Sunrise, Sunset Data = Sinusoidal Waves.

 

Want to get students involved in a project which will help them model real life data using a trigonometric regression?  Set them up to learn more about the increase and decrease of sunlight throughout the year in various locations around the world.  Arrange things so they do the whole project from the beginning by collecting data, to using a sin regression to find the equation to identifying all the parts of the transformational sin formula.

The first thing to do is to assign each student a city such as Honolulu, Hawaii,  Reykjavik, Iceland, Oslo Norway, Brownsville, Texas, Rome, Italy from all over the world. The choices should range from far north, to around the equator, northern hemisphere and Southern Hemisphere so students have a chance to see how the sin wave changes according to it's location on the earth.  This site has sunrise and sunset information from most major cities around the world. 

Students should write down the times of sunrise and sunset for same day every month.  When I tried out this activity, I chose the 15th of each month because it was the middle which I felt was an average.  After recording the data, students should subtract sunrise from sunset to get the total length of daylight.  The next step is to create a state plot of the data which can be done by hand or on a calculator.  

I've seen the data entered in two different ways.  The first way is to count the months so January is 1, February is 2, all the way to December being 12 and the second piece or L2 is the day length, or you can count the days themselves so Janury 15 is day 15 which February 15 is day 45, all the way to December. This is a great opportunity to discuss which way would be better.

This is a great point to have groups of students place several plots onto one graph so students can see the similarities and differences based on the different locations.  The cities should be grouped so one is located fairly close to the equator, another as far north as possible and two others somewhere in-between.  It is interesting to see how the data curves if it is from say Oslo, Norway compared to Honolulu, Hawaii. The one from Honolulu will have a smaller stretch than the one from Oslo.

Once all the data is entered and the stat plot is completed, students will be able to see that the sine waves will provide the best fit so this is a great opportunity to teach students how to carry out a sine regression to find the an equation for the data so it its the f(x) = asin(bx + c) + d general formula.  In addition, one can show how the a, b, c, and d are calculated individually so students see how each part is calculated.  At the end, they can create a written report comparing and contrasting their city with others.

One of the easiest things to find first is D because one just has to add the maximum and minimum hours of daylight together and divide by 2.  That would be the vertical shift and center line.  To find A which is the distance form the maximum to the center line one just has to subtract the maximum -  D to find A. In order to find B it's just 2 pi/the period such as 365.25 for the year.  Once students have A, B, and D, they can calculate C.

This exercise provides a wonderful real life application of sin waves where students learn more about how amplitude is found, how b is pretty much standard with 2pi/period, and the horizontal and vertical shifts.  Real data, real calculations and real modeling.  Let me know what you think, I'd love to hear.  Have a great day.