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Student Page 2.6.2 Giving the Tortoise A Head Start

So far, we have seen several examples of proportional linear relationships. In each case, the graph modeling the relationship went through the origin. We return to The Hare and the Tortoise to consider other possibilities. Giving the Tortoise A Head Start introduces other linear relationships.

The hare challenged the tortoise to another race. He gave the tortoise a head start to make the race seem fair. They used the same 1000-meter course and started the race together.

Figure 2.6.2.1. Empty Axes for Comparing Tortoise and Hare Races

1.

(a)

The tortoise started the race 500 meters from the Starting Line. As in the original race, she plodded at a rate of 20 meters per minute. On Figure 2.6.2.1, draw a graph of the tortoise's distance from the starting line over the time it took her to complete the race.

(b)

The hare ran at a steady rate of 250 meters per minute throughout the race. He stopped only after he crossed the Finish Line. Also on Figure 2.6.2.1, draw a graph of the hare's distance from the starting line over the time it took him to complete the race.

(c)

Who won the race? How do you know?

(d)

Suppose the tortoise started the race 100 meters from the Finish Line. Who would win the race? Explain. Draw the graph on Figure 2.6.2.1.

2.

Complete all three columns in Table 2.6.2.2 to compare the tortoise's distance from the Starting Line at the same time for these races. For each race, use a different color to highlight the time at which the tortoise reached the Finish Line. Use the same color to highlight the corresponding graph in Figure 2.6.2.3. Recall: In the original race, the tortoise started at the Starting Line.

Table 2.6.2.2. Distance and Time of the Tortoise's Travels
  The tortoise's distance in meters from the Starting Line
Time elapsed
in minutes
Original Race Race in Task 2.6.2.1.a Race in Task 2.6.2.1.d
0 0    
1 20    
2      
3      
4      
5      
10      
20      
30      
40      
50      

3.

Graph the tortoise's races from Table 2.6.2.2 on the same coordinate plane in Figure 2.6.2.3. How are the graphs related?

Figure 2.6.2.3. Time and Distance for Various Tortoise Races

4.

Study the table and the graphs.

(a)

For each graph and corresponding data set, find an equation to model the graph and data.

(b)

What do the equations have in common? Explain.

(c)

What is different in each of the equations? What accounts for the differences?

(d)

How does Tortoise's head start show up in each equation? Why does this make sense?

5.

You have studied the idea of ‚Äúhead start‚ÄĚ in linear relationships in previous mathematics classes. What is the mathematical name for the ‚Äúhead start‚ÄĚ value in a linear equation?

With your group, discuss what new information about linear relationships is gained from the student page, Giving the Tortoise A Head Start. Outline what you have learned about linear relationships through the progression of activities: The Hare and the Tortoise, Trodding Tortoise, Hopping Hare, and Giving the Tortoise A Head Start. Ask any questions you have remaining.