Illinois State University Mathematics Department
MAT 312: Probability and Statistics for Middle School Teachers Spring 1999 
Discussion Notes: Session #10 Thursday 11 February 1999 




Assignment Due Today
 Developing 17 and Developing 18 (pp 1920)
 Scatter plots from Tennessee vs Mississippi NCAA Womens' Basketball
Linear Relationships: A Review
In our study of twovariable relationships, we will begin by looking at linear relationships. Before we take the leap from exploring and describing relationships through scatter plots to more precisely characterizing them using mathematical representations, let us briefly review characteristics and representations of linear relationships.
Here are four different ways to represent the same linear relationship between two data sets
For a pair of numbers, the second number is always two more than three times the first number.
 order pairs: {(3,11),(7,23),(0,2),(5,13),(2/3,4),(1.7,7.1)}
 table of values:












Visual or Graphical Representations
The equation y = 3x + 2, where x represents the first number and y represents the second number. We could also write this as s = 2f + 2, where f represents the first number and s represents the second number.
The same relationship holds whether we represent it verbally, numerically, visually, or symbolically. What's more, there are important characteristics of linear relationships that are revealed in each type of representation.
A table of values or a graph may best reveal one characteristic of all linear relationships, that of constant growth:
 For each unit change in the first variable, there is a corresponding constant change in the second variable.
In the relationship illustrated above, the constant change is 3 units. For each change of 1 unit in the first variable, the second variable changes 3 units. This essential characteristic of linear relationships is called slope. The origin of that term is best related to the visual representation, where we can think of slope relating to the inclination of the line:
 For every 1unit change horizontally, there is a 3unit change vertically.
If we know two ordered pairs (x1,y1) and (x2,y2) that are part of a linear relationship, we have enough information to determine the slope of the relationship.
The first line shown above represents the concept of slope: constant change. The second line shows how to calculate the slope when we know two ordered pairs in the relationship. The last symbol in the second line shows the traditional symbol used to represent slope, the letter m.
Other important aspects of linear relationships include the location of the axes intercepts. That is, we typically identify the ordered pairs that describe where the line intersects the xaxis and where it intersects the yaxis. In the form of ordered pairs, that means we want to know the values a and b in the ordered pairs (a,0) and (0,b). The first is the xaxis intercept and the second the yaxis intercept. When reallife situations are described through linear relationships, the intercepts often take on important meaning. The following example is intended to illustrate that.
Linear Relatonships: An Example
Here is a table of values to show the cost associated with renting various numbers of Nintendo games. The cost includes the annual fee plus the pernight rental charge.
We can also plot the ordered pairs (number of rentals,total cost), as shown in the graph below.
In symbolic form, we can write an equation to represent the relationship between games rented and total cost. If we let g represent the number of games rented and T represent the total cost, then T = 2g + 20 represents the relationship.
We have shown three different ways to represent the relationship: numerical, graphical, and symbolic. The fourth representation, verbal, was the original description of the relationship.
How does the graph below differ from the one above? How are they similar? What is the significance of the verticalaxis intercept? Why does only one end of the line segment have arrowheads on it?
These and related questions are important to consider when working with representations of relationships. Here are some questions and answers that help focus on critical aspects of the relationship.
As we wrap up this brief review of linear relationships, consider the four situations described below. For each case:
Case I: The treadmill was set to revolve at 4 miles per hour.
Case II:
Case III: The owner of the ice cream stand determined that each singledip cone costs 36 cents to produce, and that the portable sales stand required $200 to open each weekend.
Case IV: