Another area of struggle on my PreCalc midterm is inverse trig functions:
Students had a completed unit circle (radians, degrees and coordinates – they earned them on the exam by completing one on their own the previous week) but could not use a calculator for this section. What types of mistakes do you think students would make?
Feel free to check out the exact mistakes my students made at the Math Mistakes blog.
What I noticed:
While inverse trig was a struggle all year, I was concerned how many students over-thought . When my co-worker and I wrote the midterm we thought that question would be a gimme, but student after student evaluated (to varying degrees of success) and then found the arcsin. It didn’t occur to me to include many of the correct answers I saw (π/6 and even 17π/6!) on my answer key but luckily I was paying attention when I saw the first one and added solutions to my key as I found them. While 17π/6 isn’t a mistake, I am curious why students didn’t notice that the sine and arcsine cancel.
For the next problem I anticipated students would struggle since inverse tangent isn’t easy to read directly off of the unit circle. Not only do they have to recall that tangent is , but they also needed to be able to simplify a fraction within a fraction. The red flag was students who didn’t recognize that there was an intermediate step of simplification; instead they expected there to be an angle where sine=1 and cosine= (apparently I didn’t get a picture of that mistake, sorry). The mistake I did photograph was an interesting one though. The student recognized that the -1 meant inverse, however they misattributed it to the inverse of the ratio rather than the inverse of tangent. That shouldn’t be a mistake I see on the midterm though. Especially after explaining countless times that for tangent the input is an angle and the output is a ratio, while for arctan the input is a ratio and the output is an angle.
On the final problem I saw good attempts and students who couldn’t solve it generally forgot that triangle trig was an option. I think those issues were more indicative of a lack of studying than a misconception. Or maybe an inflexibility in their approach? I’ve certainly seen students start a problem only to find that they can’t solve it. At this point they call for help assuming they made a mistake somewhere. Many times their work is accurate and what they actually need is to consider a different method. They could do with a good dose of productive struggle.
Requests for feedback:
I am continually surprised by how much difficulty students have doing anything backwards. In geometry we will study angles formed by parallel lines until they can fill in all 8 angles without a second thought. But then when I give them angles and ask if the lines are parallel, they look at me like I’m asking them if unicorns have purple stripes. The same thing happened here, students are great at finding values of sine, cosine and tangent if I give them the angle. But once I switch it up and ask for the angle instead, they start giving nonsensical answers. What is it about working backwards that students find so challenging? How do you help students recognize the structure?
p.s. Do unicorns have purple stripes?