Modeling angles of elevation and depression

Today was one of those frustrating days. It was evident that students hadn't looked at any of the problems over the weekend. In addition, the class just wouldn't settle down to think about or discuss problems.

I wanted to start by looking at the results for work on the second page of Finding the Value of a Relationship. This proved futile as students hadn't made their lists for problem 4 and had simply plugged in random degree measurements for problem 3.

We revisited that you needed to use the information given to help you answer the question. I drew and labeled the triangle and asked the class what trig ratios could be formed from the given information. Finally, someone mentioned that the tan(α) could be calculated using the ratio 12/8. I then told students to use their trig tables to find the angle that produced a tangent ratio of 1.5. The angle is approximately 56.5^o. I then asked, knowing this, what the measure of the remaining angle would be. The non-response was overwhelming; you could hear crickets chirping, it was that quiet. Finally, a student questioned that shouldn't the sum of the angles sum to 180^o? At least about half the class recognized that they should use Pythagorean theorem to calculate the third side length.

No one had bothered to make a list for the fourth problem, so I decided to move on. I might re-visit this a bit later but I didn't want to take the class time. I had hoped to use the lists to refine student thinking about trig ratios. However, since their thinking this morning was rough, at best, it just didn't seem like a good use of time.

Moving on to angles of elevation and depression, I had students read the description of the hiking situation for problem 5 and then asked them to draw and label a representation for the problem. Students were totally confused by the description. I had to demonstrate sight lines and the angles of elevation and depression.

I told students to hold one arm straight out in front of them. This was their sight line. I then told students to use their other arm and raise it up. The angle formed between the sight line and the raised arm was the angle of elevation. Similarly, if they lowered their raised arm, the would form an angle of depression. More students had an idea of what the two angles meant.

After a few minutes, I had four different students copy their drawings and labels on the board. I asked students to look over the drawings and comment on the similarities and differences they were seeing. Instead, I had a student ask which drawing was right. I pointed out that that was not the question I asked. I wanted to know what similarities and differences they were seeing among the four drawings.

We had a brief discussion, as there were some distinct differences and few similarities in the drawings. We then revisited the meaning of angles of elevation and angles of depression. At this point students saw that one of the drawings represented the situation better than the others.

We used this drawing to discuss the claims made in problem 5. First, I asked students about the two lines of sight drawn. A couple of students realized that the lines of sight should be parallel. I labeled the two lines as parallel. One of the interior angles was labeled as 23.5^o and the alternate interior angle was labeled as 66.5^o. I asked the class what type of angle pair did they form. A few students recognized they were alternate interior angles. I asked what the angle relationship was between the angles. Some other students said they would be congruent. I then asked why the angles were not the same angle. Again, there was silence. I decided to let students ponder this situation some more. I am hoping that they will come back with the conviction that the angles, indeed, should be congruent.

I asked students to complete sketches for the next three problems. There was about 10 minutes left in class and I thought we might be able to look at some sketches before class was over. Unfortunately, this activity went as poorly as the rest. Students were drawing random triangles, were not labeling what they had drawn, and, generally, made no discernible effort to represent the situation at hand.

I talked to several students and then the entire class about using their sketches to model the described situation. I left these sketches as homework. We'll take a look at the sketches and discuss the situations before revisiting the question about the relationship between angles of elevation and angles of depression.