In many situations, a linear equation may only be an approximation of the behavior of data.  Consider the relationship between the hours a student studies and the number of “A” grades received by that student.  Some students require only a few hours of study to attain high grades, while others study long hours, struggling to receive merely passing grades.  While the burning of calories by running, which we explored in this topic, is a linear function, such a function may only approximate the correlation between hours of studying and grades.

Compare these students:

• Johnetta retains all that she hears in class, has a photographic memory, and studies only two hours per week to earn an “A” grade in each of her 5 subjects.
• Paula is content with average grades and studies only in those courses she finds interesting:  for every 6 hours of weekly studies, she usually receives 2 “A” grades.
• Kirk works very hard to pass his courses, and studies an average of 12 hours per week to get a single “A” grade.
• Rayshawn knows that his effort is rewarded with higher grades:  he studies 10 hours per week and receives 4 “A” grades each term.

If we assumed a linear relationship between hours studying (the independent variable) and the number of “A” grades received (the dependent variable), which data would we use to determine the slope of our linear equation?

Data:

Johnetta:            (2 hours, 5 “A” grades)
Paula:                  (6 hours, 2 “A” grades)
Kirk:                     (12 hours, 1 “A “grade)
Rayshawn:         (10 hours, 4 “A” grades)

Table of Values:

Compare a few of the various slopes determined by selecting different pairs of values :

(6,2) and (10,4): 

(2,5) and (10,4):


(6,2) and (12,1):

In your first post, answer the questions below: As you know, the slope of a line containing two points represents the rate of change between those points. In this scenario, our independent variable is the number of hours a student spends studying. We are assuming that the number of study hours determines or causes the student to receive a certain number of A grades. (1) From the data table, do you see a general pattern on the change in the number of A grades a student receives as the number of study hours increases? (2) Do any of the slopes we calculated above match the pattern of the data? (3) Which one of the slopes, if any, would you choose to represent the relationship? Why? There are three more possible combinations of pairs of points from our data. Can you determine those slopes? Would any of those slopes be a better approximation of the relationship? Then, return to the topic several times over the next few days to read your coursemates' posts. Reply to at least two of them, being sure to offer new insights, ask questions, and encourage further conversation.