There is also a video narrative that augments this section of the lesson with an emphasis on pedagogy. I begin this lesson by asking students if they can think of any kinds of things in the real world that might have repeating cycles. (MP7) Sometimes they’re stumped initially because we’ve been working so much with the graphing calculators and trig equations, but I usually get at least a few students who think of things like seasons or phases of the moon. I list the cycles on the board as they’re mentioned, or have a student scribe write them for us all. There’s almost always someone who thinks they can embarrass me by mentioning menstrual cycles, but instead I simply add that to the list we’re generating on the board without editing. When we’ve gotten several of the more obvious cycles listed, I ask my students if they can see any correlation between what’s on the board and what we’ve been studying. Again, I’m sometimes met with blank stares in which case I give the hint/reminder that sine, cosine and tangent functions are all cyclic. This connection often generates more discussion of real-world periodic cycles, which we add to the list, then I have my scribe sit down and ask the students to decide which of the cycles we’ve listed have an equation they’re familiar with.(MP4) Most students can say that they know the period of specific cycles, but struggle with figuring out an amplitude or phase shift. I mention recent lessons we’ve had covering graphing sine and cosine functions and ask whether any of the cycles on the board might look like the graph of either sine or cosine. I allow the discussion to continue until most students can agree that some naturally occurring phenomenon can be modeled by trig functions.
25 minutes: Sharing
When all the teams have created an equation, each team passes their Weather Data sheet with their equation written on it to another team. I have several methods for this kind of sharing in my strategies folder. I don’t bother to try to make the sharing “blind” by numbering the teams or something because my classes are so small that the students know each other well enough to recognize handwriting! I ask each team to test the equation they’ve just received in their own calculators and record a score and comments following the scoring rubric on the data sheets. Each team gets to score every other team’s equation, then the teams all get their data sheets back to review. (MP3) You can also collect the data sheets, telling the students that you will review them and return them tomorrow. Even in my small school, there are still a few students who might be mean or otherwise inappropriate when scoring a classmate’s work, so if this is a potential problem, you may want to screen the scores/comments before returning them to the students. Since the goal of this lesson is to model real-world data, I close this part by asking students if they think their equations were good models for the data. Then I ask them to think of reasons for modeling data with an equation and we discuss making predictions and comparing. For example, it would be good for a farmer to be able to predict how much precipitation he/she could expect in a given month to be better able to control how much moisture the crops received.
10 minutes: Expanding
This activity is to encourage students to again think of the different types of data that can be modeled with a sine or cosine equation. I’ve included additional weather data and also a tides table that includes times for high and low tide, moon rise and set, and sunrise and sunset. If you have access to computers with internet you could have students research some of the topics they suggested earlier to find data to work with. Either way, I have students attempt to model one or more sets of data, this time working individually. (MP4, MP7) This gives me the opportunity to identify which students are still struggling with fitting an equation to a data set. When all students have successfully graphed at least one set of data, we go to wrap up
To close this lesson I give my students the assignment to find a set of data that is periodic and write an equation to fit the data. The requirements are pretty simple, but I try to be explicit in saying they must bring in a copy of the data, including its source, and the equation they wrote to fit the data.
Resources: Lewiston Weather Data, Alternative Data Sets, narrative video