Angles of Elevation

I just finished a great project with my kids in geometry so thought I'd share. After watching fellow teacher Mark Furtado create clinometers with his class, (thanks Mark!) I wanted to do the same. Click here to see how to create one. However, I knew I didn't have enough time to have kids partner up for a project...AND I wanted the kids to find out something amazing. Our school is located in Northern California (East Bay) near Mt. Diablo. Although Mt. Diablo is relatively short, the surrounding areas are flat...so it can be seen from long distances. Here's a picture of our school with Mt. Diablo in the distance. I wanted the kids to figure out how far away the school is from the center of the base of the mountain.

I did a quick lesson explaining angles of elevation and depression (note that the angle of elevation is always inside the triangle and the angle of depression is always outside...thus its depressed state at being left out of things:)..). I borrowed 3 clinometers from Mark, gave the kids a quick demonstration and sent three teams of two out on campus to determine the angle of elevation from the school to the top of Mt. Diablo. (One group asked if they could stand on the roof of the gym...umm...no...you gotta love high school kids!) An important thing to remember about a clinometer is that the horizontal distance will show 90 degrees on the protractor. So, students must point the straw at the top of the mountain and note the degree where the string is hanging, then subtract from 90 degrees to find the angle of elevation. (I tell the kids it is like measuring with a broken ruler...just find the difference between the measures.)

So three teams went out and came up with angles of elevation 5, 37, and 40 degrees. Quite a variation but that's real life. Which to use? Because two teams came up with close to 40 degrees, we tried that first. We used difference in elevation to get the perpendicular height. Here's a sketch:

So, we did the trig but the answer didn't make sense. I think the students came up with about 3.5 miles and they knew we were much farther away. I asked them how we could check our answer to see if our answer is reasonable and they told me to use google maps. However, they soon realized the driving (or even walking) distance from the school to the top of the mountain was not the straight distance. I introduced them to the phrase "as the crow flies" and printed out a map. Then one of the kids used a ruler and the scale model given on the map to determine the straight distance. Here's our map:

Students found how useful proportions can be and converted 15.5 cm to miles. So, the straight distance should be about 10.3 miles.

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Back to our original problem...we knew our calculated distance was too short. Was the angle of elevation greater or less than 40 degrees? Well, it should be less so we could get a longer distance. So, we went with the 5 degrees that one of the groups had come up with. Lo and behold, we got to 43,742.81 feet or about 8.284 miles. That was pretty close! Their was a lot of "I told you so" from the pair who had measured 5 degrees and rightly so. (The other groups good-naturedly said that it was pretty cloudy.)

The bonus came when a student piped up and asked what the degree measure should have been. Hmm...how to find the angle when two measures are given...ah yes...inverse tan!

Classroom Management:
So, were all the kids working equally on this activity? No, in fact, the whole class was working in groups on an angle of elevation and depression worksheet. I began the activity by saying "hey...let's use trig to find out how far we are from Mt. Diablo! I need three teams to find the angle of elevation." Immediately, my most active kids jumped up and begged to go outside. I gave them a quick refresher on how to use the clinometer and they listened closely. I also told them they had 5 minutes to get the measurement. So, these 6 kids were invested. I started talking about elevation and one student googled it while another argued he just finished taking numerous weather measures in physics and KNEW the elevation was 37 feet. So, we went with his measure rather than the googled elevation. Now two more kids were in. I wondered aloud how we could find the actual distance and quite a few said to google it. Now they are invested...(remember...those who weren't really interested were still working in groups...I have to say...they are really a great group of kids.) It was great having the google map up so they could see the winding roads for driving. At least they could tell the straight distance would be less than 26 miles!

What I think I liked best about this activity is that students were allowed to go in and out of it as they wanted. Some students stuck with it the whole way through...others just wanted to complete their classwork. But either way, they had a choice and they were all doing trig. We'll see how this goes next year!