Using the Difference in Daylight to Investigate Linear Relationships

Abstract

My group of students was a mix of four ninth grade students enrolled in our Algebra I course. For our project I wanted to look at what was meant by linear and to see if they could tell the difference between linear and nonlinear relationships. I emphasized that there had to be a constant rate of change in order for a relationship to be linear. The relationship we looked at was the difference in amount of sunlight between Juneau Alaska and Albion New York. We started by looking at just the month of January. The students had to put the data into an Excel spreadsheet. They then had to use a formula to find the difference in sunlight. At this point I had them graph their results. The given graph looked almost linear. We then talked about what it meant to be linear (a constant rate of change) and they figured out how to use an Excel formula to find the rates of change for the different values. What they found was that it was not quite linear. The differences were very close to being the same, but at the beginning of the month there were zero, one and two minute differences, at the end of the month there were one, two, and three minute differences. Because of this the group decided that the rate of change for the difference in sunlight might be increasing. We expanded our data to include the whole year by choosing three data points from each month. Upon graphing this difference we could easily see that the relationship is not linear. Now we discussed what kind of relationship it really was and the students had to figure out why. They used Geometer’s Sketchpad to model the earth orbiting around the sun to see why it was a cyclical (sinusoidal) relationship.