Working with angles of elevation and depression

Today's class started by looking at the representations that students had created for the three scenarios given on page 3 of the Finding the Value of a Relationship packet. I had two representations drawn for each scenario using a different student for each drawing.

We started with the first scenario. Both drawings were good representations. I asked the class the similarities and differences they were seeing and if either drawing represented the described situation. Most students agreed that both drawings represented the scenario and that there were only minor differences (such as the orientation of the ladder leaning to the left or right) between the two.

Satisfied that the class felt this was a good representation, I asked the class how high up the wall the ladder reached. Students pondered this for a few moments. I then heard students saying that the height had to be less than 10 feet since that was the length of the hypotenuse. Others started using the Pythagorean theorem to find the third side length. Overall, I was pleased from the responses and discussions I was hearing.

From here, I asked what angle the ladder formed with the floor. This question baffled students for a bit. I asked the class if they could use any trig ratios to help identify the angle. Slowly, students started considering sines, cosines, and tangents. Students started calculating the ratio values but weren't sure what to do with these. I reminded them to use their trig tables. At this point many were able to determine the angle measurement created by the ladder and floor.

We then move onto the second scenario. The drawings were not as complete or correct as for the first scenario. We discussed what students were seeing. One drawing showed 3000 feet as the horizontal distance while the other showed the 3000 feet as the altitude. It took a while and some guidance to create a good representation.

I then asked the class who had flown on commercial airline flights and how high airplanes flew. A few students had flown and most of the class knew that an commercial airplane flies at an altitude of approximately 30,000 feet. I told the class that planes would have an initial descent and then go into their final descent. We assumed the final descent started at an altitude of 20,000 feet and that the angle of descent would be the 15^o given in the scenario. I asked students how far away from the airport would the plane be when it started its final descent.

This posed a whole new issue for students. They were having trouble identifying the triangle they should work with and the relationships that they could use to answer the question. After much questioning and individualized help, I had the class moving in the right direction. Slowly, students started calculating their answers. At first, they questioned their calculations as their result was 74,641 feet. This translates to approximately 14 miles. I relayed how real flights feel like they are landing in the middle of nowhere at they make their approach since they start their final descent from so far away.

We next moved to the third scenario. Both representations were good and the class agreed. I then asked students to determine how far away the object was from the building base. Students fumbled around for this but several started to recognize this was basically the same situation as the landing airplane. For those not making the question, I asked them what connections they could make between the two situations.

We closed with looking at the next two problems (#4 and #5) in the packet. I assigned these for homework. We'll start by looking at how this work went at the start of next class.