# Inverse Trig Investigation

This was a fun one. The kids specifically mentioned this as a favorite when I asked if there was anything in particular that made them go "liked lesson, would learn again A+++++" I always like inverse trig because there is a good amount of context you can build with it and not come off sounding like you're making this all up. Prior to this lesson, I introduced inverse trig in the context of the unit circle, what angle corresponds to this coordinate? We now expand the idea to include any right triangle, and how you can determine the value of a missing angle.

### Inspiration

I did not arrive at this lesson idea overnight. There were two motivating factors: teaching inquiry and cutting off the supply of answers. Baby steps with inquiry began with lessons like this. The need to cut off answers grew out of an assignment that sucked. I cannot stress enough the importance of experimenting. Those crazy cool lessons you see online are not born of divine inspiration. Try one little new thing at a time and eventually you never know how those small learning experiences will open the door to something really fascinating.

### Opener

I put these two pictures on the board.

They quickly determine I'm trying to trick them into learning something about triangles (what gives it away?). The second picture draws a lot more interest than the first, naturally. It leads to a very interesting discussion on airports, Gibralter, and how crazy the people on that beach must be. Then I start asking some questions: "where are the triangles?" "is the landing angle of this plane important?" "should the pilot be able to dial it in precisely?" "what do you think the landing angle is?" "could we figure it out more exactly?"

Then we annotate the picture. BUT WAIT! How big is this huge plane in the first place? I field some guesses. A few of them will kind of eyeball some numbers based on the people visible in the picture. This is where we start teaching some 21st century skills. Why children, did you know that the internet will tell you how long that plane is? See this little "747-400" written on the side? Watch. Bam. Then we get into a discussion about how much of the plane is necessary to calculate the landing angle (consensus among all classes was to chop off the tail, without my prompting, and lo, the heavens opened...). A tiny inverse sin function later and we get an answer. We compare the guesses. This is a fascinating exercise is seeing what students THINK certain angles look like. You would be amazed at how many initial guesses were 30ยบ or higher.

Up the abstraction ladder and I draw a couple generic triangles to reinforce the concept.

### Their Turn

Equipment: iPads, notebooks, and pre-written instructions

The next day at every table I have taped these instructions (PDF):

They accompany a set of pictures that were pre-loaded inside a Dropbox I use for distribution:

The questions asked require them to fill in more blanks as they go along. I provided some clues as to what they'll want to research. I made sure to search for the missing information myself so I could intervene in case someone was having trouble finding it. The only big point of confusion was with the airplane scenarios. I found a lot of them basing their search around the name of the airline rather than the model of the plane. Some were trying to be far to direct with choice Google attempts of "what is the landing angle of the Jet Airways plane?" Surprise, surprise those didn't work very well. So for those of you who think our 21st century kids know everything there is to know about the 21st century just because they text message a lot should drop that assumption right now.

Every question was to be answered in their notebook with a sketch. If they had time they were allowed to pick a scenario and annotate one of the supplied pictures.

### Results

This activity took a solid hour or so. Students remained focused nearly the entire time and there was minimal Facebook-ing. If you aren't fortunate enough to have a block day somewhere in your schedule, the scope could be reduced pretty easily. Very little prompting was necessary to get their research headed in the right direction. Very little prompting was necessary to get them to produce a sketch on the iPad. We've done it enough now that a lot of them remember how to import a photo or at least take a screenshot and import that. I had enough iPads on hand for 2:1, and maybe one instance of 3:1. School officials planning iPad rollouts: anything above 3:1 is pointless.

At the end we did a little debrief to compare results. We made sure everyone agreed on the researched distances. We compared the results of the actual inverse trig to make sure everyone was in the same ballpark.

The bigger lesson I wanted them to learn goes beyond inverse trig. They have all these exciting toys around them. People have spent an insane amount of time putting everything imaginable on the internet. But you have to know how to use it. Teaching the difference between asking "what is the landing angle of the Jet Airways plane?" and "what are the dimensions of a Boeing 737-100?" is HUGE. Inverse trig was just a supporting player in the objective of this lesson.

Posted