Modeling with Trigonometry
Lesson 8 of 10
Objective: SWBAT model with trigonometry. Students will understand how right triangles may appear in real-world contexts.
The lesson opener is intended to be used as a formative assessment. If students, working in teams, are not able to correctly solve for the missing side lengths of both triangles, I am prepared to give some remedial instruction. Key mistakes to look for: students fail to identify the correct trigonometric ratio to work with; students do not know when to multiply or divide a given side length in order to find the unknown side length (possibly because they have resorted to recognizing patterns in the problems instead of trying to understand the definitions of the trigonometric ratios.) This is the point where I am ready to demonstrate a ‘preferred method’ of solving right triangles, using Example 3 in the Properties of Sine and Cosine notes (uploaded with the Properties of Sine and Cosine lesson) (MP5).
This activity follows our Team Warmup routine, which is described in my Strategy folder.
While students are working on the lesson opener, I complete administrative tasks. These include taking attendance and noting which students have not completed their homework or brought required items to class.
Modeling with Trigonometry
Students work in teams to answer a set of real-world problems (i.e. word problems) that can be solved using trigonometric functions or inverse functions. To encourage students to discuss the mathematics involved and to check each other’s work for reasonableness, I use our Team Jigsaw routine, a cooperative learning format. In this case, positive interdependence is created by assigning individual roles to students. One student, filling the role of ‘Picture Master’, is responsible for interpreting the problem by drawing a picture to represent the situation and labeling it with all the information given (MP1, MP4). I tell students to forget about depth, texture, shading, etc.; instead, they should make clear diagrams so that the right triangles really ‘pop out’. The other student, filling the role of ‘Math Master’ observes, offers hints or encouragement, and asks a question if his or her partner’s drawing does not make sense of seems to be in error (MP3). The Math Master then takes the Picture Master’s drawing and uses it to set up an equation or set of computations in order to solve the problem. Teams can only earn full credit by following the ‘rules’, which are displayed using the slideshow.
As teams work, I circulate through the classroom observing what they are doing. I look
out for the following common problems:
- Students misinterpret the problem, as revealed in their drawing. I make sure that the student is really working with his or her partner. If they are both confused, I walk them through the problem, pointing out clues in the wording.
- Students may set up the problem incorrectly. I explain that, to solve for an unknown side length, it must be compared in a ratio with a known side. That ratio must be set equal to one of the three trigonometric ratios in a proportion. I review with students how to determine which
trigonometric ratio to use (perhaps using ‘SOHCAHTOA’) and how to set up a valid proportion.
- Students may solve for the unknown incorrectly. I stress that, once they have written the equation correctly, the rest is applying rules of algebra. Referring to a proportion that has been set up correctly (perhaps with x, the unknown, in the denominator), I ask the student how he or
she would solve that sort of problem. Many of my students tell me that they would cross-multiply. I normally discourage cross-multiplication, but in this case I refrain. I want students to make the connection between solving a triangle using trigonometric ratios and solving proportions, so I encourage them to do whatever they would normally do to solve a proportion.
Before class: I print the resource for the activity. I make one copy for every two students and
cut into half-sheets. I generally make a few extra copies, in order to allow students to start over with a fresh sheet if they go down the wrong path.
I display the lesson close question on the front board using the slideshow. I have the students brainstorm in pairs, then in teams, before writing their answers in their learning journals. The purpose of the learning journal is to encourage students to reflect on what they have learned (as well as to provide individual accountability). Time permitting, I also ask one student from each team to write a team answer on the white board. This gives me immediate feedback on what students learned from the lesson.