Some preparation was necessary before beginning this activity in the classroom. Throughout the year, Tara provided students many hands-on learning opportunities. Students had participated in
many weeks of team building and got along as a class very well. They were used to working with a partner or in small groups on a daily basis, but they could also be productive when asked to work individually. As for the content, Tara made sure students had a good understanding of the Cartesian coordinate plane, were able to compile tables or t-charts, and were able to solve an equation in the form of ax + b = c. She also introduced them to the graphing calculator and the calculator-based ranger (CBR).
Day 1: Runners, Take Your Mark
Throughout the year, Tara begins each class asking for a recap of the prior day’s events. Before beginning The Tortoise and the Hare, she asks the students to recall the previous day’s events with the CBR (a motion detector attached to a graphing calculator). The students had spent the prior day duplicating various graphs using a CBR to graph their walking. They experimented with different walking rates, starting points, and directions. They created simple linear graphs first, then absolute value functions, and finally a step function.
The students were not aware of the names of the graphs they were creating; they were just walking to discover how to create graphs made up of straight-line segments. They also experimented with different starting points, directions, and speeds. Discussion focused on how the students made the different graphs, with special emphasis on how to create different steepnesses. Tara begins her review by asking review questions based on Figure 11.1:
Tara: How do you determine where to start walking for each graph?
Marie: Start close to the motion detector and walk away from it when the graph is going up, and
do the opposite when the graph is going down. (Tara was referring to the absolute value graph students walked.)
Tara: The graph that most of you walked and that is on your calculators has a round top. How
can we make it more like an angle?
Mimi: To make a sharper corner, change directions quickly.
Tara: OK. Now, how can we make an absolute value graph? And how can we be sure that the
steepness is the same in both directions as in graphs d and e?
James: To make the absolute value graphs, keep the same pace as you walk forward and backward.
Tara: Who can show us how to make the step function graph shown in graph h?
To this question Brandon, Tony, Justin, and Joe immediately get up to demonstrate. They stand in a single-file line in front of the CBR and jump out of the range of the motion detector at certain time intervals. At first, Marie asks, “What are they doing?” Tomeca replies, “I don’t know, but that’s not going to work.” But after seeing that the solution is correct, they nod their heads in agreement saying things like, “OK, that makes sense” and “Oh, I get it.” Tara asks, “Can you explain what you understand now?” “Well,” Tomeca says, “whenever one of them jumps out, then the motion detector picks up the next person standing further away. That’s why we see those steps.”
After the recap of the CBR activity, Tara feels ready to launch the challenging problem for the unit. She begins by asking one of the students to read Aesop’s fable of The Tortoise and the Hare:
The Hare was once boasting of his speed before the other animals. “I have never yet been beaten,” said he, “when I put forth my full speed. I challenge anyone here to race with me.” The Tortoise said quietly, “I accept your challenge.” “That is a good joke,” said the Hare. “I could dance round you all the way.” “Keep your boasting till you’ve beaten me,” answered
FIGURE 11.1 Can You Walk These Graphs?
the Tortoise. “Shall we race?” So a course was fixed and a start was made. The Hare darted almost out of sight at once, but soon stopped and, to show his contempt for the Tortoise, lay down to have a nap. The Tortoise plodded on and plodded on, and when the Hare awoke from his nap, he saw the Tortoise nearing the winning post, and could not run up in time to save the race. Then said the Tortoise, “Plodding wins the race.”
“What do those last words mean?” Tara asks. Tiffany replies, “That if you just keep at it, you’ll win.” Tara asks, “Can anyone tell us a bit about Aesop?” The students vaguely remember having heard Aesop’s name, so Tara adds, “Aesop was a Greek philosopher who lived in the 6th century bce. He wrote hundreds of tales like this. They are called fables, and each has a moral. If you search for Aesop on the web, you will find over 600 of his fables.”
Tara next distributes the handout, Runners Take Your Mark (see Figure 11.2) that requires students to compare the movement of the Tortoise, who always plods at the same steady rate, to that of the Hare, whose rate, though constant over intervals, is less consistent. The students are presented a variety of scenarios to compare, some with positive and some with negative slopes. In all of the
FIGURE 11.2 (Continued)
scenarios, the students build intuition for the concepts of slope and linear functions without actually finding values for slopes. As a result of this activity, the students are able to talk intelligently about the relationship between the relative steepnesses of two linear graphs and the relative speeds of the characters each represents.
Tara: How can we determine which animal is moving the fastest from a graph?
Jessie: By the steepness of the graph.
Tara: Tell me more. How does the steepness do that?
Marie: The Hare’s graph is steeper than the Tortoise’s when the Hare is moving faster.
The next step in Runners Take Your Mark is to graph the motion of each of the characters throughout the race. This is done as a whole-class activity. Tara asks the students what to graph, and she displays what they tell her to draw on a whiteboard with the Cartesian plane displayed in the front of the classroom. After the axes are labeled, Tara asks the class how to draw the Tortoise’s graph. Tim responds, “Draw a straight line from corner to corner.” Tara then names a line, y = W, where W is a constant, as the winning line before graphing the Hare’s run.
Tara: Let’s list some of the properties of the Hare’s graph.
Ryan: The Hare’s graph needs a flat part in the middle for when he’s taking his nap.
Jessie: When the Hare wakes up and runs again, his graph has to be steep.
Justin: The graph crosses y equals W after the Tortoise’s graph did.
Tara then reads through a series of racing scenarios between the Tortoise and the Hare for which students are to draw the graphs and then write a few sentences explaining how the graphs pertain to the scenario (see question 2, Figure 11.2). Working with partners, the students work through the graphs and sentences about the first two scenarios with ease (parts 2a and 2b). They begin creating their own scenarios for the racers in class (part 2c). Before class ends, Tara asks the students to share their stories aloud. One of the more creative stories follows:
The Tortoise put sleeping powder in the Hare’s drink. So the Hare takes off, and before the finish line he falls asleep. The Tortoise comes and eats the Hare, and because he ate too much, he dies. So the Tortoise would have won.
Questions 4 and 5 of Figure 11.2 are assigned for homework. In question 4, students are to compare the Tortoise’s actual motion with several alternative scenarios, graphically and in words. In question 5, students are to create their own characters and race scenario, graphically and in writing.