## Learning Math: Patterns, Functions, and Algebra

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

## Session 4, Part D

In Parts B and C we examined the proportionality of mixtures and scale models. In this activity, we will take a close look at graphs to learn more about the relationships between distance and time, and between steepness and speed. **Note 8**

Suppose seven cars are all near an intersection. The graph below show the distances between cars and the intersection as time passes. Study this graph carefully and then try to answer the following questions.

**Problem D1**

In what direction is each car moving in relation to the intersection?

**Problem D2**

Compare the cars’ speeds. How do their speeds relate to the steepness of the lines?

*Tip: How could you approximate the cars’ speeds using the labels on the axes?*

**Problem D3**

Does the distance vary directly with time for any of the cars? That is, is the relationship between distance and time proportional for any of the cars? How do you know?

**Problem D4**

Do any of the cars stop during their trips? If so, which cars?

**Problem D5**

Choose one of the seven cars, and describe your trip in this car. Use your imagination! Think of your observations as a passenger or driver in this car, and give the highlights of your trip for these 15 seconds. Include where and when you started the trip and what you saw going on around you — in front of the car, to the sides, and through the rearview mirror.

###### Car problem taken from IMPACT Mathematics Course 2, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 328-329. www.glencoe.com/sec/math

### Notes

**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.

### Solutions

**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.