Day 3: The Hopping Hare
Tara begins class by checking students’ understanding of slope.
Tam: How did you find slope to fill in the table in question 6 of Figure 11.4?
Kelly: Divide the change in y by the change in x.
Tara: So let’s go through the table.
Tara displays a copy of the table on the whiteboard and calls on students at random to fill it in. After the table is complete, the students notice that the speed of the Tortoise and the slope are always the same.
Tara: Were you surprised that the slope and speed are the same?
Tim: At first I was, but then I realized that the change in y is the change in distance and the change in x is the change in time, so the slope is just the speed the Tortoise traveled.
Tara: Wow! I’m impressed, great answer!
In The Hopping Hare (see Figure 11.5), the students examine the Hare’s race more closely. They use information about the speed of the Hare to complete a table. They plot a graph of a piecewise
FIGURE 11.5 (Continued)
linear function from the table and find the equation of a linear function to fit the data for the first two minutes of the Hare’s travel. They also determine that for the portion of the race when the Hare is sleeping, the slope of the graph and the Hare’s speed are zero. This activity provides more practice for students to find slopes and relate them to speed.
The students are all given The Hopping Hare handout, and Tara calls on students at random to read the three observations that are known about the Hare’s race. After the observations are read, Tara leads the class through a reenactment of the situation. A course is taped on the floor with marks indicating the starting point and the winning post. Since there are three main parts to the Hare’s race, the course is split into thirds and the sections marked on the floor. Raphael volunteers to be the Hare. He sprints to the first mark, falls down, and lies there until virtually the whole class starts yelling, “Get up!” “Run!” “Move it!” He jogs the third part of the race. Meanwhile, Tara times Raphael’s race using a stopwatch. Kristen records the time and distance on the board as ordered pairs, with the first pair clearly identifying the variables:
Start of race: (0 seconds, 0 meters) or (0, 0)
Beginning of sleep: (4, 1.5)
End of sleep: (27, 1.5)
End of race: (30, 4)
Tara wants the students to recognize how many meters per second Raphael was traveling during each part of the race. The following dialogue ensues:
Tara: How can we get the Hare’s speed during the part of the race before he fell asleep?
Marie: We need to divide distance by time to find the Hare’s speed.
Tara: What distance and what time? Come show us.
Marie: We look at (0, 0) to (4, 1.5). To get the distance, we do 1.5 meters minus 0, which is 1.5. To get seconds, we do 4 seconds minus 0 and get 4 as the number of seconds to go 1.5 meters. Now, to get speed, we do 1.5 meters divided by 4 seconds to get 0.375 meters per second.
Tara: What was the Hare’s speed for the last part of the race after he woke up? Which ordered
pairs do we use?
Tasha: We use (27, 1.5) and (30, 4). Four meters minus 1.5 meters divided by 30 seconds minus 27 seconds. That gives . . . 2.5 meters divided by 3 seconds, or 0.83 meters per second. Tara: What about the second part of the race? How fast was Raphael going then?
James: He didn’t move because he was sleeping through it, so how could he have any speed? So
his speed was 0.
After finding the speeds for the Hare’s race as enacted by Raphael, the students are asked to work together with a neighbor to fill in the first table of The Hopping Hare (Figure 11.5) to determine the distances the Hare had traveled by various times. Char and Tara circulate and help students fill in the table. The students find the distances corresponding to the whole minutes fairly easily, but they have difficulty finding the distances associated with the half-minutes.
Tara: Tell me how you filled in your table so far.
Joe: I know 0 minutes is 0 meters. I also know 1 minute is 250 meters because his rate is 250
meters per minute.
Tara: Good. What are you going to put for half a minute?
Joe: Well, half of 250 is 125, right?
Tara: What do you think?
Joe: That’s right!
Meanwhile, Char is helping Jessie. Jessie has the first part of the table filled in, but she is struggling with the last part.
Jessie: What do I put for 49.5 minutes?
Char: Think about Raphael’s race. Describe the race.
Jessie: He ran fast, slept, then ran faster.
Char: How does Raphael’s race relate to the Hare’s race?
Jessie: He’s just waking up at 49.5 minutes because his nap was 47.5 minutes long. He’ll be at the same place he was at 2 minutes when he went to sleep.
Char then challenges Jessie to think about the last two cells in the table, reminding her that the Hare’s rate was 500 meters per second for that part.
Tara and Char constantly refer back to Raphael’s acting to get students to remember what had happened in the race. It is evident that when students can refer to Raphael’s race, they understand the Hare’s race better. Questions like the following are asked of the students to help them understand what the table is asking: “When the Hare woke up, how fast did he travel?” “How far did he go in 0.5 minutes?” “How does 0.5 minutes compare to a whole minute?”
To help students prepare to graph the data in their table, Tara asks the class, “What do x and y represent in our story?” Justin answers, “The y is the distance, and the x is the time.” Tara then asks, “What is the distance measured in, everyone?” The whole class shouts, “Meters!” After completing the Hare’s graph, the next step is to discuss the meanings of the change in x and the change in y with respect to the Hare’s travel. Tara asks the class questions to summarize the important points from the previous day’s activities and to lead into question 2 of The Hopping Hare.
Tarn: Can anyone tell me what slope is?
Brandon: Change in x over change in y.
Kristen: No, you mean change in y over change in x.
Tara: How are we going to remember if slope is change in y over change in x or change in x
over change in y?
Kelly: Slope is speed like meters per minute.
Marie: And meters is distance—that’s our y.
Nick: And minutes is time—that’s x.
Tara: So slope is . . .
Brandon: Change in y over change in x.
Tara: And slope and speed are related how?
Kelly: In The Trodding Tortoise, every slope we found was the same as the speed.
The students easily determine changes in distance and time, finding corresponding slopes and speeds for the Hare’s travels. Questions 3 and 4 in Figure 11.5 are assigned as homework. In question 3, students are to find an equation relating elapsed time and distance for the Hare’s first two minutes of travel. In question 4, students are to use the equation to determine the Hare’s distance for a few different times and to determine the time it would have taken him to complete the race had he continued to travel at the same speed. For the remaining lessons, students reenact different scenarios to reinforce properties of slope, y-intercept, and their connections to the graph of y = mx + b.