To introduce this lesson, the class should engage in a review of the meaning of slope, as the change in y over the change in x or rise over run, and the meaning of the y-intercept , as the point where a graph crosses the
y-axis. Students should also discuss least squares regression lines and correlation coefficients. Guiding questions are as follows:
- What is the slope of a line?
- What does the slope tell us about the direction of a line? What does the slope tell us about the steepness of a line?
- What is the y-intercept of a line?
- What do we know about the y-intercept of any graph?
- What is a least squares regression line?
- What is a correlation coefficient?
- What does the correlation coefficient tell us about the points being plotted and the resulting least squares regression line?
During the remainder of this lesson the teacher begins to help students make the connection between the slope of a line and its meaning as the rate of change in a real-life application. The meaning of the y-intercept in the context of the same application is also investigated.
A discussion of a real-life example of a linear function should now take place. One example might be to ask the class about traveling distances. “If you were traveling 50 miles per hour, how far would you travel in one hour? Two
hours? Three hours?”
The teacher should go to the Web site: http://illuminations.nctm.org/index_d.aspx?id=454 and demonstrate how to change the window settings and graph the responses given to the questions regarding traveling distances above. The points that are plotted should be the number of hours traveled vs. the number of miles traveled.
After the points have been plotted, the teacher should stop to direct the students’ focus to some of the mathematics involved in the plot.
- Do you notice any pattern in the shape of the plot?
- What type of function do you think will fit this data?
- Do you think the slope of the line will be positive or negative?
- What do you think the y-intercept of the line might be?
The teacher should now have the applet draw the least squares regression line and calculate the correlation coefficient. Direct the students’ attention to the slope of the line and ask what relationship it has to the original questions they were asked about traveling distances. The teacher should direct the students to interpret the slope of 50 as the 50 miles per hour in the original problem. This value indicates a rate of change; it indicates how many miles per hour the vehicle traveled. Ask what label would have been appropriate for each axis and how this information could help them calculate the rate of change. Tie this information into the formula for slope, the change in y divided by the change in x.
Next, examine ideas related to the y-intercept of the line. When the points (1, 50), (2, 100), and (3, 150) are plotted, this applet gives a y-intercept of –0.89. Ask the students what this number means in the
context of this problem. Direct the classroom discussion so that the students see that the y-intercept in the applet means that if the vehicle traveled for 0 hours, it would travel –0.89 miles. Does this make sense? Explain that this is the predicted value, not the actual value. Ask how the appropriate labels on the axes can help the students interpret the meaning of the y-intercept in the context of this problem.
The correlation coefficient should be discussed next. Ask the students to explain what the correlation coefficient of 1 means in terms of this example. Direct the questioning to get the students to see that the line passes through the data points. Focus in on the fact that a value of 1 indicated a linear relationship exactly.
The students should now be given a copy of the handout Graphing Real-Life Data. They should be given the opportunity to label the axes, indicate the scale, plot the points we had used on the applet, draw the line through the
data, and fill in the remaining information for the first graph on the handout. Grouping the students in pairs while they do this part of the lesson will allow them to discuss what they are doing and help reinforce the ideas in the
lesson.
As the students work, the teacher should circulate among the students to observe their work and answer any questions they may have at this point. After they have finished their work on the first graph, the teacher and students should summarize the lesson by completing the items next to the graph on the handout. The students should then be directed to save their handout for the next day’s lesson.
