least 140 centimeters in height.
I ended by previewing the following day's lab: Slinky Stretchers.