Chapter 1
Relationships
1.2 Mathematical Models
1.2.3 Representing Relationships
We can display or describe the relationship (if any) between two quantities or variables in many different ways. Here are some of the ways that are already familiar:
List all the pairs of values that occur together.
Make a table or chart that displays some or all of the pairs in the relationship.
Give a formula or equation that relates the two variables.
Draw a graph in a rectangular coordinate system with the dependent variable on the vertical axis and the independent variable on the horizontal axis.
You will spend a fair amount of time in this course studying graphs of relationships between two variables. So far, you have seen scatter plots and line graphs of data. We will often use graphs of formulas, whether derived from known relationships or from fitting formulas that approximate data. And a graph can also be a conceptual picture of a relationship for which we know neither data nor formulas.
For example, Figure 3 shows a typical learning curve for a concept such as “function.” The independent variable is time, and the dependent variable is the amount learned. (Note that time alone is not sufficient to cause learning, but learning certainly depends on time spent working at it.) Initially, the learner may know nothing about the concept. With any new learning challenge, the pace of learning is likely to be slow at first, then pick up (the steep part of the curve), then slow down again as there is little more to be learned in order to master the concept. The shape of the curve in Figure 3 captures these features of the learning process. Of course, you can imagine variations of the process that would lead to curves with somewhat different shapes.
Figure 3 Learning curve for a concept
Activity 3
Sketch a graph to represent the relationship between the number of customers checking out at a grocery store and the average amount of time each person must wait in a check-out line.
Does the number of customers determine the waiting time, or is it the other way around? Which is the independent variable in this relationship?
What characteristics of the relationship can you determine even without specific information about the time required for a customer to complete the check-out process?
As the number of people waiting to check out increases, what can you say about their average waiting time?
