Day 2: The Trodding Tortoise
The second day begins with a review of Runners Take Your Mark, which required students to draw graphs of different race scenarios for the Tortoise (see question 4 in Figure 11.2). Tara displays the graph in question 4 on the overhead. The following conversation ensues:
Tara: How fast is the Tortoise going in part a?
Santana: The same as before.
Tara: So how should the graph look?
Santana: The same steepness.
Tara: Will the graph start in the same place?
Ryan: No, the Tortoise started closer to the winning post.
Tara: How can we show that on the graph?
Ryan: Since it’s one-fifth of the way, go up one tick mark on the y-axis.
Tara: For part b, how can we change the graph to show the Tortoise is going twice as fast?
Mimi: The line will be steeper.
Tara: How steep?
Mimi: Twice as steep as the original.
Tara: For part c, what changes do we need to make on the graph?
James: He started at one minute.
Tara: So how can we show that on the graph?
Joe: Shouldn’t it start at one minute and go up the same as the other graph?
Tara: Class, what do you think of Joe’s idea?
Kristen: I agree!
Alecia: He’s still going the same speed, so the graph is the same steepness. He’s just starting later.
As the students answer the questions, the graphs are drawn on the overhead. To answer question 4d, the students look at the graphs and notice that the graph in part 4b reaches the winning post in the shortest amount of time.
Both correct responses and misconceptions are found in the homework that the students turned in for question 5. See Figure 11.3 for a sample story written by Kelly.
Kelly’s story shows Tara and Char that some students are starting to understand the connections between steepness and speed and that the horizontal sections of the graph show that the character is resting. But some students turned in work that shows misconceptions. For example, one student drew a graph resembling nearly vertical zigzags and wrote a story about an animal climbing up and down a range of mountains. Others simply wrote the story of The Tortoise and the Hare and merely changed the characters’ names. A cursory review of students’ work leads Tara and Char to give more examples and to keep working to help students acquire understanding of concepts of constant rate, zero rate, and change in direction versus accumulated distance.
After collecting homework, Tara started the class working on The Trodding Tortoise activity (Figure 11.4) where students use information about the speed of the Tortoise to complete a table, plot a graph from the table, and find the equation of a linear function to fit the data and the graph. In this first experience with finding an equation of a linear function, the y-intercept is zero. Slope is
FIGURE 11.3 Kelly’s Story
defined with students realizing that the slope of the line is the same as the speed and also the same as the coefficient of x in the equation of a linear function.
Tara distributes The Trodding Tortoise handout and asks the students to follow along as she reads the details of the Tortoise’s race to them. While finding distances corresponding with various travel times for the Tortoise (see the table in question 1 of Figure 11.4), the students have an easy time finding the recursive relationship relating subsequent distances traveled, but they have a hard time finding the explicit formula relating distance to elapsed time. For example, Brandon determines that the Tortoise traveled 100 meters in 5 minutes, 200 meters in 10 minutes, and 300 meters in 15 minutes. He can extend the pattern for distances for subsequent 5-minute time intervals. However, Brandon is unable to find an equation that will tell him the distance traveled directly from any given time.
To help the students see the importance of the relationship between distance and time, Tara says, “Let’s include one minute and 11 minutes in the table.” With these changes now reflected in the table (see question 1 in Figure 11.4), Tara continues to ask questions to guide the class toward a focus on the relationship between distance and time.
Tam: What distance did the Tortoise travel in one minute? 11 minutes? How do you know?
Brandon: In one minute, the Tortoise goes 20 meters because his rate is 20 meters per minute. For 11 minutes, I can add how far the Tortoise goes in one minute to how far he goes in ten minutes, so he goes 20 + 200 meters or 220 meters in 11 minutes.
Huey: Or we can just multiply the number of minutes by 20 meters per minute to get how far
The students are assigned the last table (see question 6 in Figure 11.4) as homework.