Proportional Reasoning Part D: Speeds, Rates, Steepness, and Lines (45 minutes)

Note 8

The graph in this section is designed to uncover a variety of common misunderstandings. First of all, some may interpret the graph as a picture of the trips the cars took, a flawed interpretation. If the graph has a segment that is horizontal, there may be more than one interpretation of the car’s trip. The car could be stopped, or it could be going in a circle around the intersection. This is often an interesting insight for people.

Some may assume two cars that are the same distance from the intersection are side by side. Others may think that the two cars in this situation are on opposite sides of the intersection, facing each other, or that they could be approaching the intersection along intersecting streets.

A negative slope indicates that a car is moving toward the intersection. A positive slope indicates that the car is moving away from the intersection, since the distance is increasing over time. Some may wonder where the intersection is. In fact, the graph is not a picture of the car’s path, but a representation of the relationship between distance and time.

Keep in mind that speed is always a positive quantity, while velocity can be positive or negative, corresponding to positive and negative slopes.

Groups: End the session by discussing Problem D5. Each group can share the “story” of their car’s trip.

Problem D1

First, the graph only allows us to decide whether a car is moving toward or away from the intersection, and does not tell us any specific direction. In relation to the intersection, the movement of each car is as follows:

• The black car moves to the intersection, then stops there.
• The red car does not move at all.
• The orange car is moving away from the intersection at all times.
• The green car starts at the intersection and moves away from it at all times.
• The blue car moves away from the intersection for 12 seconds, then stops.
• The yellow car stays at the intersection for 5 seconds, then moves away from it for 7 seconds, then stops.
• The purple car moves toward the intersection, then stops a distance away from it.

Problem D2

The steepness of the line gives the speed of each car, in meters per second. For example, the green car starts at the intersection, then is 120 meters away after 4 seconds. Its rate of speed is then 120 / 4 = 30 meters per second. Using the same technique, and any two points on the line, we can find the speed for each car.

• The black car moves at 20 meters per second, then stops.
• The red car does not move at all.
• The orange car moves at 20 meters per second.
• The green car moves at 30 meters per second.
• The blue car moves at approximately 12 meters per second, then stops.
• The yellow car starts at rest, then moves at approximately 23 meters per second, then stops.
• The purple car moves at approximately 13 meters per second, then stops.

Problem D3

Yes, the green car’s distance varies directly with time. Its graph is a straight line passing through the origin (0, 0).

Problem D4

Yes, any horizontal line represents a stopped car (because the distance from the intersection is not changing). The black, red, purple, blue, and yellow cars are stopped at some time.

You could argue that the red car, or any car that stays the same positive distance away from the intersection, is moving in a circle around the intersection! It may be true that the blue, yellow, and purple cars all entered a traffic circle at the same time; as mentioned in Problem D1, the graph does not give us enough information to tell us in what direction the cars are moving, only their distance.

Problem D5

Answers will vary. Here’s one scenario: The yellow car started out smoothly from a red light, and continued moving away from the intersection at 20 meters per second. Then, suddenly, the yellow car was stopped short by a collision just ahead between the purple and blue cars.