When Will We Ever Use This

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SWBAT develop an appropriate mathematical model for inverse trigonometric data.

Big Idea

Want a good answer to “When will we ever use this?” This lesson gives students an opportunity to explore applications of inverse trig functions.

Set the Stage

10 minutes

Students will need their graphing calculators. I begin the lesson by reviewing the previous day’s activity relating to graphing inverse functions.  I ask my students to explain in a pair-share why the inverse functions need to be restricted over a specific domain. (MP2) While they’re talking, I walk the room to make sure all the groups understand.  I then tell all my students to set their calculator windows to the restricted domain we’ve discussed (xmin = -π/2, xmax= π/2) and to graph the inverse sine function (y = sine-1(x)).  Some students will need help adjusting the rest of the window dimensions, but I address this by asking my students to help each other get a “good view” of the graph.

Once everyone has the graph set, I challenge them to transform the amplitude of the graph by a factor of 2 (y = 2sin-1(x)).  We’ve done enough work with transformations that I expect the majority of my students to be able to accomplish this fairly easily, but they may need a little direction in re-setting the view window.  Again, I ask my class to help each other while I walk around giving additional assistance as needed. I ask my students for suggestions for a few more transformations of the inverse sine, including changes to amplitude, period and phase, and we try them as a class until everyone is reasonably comfortable with them. 

I repeat this procedure with inverse cosine and inverse tangent functions until my students have viewed at least two dozen inverse trig graphs and can adjust their calculator window as needed. (MP5)I always seem to have one or two students who continue to struggle with determining the window, but rather than holding the entire class back, I set up a time to work independently with these kids and for this lesson pair them up with someone who is competent at the skill and also patient and mature enough to help.

Put into Action

25 minutes

There is a video narrative that accompanies this section of the lesson and addresses some of the pedagogy. You will need copies of the Problems handout and graph paper for this section.  I tell the students they will be working in teams of three to complete this activity, but that each student needs to complete all the graphs and written work.  I choose to have them work in teams of three rather than independently or in pairs because this activity asks the students to develop models for real world data which is fairly challenging.  I’ve found that putting my students into teams for this gives them enough support to preserver in solving these models.  I give each student a copy of the handout and explain that their challenge is to find the best model for each set of data, record the model as both an equation and a graph, and explain why they think their model is the best fit. (MP1, MP3, MP4) I also say that team members do not necessarily have to agree on the models they choose, but they do have to be respectful in their disagreements. I walk around checking progress and giving encouragement and assistance as appropriate.  Some students are really enthusiastic about tackling problems like these but others are somewhat intimidated about how to begin.  
My seating groups of three are generally set up to be heterogeneous in terms of both aptitude and attitude, so I don’t have to do too much encouraging, but when necessary, I try to encourage either by asking leading questions that they can easily answer or by reminding them of a similar challenge that they successfully completed. The biggest hurdle is usually figuring out what to do with the data to find x and y values.  
After about 15-20 minutes I advise the students that they should be finishing up the first challenge and beginning on the second.  I continue to walk, observe and encourage until all the teams are done or until we are down to our last five minutes of class.

Wrap it up

5 minutes

I use notecards as a part of this closure activity, but you could also have students turn in their responses on their own paper.  As teams finish, I ask them to be ready to share their results with the class via their student whiteboards (I have a classroom set cut from shower board) All the teams post whiteboards with their equation for problem #1 and for problem #2 around the room so the entire class can see and compare the results.  Because there are multiple equations displayed, those teams that did not quite finish can still participate in the viewing.  I also have them turn in the work they’ve completed along with everyone else, so I can carefully review what they’ve completed and what stumped them.  Sometimes the problem is with my groupings, which becomes evident as I walk the room, other times it’s simply confusion about this specific lesson.  If that’s the case I make sure to connect with that team or teams that need additional instruction.  As the final piece to this activity I give each student a notecard and tell them to review the equations and look for similarities and differences for each of the problems.  I then tell them to select either problem #1 or #2 and to write an equation that combines the similarities they’ve noticed into one “optimum” equation on the notecard, along with their name, and turn it in.  (MP4, MP7)