Mis-representations of solving right triangle scenarios

Well, today was an interesting class. I started by having two different students draw their representations for the first problem of the Solving Right Triangles packet. I had two different students do the same for the second problem. Once these were up on the board, I asked the class to comment on or to ask questions about what they were seeing.

I waited, but nothing was forth-coming. I asked students to focus on the problem 1 representations. I wondered aloud why the wall heights were different for the two drawings. Students were a bit confused about why the drawings were different. Finally, a student said they thought the side opposite the 65^o angle would be smaller. We then had a discussion about which side should be longer. I asked students how they could determine this. There were a few suggestions but finally someone said that the tangent represents the slope of the side and that a larger angle would have a bigger slope and therefore would be taller. It was a nice, mathematical response that the class could agree with. We also discussed the labeling of opposite and adjacent sides as the two drawings had the labels flipped-flopped. This difference was quickly resolved by the class.

We then moved to the second problem's representations. In both cases, the drawing indicated that the hypotenuse was 15' long. A few students questioned the drawings and a student drew what they thought the scenario looked like, showing the 15' length as the horizontal leg. Many students agreed with this representation and we had a brief discussion about why this drawing worked.

I then had different students come to the board and provide their solutions to each problem. There were some questions about the labeling of opposite and adjacent sides, about which trigonometric ratios were most appropriate to use, and about whether the calculations were correct. We were able to resolve these issues without too much difficulty.

At this point I asked students to label their opposite and adjacent sides for the remaining problems and work through solutions. Students still struggled with some of the representations, especially the third problem. They also struggled with actually performing the algebraic calculations necessary to derive a solution.

For problem 3, the issue was which length represented the 6 mile stretch. The first reading might lead one to believe that the horizontal length is 6 miles. I'll admit that was what I initially thought. But upon reading the description again, the road segment is 6 miles, which is the hypotenuse in the representation. This was confusing for students.

I asked students to complete finding solutions for the remaining problems. We'll go through these solutions next class and then proceed with working on the next set of problems.