Writing Trig Equations

While most of my students are fine when I give them a trig function and ask them to graph it, they struggle with the converse- writing an equation from a graph.  For most of the quizzes and tests I give I only need to create one version of the retake, some students will need to retake a standard twice.  But for the standard that required students to write a sine or cosine equation from a graph?  I made four versions and more than one student needed them.  (Kudos to those kids for caring enough to put in the time and energy to master the standard.)  When students have an equation in front of them they do just fine reading off the coefficients and constants, then deciding what each one does.  The graph has each of those numbers hidden inside it, but there are four different things to check for (amplitude, period, horizontal shift, vertical shift - plus choosing sine or cosine) and students inevitably forgot one of them.

I introduced writing equations with the standard Ferris Wheel Problem (although I really need to check my numbers for accuracy, the original was in seconds so I know it wasn't based in reality).  Students had a very hard time with this problem, which surprised me since they all had graphing calculators.  I expected students to do plenty of guess and check until they found an equation that matched.  Instead, I got equations that were completely off, heard frustration from students (and parents, poor timing assigning this the night before conferences) and had a crew of kids stay after school to figure it out.  A couple students even attempted the bonus after struggling through the first section.  That was my first real insight to the dedication of some of these kids, it was awesome to watch how excited they were that they succeeded and hear them brag about staying after for two hours.

So, when I assigned the hours of daylight problem during midterm review week I didn't expect it to go perfectly.  It turned out, I didn't correctly anticipate any of what happened.  First, it took forever to set up the graphs.  This was Ashli's project (that she got from someone else, thanks mystery original author!) and she used excel, smart lady.  Second, students had great intuition about what the data meant, they have a deep understanding of daylight, the tilt of the earth and the effect latitude has.  Impressive!  Third, some students forgot the difference between sine and cosine graphs - I expected them to need reminders about amplitude or period, but to be mad at me for asking them to figure out the difference between sine and cosine (with notebooks, graphing calculators and unit circles at their fingertips) was surprising.  But, we survived it all.  I wish that I'd given this assignment when we had more time to examine it.  Since it took so long to graph the original data I just asked students to describe the shape of the graph of the changes.  Most students picked up on the fact that it was another curve, but I didn't get the chance to expound upon how strange and exciting that was.  Next year I'll plan things better.  I'm not sure yet what better will look like, but it will be better!