Conjectures about sines, cosines, and tangents

Today's focus was on working through the conjectures made on the second page of the Relationships with Meaning packet. Through my observations of and discussions with student, I knew there were some misconceptions out there. I wanted the class to uncover the truths of the given conjectures for themselves, as best they could.

After giving students a couple of minutes to share what they had concluded over night, I wrote the numbers 5-13 on the board. I then went around the room and asked different students whether they thought the conjecture for that number was true or not. I didn't ask for any explanations or make any comment as to the correctness of the conclusion. I find students are more willing to share if there isn't any immediate judgement being placed on their work. Oftentimes, even students who normally don't want to say anything in class are willing to share a simple answer.

After writing either a T or F next to each number, I opened the floor up for discussion. Almost immediately, students wanted to discuss the problems. We worked our way through the conjectures. There were some good explanations about why a statement should be true or not. As we progressed, there were a few statements that baffled students. In particular, conjectures 7, 11, 12, and 13 showed students were not in agreement or were totally incorrect in their thinking.

I had anticipated from yesterday's work that some of the conjectures may pose a problem. I had copied a trig table for each student's use and passed these out. I briefly explained how to use the trig table and then asked students to try out the conjectures in question.

I did have to walk around and explain to some students how to make use of the table in more detail. I also had to watch out for students misinterpreting conjectures 11 and 13. I found that this assistance and using the trig table made the squaring of sine values much more understandable.

Students readily could see that the sin(A) = cos(90^o - A) and that sin²(A) + cos²(A) = 1. There was still a bit of confusion on the meaning of sin²(A) = sin(A²), but I was able to provide concrete explanation using the trig table.

We were running out of time, but I asked students to verify conjectures 11 and 13 if they hadn't already done so. I also asked students to complete the next problem about calculating all the trigonometric ratios when a triangle has sin(30^o) = 1/2. I also asked students to practice solving the four equations and proportions that followed.