So today I had this experience where this precalculus test I gave was a bit of a bloodbath. Not for everyone, but for more than usual. In a way where I cringe, cry out to the high heavens, and scream:


The reason is because I felt pretty proud of the way I have been introducing the material. You see, in precalculus this year, the kids are coming up with everything on their own. I don’t give them anything.* And thus far they’ve been doing well with this. And during this unit, even though I didn’t quite have the same amount of time to create everything to my best ability (I relied a lot on the textbook for this stuff), I felt pretty confident about my kids’s understanding.

So I have to wonder: Where did I go wrong as a teacher? What was different about this unit than the others?

First off, this unit was some pretty heavy stuff. We were deriving and applying the trig formulas, and then we were solving more complicated trig equations (they had done basic trig equations previously). All in all, we took a total of 8 days to do this. I should also note that this is an advanced class, and they have been doing a lot of collaborative work this year.

FYI: these were the trig formulas we derived and applied… this is the “trig formula family tree” I made for them.

And for this unit, I led class in a pretty routine way. Each day I had a packet for the kids to work on. They would work on the warm ups with their groups (which were designed to activate prior things they knew but forgot, and have kids make some connections on their own). After 5-10 minutes, we would all talk through the warmup problems together.

Then I would let each group work on their own. I would walk around and facilitate, nudge, question, and answer questions. On some packets, I would have special places where I told kids to “draw in a heart, and call me over when you get to the heart.” (But to be honest, overall, I think I was throwing myself into the groups less than I usually do this unit, as I’ve been trying to let go.)

Then class would end. Most groups were where they should have been… close to done with the packet, and ready to start working on the book problems. These book problems varied in difficulty from the routine “can you do something simple?” to the “okay, apply this in a moderately deep way.” For this unit, I did assign more nightly work than I normally assign, because I knew that to get good at this stuff involves a lot of practice. (I don’t think that is true for everything in math, btdubs.)

Then at the start of the next class, I would have one set of my handwritten solutions per each table (that way, three kids have to share, and thus talk!). I would give kids 5-10 minutes to compare their answers, talk with their groups to figure out things they were doing wrong, and then we would come together as a class and I would field questions that groups couldn’t answer. Then we started a new packet, and the process continued like that for most days. [We did have a bit where we did a paper folding activity, which was pretty cool.]

To see what these packets look like, I combined all of them here so you can scroll through them. I highlighted some of the problems/questions which I thought were good at getting at something hard/interesting/conceptual:

As you can see, these aren’t really great. Not bad either, though. [1]

So where did things go wrong?

When I look through the tests, here are some things I noticed as a trend:

  • Kids struggled with some of the basic “apply the formula” questions
  • Kids had trouble figuring out which sign to use when using the half angle formulas (e.g. \cos(\beta/2)=\pm\sqrt{\frac{\cos\beta+1}{2}}) [2]
  • Kids really nailed the conceptual explanation part of “how many solutions does this trig equation (e.g. \cos(24\theta)=-1) have on the interval $\latex 0\leq\theta<2\pi$?” question
  • Kids struggled with remembering that when you take the square root of both sides, you get two solutions (so \sin^2(\theta)=1/2 is really two equations to solve)
  • Kids did a pretty good job of deriving the trig formulas
  • Even though kids did a pretty good job on the “how many solutions does this trig equation have?” they didn’t find all the solutions to the basic trig equations given.

As far as I can tell, here were the contributing factors (in no particular order):

(a) Lots and lots of sickness. I still have 5 kids who haven’t taken it (out of 19).

(b) I thought I was getting formative feedback when I gave regular little mini non-graded “do you remember the trig formulas we’ve derived” at the start of some classes…


And honestly, I felt proud that I have been making a conscious effort to collect this formative feedback. But now I see it wasn’t the right formative feedback.

(c) I usually get a good amount of formative feedback in Precalculus. Mainly I do it by collecting of the nightly work, marking it up, and handing it back. Thus I usually know what students are understanding and what they are not, and they also know what they understand and what they don’t. However, because I was swamped, I didn’t really do that. Maybe once in eight days? So each day, kids got to compare their own work to my solutions, which I thought would at least give THEM feedback… But I never got to see what kinds of mistakes they were making, or where they were getting tripped up, not in detail and not in a big-picture way. So I didn’t build these things into the lessons… which is important because…

(d) This material is hard. Harder than some of the previous units/ideas. That’s because this unit required conceptual understanding, juggling a lot of memorized formulas, a bunch of intuition (as to how to start solving the trig equations), and a lot of “fact” information (like where in the unit circle is \sin(\theta)=-1/2?). It’s just pulling a lot of stuff together.

(e) I should have spent more time reminding them of the trig equations they had previously solved. I assumed that they remembered all of that stuff we did weeks ago and could apply it. I jumped in too fast.

(f) The test was a bit too long. The kicker is, I thought it was too long, so I cut some stuff out. I was trying to be conscious of that. Well, the road to hell…

So there we are. Surprisingly, typing this out has made me feel a lot better. I feel like I now have a better grasp on why something I thought was going pretty well was actually not going as well as I thought. I also have some concrete ideas on what to do next year. The main takeaways for me are: go slower, bring in more visual understanding for trig equations, don’t mess around with the harder stuff, get a lot of formative feedback on the basic types of problems, and make the assessment shorter than my intuition tells me.

*Okay, to be fair, I have given them two things — one which we proved later, the other which I never proved. (The former was the sum of angles formula for sine and cosine, the latter was Heron’s formula.)

[1] It was a stressful time when I was doing this unit, and so I just didn’t have time to come up with anything better. But still, I think they get at good stuff. Even if there needs to be A LOT MORE GRAPHING next year. We did a lot of graphing when we did basic trig equations. We should have done graphing here too.

[2] The kicker is that I said in class that figuring out the correct sign is the most important thing about applying that formula. Multiple times. But me saying it until I’m blue in the face the same as them totally understanding it. Next year I need to build in some warm up questions like: if \alpha=200^o, what quadrant is \alpha/2 in? Draw a picture. If \beta is in the fourth quadrant, explain in words and with pictures why \beta/2 is in the second quadrant..



  1. Awesome analysis of the unit–thanks for sharing all of that with us, Sam! I think those packets are better than you give yourself credit for. (I especially like the hint “don’t read past this… don’t read past this…”). I don’t know how you found the time to get all of that material together, but it sounds like a great way to do the unit because it involves them problem solving at a deep level for about as much as students can in such a short time.

    My students did much better this year than last year on the trig identities unit, but there are some significant differences in what and how we taught and I’d like to reflect on those here if you don’t mind.

    (1) We didn’t get nearly as far. I would love to go deeper with the students, but the way I taught it (mix of lecture, group work, and random activities that try to hide the fact that they’re actually problem solving trig identities) doesn’t seem nearly as engaging. Would you say that most of the students were engaged for most of the time? If my students were into the problem-solving process, I would have taken them much further, but it felt to me like there was only so much abstraction they could take, and you can see them starting to resist and tire (even the good students, of which I have many!).

    (2) I did not make them memorize hardly anything. I would give them a cheat-sheet and they could always use their calculators (not that that helped them on the proofs *evil laugh*). I totally agree that this is hard, but I wanted to focus on the problem solving. And yes, I know that they will be better problem solvers, at least when it comes to figuring out trig ID proofs, if they have the necessary information memorized and ready to be used, but when there is SO much being memorized, I just don’t know if it helps over having a sheet of paper in front of them. (Perhaps part of the problem is I didn’t enjoy memorizing as a students, even though it was essential for so many classes… I shouldn’t let that affect the way I SHOULD teach, though.) What do you think? Why do you have them memorize those formulas and the Unit Circle? (I’m assuming you do?)

    (3) I tried to use a lot of different approaches to getting them to think creatively–or at least divergently. One thing I attempted was to have them make “web diagrams” where they start with the expression in the center and they write down all the equivalent expressions they can think of, branching out in different directions. After that, they choose one they would like to pursue. Sometimes they end up at dead-ends, but sometimes, if they can think of enough possibilities, they manage to make it to the end. I realized that this is what I do in my mind, and so I thought they should get to try visually and see if it helps them. I think it helped some of them, but the downside is that it is such a tedious and time-consuming process, especially writing everything down, that they end up spending a whole class period on one or two problems (yuck!).

    (4) I confess that I’ve sometimes given problems very, very similar to ones we did in class. Not a practice I’m particularly proud of, but if something can boost their confidence to get them to really think about the harder problems, then it might be worth a try? I was sorely disappointed, however, with how poorly students would react/interact with a truly new type or problem: perhaps that’s where I should turn more to your style of teaching this unit. Of course, I don’t feel like I even have the time to read through all of your worksheets and really analyze how I should approach teaching them, let alone create them as you did. Props for being swamped and yet providing so much for your students.

    And yes, I agree that this unit is HARD. I start off this unit each year (well, for 2 years now) by saying “some of you who were good at math are now going to suck at this and will have to work hard, but some of you who were bad at it, will do really well at this type of math!” I emphasize the 2nd part for those students who feel like failures in math in the past, and I think it works for a few students each year.

    You’ve inspired me to work harder at getting farther through this unit, so thank you for that, Sam. I’m not going to thank you for all the extra-hard work I’m going to have to put into the class next year, but I know I’ll appreciate it when it’s all over, and hopefully the students will.

    Thanks again for your reflection! Oops, now I went and wrote me an essay so long nobody will read…

    1. Hihi! Thanks for the long response!

      I should probably say that this was not our trig identities unit. We had done trig identitites, and this was the “trig formulas” unit… and I threw in a few trig identities involving the formulas. But it mainly focused on trig formulas, application of trig formulas, and solving advanced trig equations.

      I did make them memorize the trig formulas. I have heard (and I agree) that the half angle trig formulas are archaic and I think (if I can convince the teacher next year, I will eliminate it). But I think it’s important that they can memorize, derive, and use these. Why? Well, my kids are advanced kids. And some of the nicest things that can happen in calculus is when you can solve some crazy integrals simply by a simple trig observation. But I also don’t see the point in having them use formulas if they don’t know where they come from. Then math is just magic.

      When I do trig identities next year, I am going to do this: “One thing I attempted was to have them make “web diagrams” where they start with the expression in the center and they write down all the equivalent expressions they can think of, branching out in different directions. After that, they choose one they would like to pursue.” I might even have them pick one direction and explain in words why they think it’s worth pursuing, and pick another direction and explain why they think it will lead to a dead end. So they can articulate their intuition, and see if it worked or not.

      “I confess that I’ve sometimes given problems very, very similar to ones we did in class.” Heck – on my test that they bombed, I gave students TWO identities that they had done for nightly work, and gave them the choice of completing one. (So if they got stuck, they could do the other one.)

      Thanks for your thoughts and commiseration!

      1. That’s reassuring about “giving them similar problems”. I still feel a little guilty about it, probably because way too much of my test was just copied/barely altered from previous problems.

        I really like the idea of “discussing why they like that direction”. I’ll mention my reasons for choosing that path a few times, but to have them discuss and dialogue avoids the “right thing said once” problem (

        It’s also interesting that you said this is a separate “trig formulas” unit. Does it come right after the trig identities unit? Or is there a gap of time (and other therefore information) between the units? And how did teaching the trig identities unit differ from the trig formulas unit? Do you think you’ll teach the trig identities unit more like this next year?

      2. Our last unit included trig identities. This unit started off with coming up with trig formulas, and then doing some more identities with the formulas.

        I hopefully will come up with a better way to teach this stuff next year…

  2. Last year, due to a scheduling issue I was asked to assist in teaching a PRecalc b class with almost 50 students. I had never taught the material before, so I let the other teacher take the lead. (I took the lead the next topic)

    Trig identities is where we started. More so than most topics, trig identities are not necessarily solvable in just one best way. We would pick a couple a day and run a challenge as to who could solve it in the fewest steps. It was fun, and the kids then saw many ways to try these problems.

    This is definitely a topic where the sage on the stage needs to exit. Give answers for some, but not all. We had Home Depot whiteboards all around the room and when the kids wanted to know about a problem, we’d have 2-3 groups put their work up and spitball who got closest and what flashes of inspiration existed in each of them.

    Right now I’m teaching trig in algebra 2 (which in my school is the first time they’re seeing it, as geometry is their next course).

    Thank you… (Cause you rock!)

    Scott Hills

    1. I love love love this idea… “We would pick a couple a day and run a challenge as to who could solve it in the fewest steps. It was fun, and the kids then saw many ways to try these problems.”

      I think the multiplicity of solutions is something that I need to highlight. Like see how many fundamentally different ways we can solve some rich identities…

  3. Hi Sam,

    Every time I think I’ve found a new way to assess and teach that doesn’t require me to collect work regularly to look at between classes, I get burned in the way you describe in this great post. There is just too much good information that comes from my looking at their written process and interpreting what they do and don’t understand. Even with small skills quizzes and moving fully to SBG this year, I still get a lot of insight and ideas for tweaking the class to match their abilities seeing their work.

    I still push Wolfram Alpha, Geogebra, and each other as great ways to get feedback and hints when they get stuck, but there is sometimes no substitute for a teacher with experience looking at your work. Furthermore, there is nothing as fulfilling/enlightening to growth as a teacher as taking the time to dig into student work. While having more time to do other things is sometimes worth giving up the non-trivial time it takes to collect, comment on, and turn back work, I haven’t found anything to be as useful for helping students progress. Darn you, moral imperative!


    1. You know, this is the first year that I’m collecting work and marking it up. I have to say, we all talk about formative feedback (for the student and teacher) and I’ve found this to be the best feedback for me! It’s so so so much work (darn you , moral imperative! seconded), but I’m finding it insanely useful.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s