Which Way Do We Go?

Select an appropriate video for the introduction of this lesson well in advance of the class so you can slant your lesson toward that video if you choose.  Please see the resource entitled “data sources” for possible videos or choose one with an explicit or implied trig function that is particularly relevant to your students. I begin this lesson with a short video clip about farm produce that demonstrates the value of being able to model data with an equation.   This gives my students an immediate connection to why modeling data accurately is important since we are a farming community.  My students have already explored weather data, so they’re comfortable working with real data and recognize that it’s not usually possible to make a model that is a perfect fit.  After the clip I ask my students to pair-share with their right-shoulder partner what information they need to plot data that will allow them to answer questions and make predictions about at least three different farm commodities of their choosing.  These might include corn, wheat, rye, barley, soybeans, canola, or garbanzo beans to name a few.  (I have them choose three so that if they don’t find enough information about their first choice easily, they already have alternatives selected.)

Because the primary goal of this lesson is to help students develop a confidence about selecting and using different models, I spend a great deal of class time on discussion about making choices.  After the partners have shared their ideas about which items to study and what they need to know, I ask random students share with the whole class, but I don’t ask them to share what they chose, I ask them to share how they decided which information they needed. (MP1)   This is really tough for some of my students who may respond with a shrug or “I don’t know”, in which case I ask directed questions like “You said you wanted to work on corn prices.  What made you choose corn?”  That’s an easier question for most students to field and can lead to additional questions like “Why do you want to study the prices of corn?” and then “What else do you need to know besides the price to look for a pattern?”  I have found that walking a few students through the process gives the whole class a better sense of what they’re thinking.  I make a point of selecting students for this first step that are comfortable being the focus of class scrutiny.  After several students have shared their reasoning, I ask if there are any students who need further clarification of the process of choosing data.  I answer those questions then tell my students they will be working with their partner for most of today’s lesson.

• 12 minutes:  For the first part of this section of the lesson, I have time reserved in our computer lab/library which is only two doors down from my classroom. If you don’t have that option, you might want to collect data yourself before class or assign the data collection as homework. Now that each team has selected some commodities to research, I tell them they will have ten minutes to find data about one of their selected commodities to help them answer their questions and make predictions. I remind them that they have just discussed why they chose which data to find and that they need to keep their “eyes on the prize”, that is being able to use the data to answer questions and make predictions about their commodity. I give each team the web address for the USDA (it’s in my Data Sources resource)to get them started, but encourage them to find other sources if they choose. As my students work on the computers, I walk around and offer suggestions and assistance as needed. (MP5)
• 8 minutes: Your students will need graph paper for this next activity. When we return to the classroom I tell my students they will be plotting the data they’ve collected first on graph paper and then on their graphing calculators. There is always at least one student who asks why they have to do both and I explain that they will be marking the paper plot in a way that they can’t mark their calculator, and then using their calculator to find a good model for their data. I do not allow them to have their calculators out for this section, because I want each student to have the opportunity to experience plotting and then marking their data. I ask for a volunteer to explain what data they will be plotting on which axis and why. Again, the focus of this lesson is on why choices are made as they are, so if there are no volunteers or if the responses are less in-depth than I’m looking for, I ask leading questions, like “Which variable are you going to be predicting?” or “Which variable is dependent on the other?” After my volunteer gives his/her reasons for assigning variables to the axes, I ask each team to take a minute to determine what they will do with their chosen variables. (MP6, MP7) I ask if there are any questions then tell my students that each partner needs to graph the data so that they can both try different models in the next step. I walk around while they work, looking for students/teams that are having difficulties and listening to the discussions about what they’re seeing as they plot their data.  
• 15 minutes: Your students will need their graphing calculators for this section. The final part of this activity allows my students to explore how to fit a model to their data while at the same time strengthening their competence and confidence in selecting a model. I begin by sketching a plot of data representing wheat prices over time. I then ask for suggestions about what function might best model this data and why. I’ve included a graph of the data with the highest and lowest points marked and a copy of the data that was used in my resources as “graph” and “wheat prices data”. I generally don’t get any ideas or the students who make suggestions can’t articulate why they’ve chosen the function they have. To help my students understand that the process is neither random, nor in some way reserved for those who ”get” math, I select a student who I know is not considered to be a math whiz by her/his classmates to help me through the steps.  I make sure that the student I select is comfortable enough with me to stand in front of the class for this step.  Selecting a student not seen as mathematically talented by the other students bolsters their sense of ability and strengthens the confidence of the student selected. As we look at the board as a class, I ask my student “assistant” to mark all the highest points with a colored marker. I then have him/her mark the lowest points.  When the task is completed, I tell the rest of the class that they should now repeat that same process on their own graphs, marking highest and lowest points. I allow my assistant to work with his/her partner for this activity and walk around while the teams work on marking their graphs. When all are finished, I select another assistant using the same criteria as before to join me at the front board. I explain that these steps will help them determine what function might be the best model for their data. I have my assistant write the approximate distance between each high point and between each low point. These distances all should be similar and represent the period of the function, but rather than telling my students that, I ask them if they see anything interesting about these values. If necessary I ask additional questions until someone suggests that they are all similar and seem to be part of a repeating cycle. (MP7) Then we can determine that they represent the period of the function. Next, we look at the distance each of the high points is from its nearest low-point neighbors. I ask for suggestions about what this distance represents and help my class decide that we are looking at twice the amplitude of the graph. (If necessary I remind my students that the amplitude is the measure from the center line to the maximum or minimum, or half the distance between those two points.)  I have my assistant again return to her/his partner and tell my students to determine the period and amplitude of their own data. As they finish this task, I select another assistant and have this student mark the point nearest the y-axis and the first point that intersects or is very near to the x-axis. I ask what these points represent and ask additional questions to help my students conclude that these help them determine which trig function their data most closely fits and whether or not there are any phase shifts evident. Finally, after all the teams have reviewed their own graphs for starting points and shifts, I tell my students they can use the information they’ve just gained and their graphing calculators to find a graph that seems to model their data. I walk around as they complete this final step in creating a function to model their data.  (MP4)

There is a video narrative that supplements this section of the lesson in my resources.  In order to cement this lesson for my students, I give each one a notecard and ask them to write a brief summary (three to five sentences) of why they selected the function they finally chose. (MP3) I also assign as homework the challenge to come up with two different conclusions/predictions that can be made about their commodity and supported by both their data and the function they chose to model it.