Today's Warm Up again spirals back to scientific notation. Turtle & Snail Part 2 Warm Up contains two problems that require computation. Again I have chosen problems that I expect will challenge my students.
The third problem gives the students a horizontal table of values and asks them to find a rule describing the pattern. The decision to display the data in a horizontally aligned table is intentional. I want my students to make sense of the data displayed in this way, as well as in a vertical table.
Building on the previous day's lesson, Turtle & Snail Part I, today's lesson also employs the Rule of 5. There is one small, but important change to the problem: Snail gets a head start. The head start is big enought that it keeps most students from knowing instantaneously who will win the race. This fact motivates modeling the problem using the Rule of 5. The changes to the data tables, the graph, and the function rules provide opportunities for rich classroom dialog as students model the problem and determine the winner.
I begin the lesson by again asking for volunteer actors to play the various roles in the story. I share the new story and we act it out, just as we did the day before, this time allowing snail to start at the six feet mark. Initially, some students want Snail to begin at the three feet mark, so I ask, "How far will Snail travel during the three minute head start? We need to make sure we are focusing on the distance traveled not the time." This misconception will likely reappear when we create the tables, so I do not dwell on it here.
After role-playing through three minutes of the race, some students begin to guess at the winner. When we finish, the students actually see that Snail wins by a foot.
I then remind students of the Rule of 5 that we used the day before to model this story and tell them to turn to the next clean page in their journals to draw the picture of what is happening in the story (see Turtle & Snail Part 2 Notebook). After they begin drawing, I quickly sketch a representation on the SmartBoard. I call their attention to the fact that I drew Snail six feet ahead.
Once we finish drawing, I ask, "What do we need to do next according to the Rule of 5?" I then show the tables. On the smartboard file, I have the first two cells of Snail's table blocked with cell shades. We start with Turtle's table. I ask, "Will Turtle's table today be any different than yesterday's?" I will call on a student to answer, expecting him/her to explain that this table will not change because Turtle is traveling at the same rate as yesterday.
After completing Turtle's table, we move to Snail's. It is critical that I focus on a common misconception during this part of the lesson: students want to write a "3" in Snails distance column as an output for a time of zero in order to represent the head start. When I see this occurring I stop students and ask them to tell me what "t" and "d" stand for in the t-table. Once a student tells me, I ask, "So is '3' what I want to write for the distance at 0 minutes? I thought his head start was 3 minutes, so how far was his head start in distance?" A student responds 6 feet and I ask her to clarify. "You know Snail travels at 2 feet per minute, so if Snail gets 3 minutes head start, that is 6 feet, 2, 4, 6 feet." As a class, we complete the rest of Snail's table. I ask, "So, from the data in the tables, who wins the race?" I select a student who explains that Snail wins because he passes the finish line just after 4 minutes, but it takes Turtle 5 minutes.
Next, I review where we are in the Rule of 5. I say, "We now have done the story, the picture, and the tables. What comes next?" We then determine the rules for each of the critters. Again, I ask if Turtle's rule will change at all for this story. Once a few students tell me no, we move to writing a new rule for Snail.
I ask students to think about what rule we used for Snail the day before. I then ask what is different in today's story. A student responds that Snail gets a head start. I ask how we might represent that head start within the rule and one student suggests, "2t + 3". Again, I see the misconception that the head start is represented as time instead of distance. I clarify by asking, "Was the head start in our table in terms of time or distance? So what makes sense for our rule. One student volunteers: "d=2t + 6". I ask for thumbs-up, thumbs-down, or thumbs sideways. I expect to see the majority of my students agree, but also to have a few thumbs sideways. I make a mental note of these students, so I can track their understanding in the coming days.
After creating the rules, we move to graphing the stories. I copy the tables to the graph page and select a student to translate the data graphically for the Turtle. I ask, "Is the data still linear?" I then ask for a volunteer to graph Snail's data in blue on the graph. After she plots the first point (0, 6), I ask, "What does this mean?" At this point, I expect students to respond, "It means Snail got a head start of 6 feet." As she continues plotting all the data points and I ask, "Is Snail's data linear?" I then ask, what part of this graph shows us that Snail won the race?
Next, another student comes to the board and points out that Snail's line reaches 15 feet before Turtle's line does. I ask, "How long does it take Turtle to reach the finish line?" and she answers, "5 minutes". I then ask, "So how long does it take Snail to reach the finish line?" and she answers, "4 and a half minutes." Often at least one student will want to discuss the fact that Turtle should have just given Snail a 2 minute head start. Then, he still would have won the race!
For closure, I wanted students to answer some lesson reflection questions in writing in their journals. I posed three questions for the students to answer in their journals (Turtle & Snail Closure).
Creating a written response requires much deeper understanding and also poses a challenge for some students find it difficult to get their ideas on paper. I remind students to use the strategy of restating the stem of the question to get them started. For example, on the first question, "Using the Rule of 5, explain two different ways you can prove who won the race", a student response would be: "Two different ways I can prove who won the race are..." Teaching writing strategies in math class helps to reinforce what is already being taught in Language Arts.