To introduce this lesson, the class should engage in a short review of the meaning of slope, as a rate of change, and interpreting the y-intercept of a line in the context of a real-life application. The teacher should refer to specific examples used earlier in this Unit Plan. The meaning of the correlation coefficient should also be reviewed. Guiding Questions to help you with this discussion follow.
- How does slope relate to the graph of a line?
- How does slope represent a rate of change?
- How do the labels on the axes of a graph help you determine the rate of change?
- What does the y-intercept indicate about the data on a graph?
- How do the labels on the axes of a graph help determine the meaning of the y-
intercept in the context of a real-life application?
- What does the correlation coefficient indicate about the least squares regression line?
- Why are some correlation coefficients positive and others negative?
- What can we conclude about correlation coefficients with very small values (close to zero)?
- What does the sign of the correlation coefficient indicate about the slope of the least squares regression line?
Today’s activity should begin with a general discussion of what students know about the year in which an automobile was produced and the mileage they would expect to be on the automobile. The teacher should consistently refer to the
year in which the car was produced and not the age of the automobile.
Students should be divided into teams of two to work at the computer and given a copy of the handout Automobile Mileage—Year vs. Mileage (or a similar handout of student-produced data). They should visit the Web site: http://illuminations.nctm.org/index_d.aspx?id=454.
Working together, the partners can share the responsibility of making sure the data is plotted correctly. One student should plot the data, while the
second reads out the data and makes sure it is plotted correctly. This is a good time to bring independent and dependent variables. Explain to students that x
is the independent variable and y is the dependent variable and ask them to differentiate which is which for our year vs. mileage situation. Are students clear on the fact that the year is associated with the x-axis and is the independent variable and the mileage is on the y-axis and is the dependent variable?
Students should click on the applet and make the changes in the viewing window indicated on the handout. As the students begin to plot the data, the teacher should walk from group to group, making sure the students are plotting
the data correctly. Encourage students to think about the data they are plotting and the resulting plots. Ask questions such as the following as you monitor and
facilitate the group work:
- Are you beginning to 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 regression line might be?
Allow the students to complete the plot and answer the questions on the handout. Continue to circulate and facilitate discourse between the partners.
After completing the questions on the handout, students should be given the opportunity to discuss their findings as a class. The questions on the handout can be used to help guide this discussion. This will encourage students to
reflect on what they have discovered about the graphical and algebraic estimations of the real data and allow them to strengthen their understanding of slope as a rate of change. The teacher should pay particular attention to the
students understanding of the units used on the axes.
