[2016 Blogging Initiative] Week Four: A Lesson Introducing the Unit Circle

My Relationship with Textbooks: “It’s Complicated”

My first several years of teaching I avoided the math textbook as much as possible[0]. One year I even waited to hand out textbooks to students until the second quarter. I assumed (incorrectly) that using the textbook would make me a lazy, bad teacher. However, at the start of this year I decided to embrace the textbook for the good resource that it can be: a bank of practice problems[1] not a replacement for my teaching[2].

Background: My Classroom 

One other thing I’m doing this year is flipping my classroom. The flip, however, isn’t just lecture. I’m trying to challenge my students do problem solving through the vidoes, and I hope to show how I’m trying to achieve that in this lesson. For one thing, I provide guided notes for the students to fill out as they watch the lesson. I also don’t do every problem: I ask them to pause the video and try some in the middle of the video. To that end, I’m also using EDpuzzle which pauses the video and asks them questions that I’ve created at a variety of levels.

When we get back together in class the following day, the students are randomly assigned into groups of 3 or 4. Students spend about 10 minutes going over the notes and making sure each students’ notes agree with one another and that students understand the topic. After that students work on practice problems, from the textbook, on the same topic. [3]

The Challenge

So we’re chugging along and we get to the Unit Circle. This is the first lesson that I disagree with how Blitzer (our textbook) approaches it. I’ve had success with students in the past by teaching special right triangles first because students see them in the Unit Circle. So I decided to create my own “chapter” and left the textbook, like old times.

The link below is a short (<13 minute) video so you can see what the students will do for HW prior to class. But you should watch it because that’s the interesting part of my lesson. 🙂


Here’s the handout:

As you can tell from the video, I show students the special right triangles and where the values come from. My hope is that they use the Pythagorean theorem if they ever forget the shortcuts in the future, but most students will, unfortunately, probably forget that. I’m not sure how to share that with them differently.  However I only give students a few points from the Unit Circle, and ask them to “figure out the rest”. If they can figure it out on their own before coming to class, and they understand the special right triangles, then I think that it will be more likely that the Unit Circle will stick.

Since I’ve deviated from the textbook here, I had to find practice problems online, but that wasn’t too difficult. Students will go to my website and simply click on the worksheet links (complete with answers) to practice this in class. I’ll only print out the sheets for those students who want more practice beyond class and have no internet at home.

I’ve assigned the video (only 2 students have watched it so far), but we’ll meet in class Monday to see how well they did filling out the rest of the Unit Circle.[4]

Request for Feedback

How can I improve this approach? How can I teach special right triangles in the video so that they do more of the “heavy lifting”?

How are the quality of the questions in the EDpuzzle video? Are there others you thought of that I could do?

Is there a better way to approach the Unit Circle that you’ve seen/used other than special right triangles?

If you could answer any of the questions above, I’d greatly appreciate it. Thank you for reading! (and watching??)


[0] I still avoid it in Physics–I haven’t handed out a textbook in 3 years, with the exception of one student who begged for it. It didn’t help her.

[1] It’s also a good resource for ideas for 3-act lessons.

[2] I’ve seen some teachers teach how to read a textbook, which is a valuable skill, but one that I’ve decided pass on for now. I want my students to understand the math first and foremost. I’m still not sure how I feel about not teaching students to use a textbook effectively and efficiently.

[3] Because I believe that HW is practice, earlier this year (before I flipped), I don’t grade HW. Students also didn’t do the HW (with very few exceptions). Now, I still don’t grade that they watch the video, but I’m not afraid to email or call home if students are missing it chronically. Also when students get to class, they recognize that they’re responsible for learning the material at home, and so will work harder at the start of class to understand what they didn’t watch. It’s amazing how much more “HW” (practice) they’re doing now just because it’s happening during class.

[4] And if I’m on my blogging game, I’ll blog about how it went. Unfortunately it’s tennis season, so I probably won’t find time to soon.



Filed under Teaching

12 responses to “[2016 Blogging Initiative] Week Four: A Lesson Introducing the Unit Circle

  1. Boy, you make creating the videos look easy. I really should start that with my students…
    This is exactly the exposure to unit circle I want my students to have. I do the same exact arrows when I review, except I draw them on the outside of the triangle, and they’re connected to a broader context of using arrows to represent operations/functions, which means I also get to draw an arrow from 45 degrees to 1/sqrt(2) [not inside the triangle; outside] and label it “cos”

    I’ve never really liked the layout of the unit circle with the special angles on it. The 45 is clear enough, but the 30 and 60 you just have to know that’s what’s being represented, and unless you mark every 15 degrees there’s no way to really figure it out. I almost always draw separate circles for each triangle I need, unless it’s in a different quadrant. Then I also draw the first-quadrant version — but with all the other lines, I fear it gets really cluttered and turns into a memorize-placement task.

    • That’s a good idea about separate circles for 30 degree increments vs 45 degree increments. I’ll remember that in the future! And making a 13 minute video + EDpuzzle takes about 2 hours of my time, so it still is by no means “easy” for me 🙂

      I also really like your idea of using an arrow to represent a function. I would have to change a lot of habits because I’ll draw arrows and not even think about it (I had to really think about what you were saying with an “arrow” in that context because I had forgotten that I had drawn them!). I like the connection to the trig functions on top of that, using the arrows. Thanks for reading and leaving ideas!!

  2. When I teach the unit circle, I try to break it down into the three types of triangles, just like you did. I call them a short, medium and tall. I think we fill in that same blank unit circle, but just the one time. I am a big believer in not memorizing the values – too easy to get mixed up. So pretty much any time we need a value we draw a quick sketch in the appropriate quadrant. As they get better at the values, we usually just end up “drawing” the triangles in the air or by tilting a pencil at the appropriate angle. I have had colleagues show a “trick” using their hand and other things, but drawing those pictures (however we do it) always seems to have the best results. Students even comment when they get them wrong that they didn’t draw and that was the reason
    P.S. Love the video aspect and the questions in between allow some checks for them and for you too!

    • Yes, problem solving instead of memorizing–that’s exactly what I’m hoping to achieve! When I was in HS my precalculus teacher gave us a timed test on this so that we HAD to memorize it instead of “figuring it out for each value”. I much prefer our method, though, because it gives the students problem-solving skills that they will take with them even if they never use trigonometry again.

      Thanks for sharing!

  3. Thanks for this. You helped me plan. I’m in triangles now and doing special rights tomorrow after Pythagorean triage. I love what you did for efficiency’s sake. Making the development more sticky is the hard part. This year I am going to try to do a better job of tying in reflections across the axis and rotations to the unit circle. I think that will help get those coordinates in quadrants II, III and IV better. I’m starting to think about 1 circle at a time as suggested by one of your readers, then stack the circles and put a brad in the center to spinORmaybe, 3 quadrants each with their own unit circle and we peak and reveal one special triangle at a time. Ok, O need to get cracking on this. Thanks for the inspiration.
    When I do special rights, I start with Pythagorean Theorem, them move to special rights without the kids realizing it. I may do that via constructions review so they come up with the 45 and 30/60 themselves. rather than handing them the angle. Woah! Crazy talk.
    Anyway, I make the 45-45 and 30-60 painful with integers > 1 and then let the students come up with the “rules” by eventually giving them side lengths of m or z. I do that backwards too with just the hypotenuse and one angle defined.

    • That’s exactly the kind of approach that I would like to do, I’m just worried we don’t have the time to spend it in my class. Since my students are in Precalculus, I’m hoping they’ve seen this once (maybe twice?) before in Algebra II and/or Geometry, and my hope is that those teachers are spending the kind of time that you are on special right triangles: having the students come up with the rule themselves so that I don’t have to reteach it.

      I especially like the “painful with integers >1” comment 🙂

      Thanks for sharing!

  4. Thank you for sharing how and what you use to flip your classroom. I like Edpuzzle, and have never seen it before.

    For special right triangles I like to assign each group of two or three students a length of a side of a square and figure out its diagonal. When they all start posting the sise Times Square root of two, they get really excited. Also, when you get them to reason backwards with a whole number for the hypotenuse, I love that dome never divide, they reason through multiplication from the given side. Same goes for 30-60-90. You inspired me to approach the equilateral triangle as a reflection.

    I think developing the unit circle in class, using a template and either a cut out triangle, or an orientation of reference angle is so beautiful, I wouldn’t want to give it away in a video. The discovery embeds the reasoning.

    Again, admire your invitation and hope to get my 4th challenge written for same kind of feedback!

    • Yes, developing the unit circle in class was so wonderful, perhaps in future years I’ll just make a video with special right triangles and leave out the unit circle until they get to class. Doing it in class I get the added benefit of, once they finish the entire thing, going “Oh, this is that ugly thing that we learned last year! This was so easy!!” That’s selfishly rewarding as a teacher 🙂

      Finding the diagonal in a square is a great way to frame the problem, as it seems more useful and accessible than the super specific “isosceles right triangle”. I never considered how both it and the 30-60-90 triangle are better framed as a very common geometric shape split in two!

      Thanks for sharing!

  5. Wow, the EDpuzzle video was great! How have your students taken to the flipped classroom model this year? And do you feel that this model has led to more/deeper understanding of the material than in previous years?
    Finally, do you make all of your EDpuzzle videos on Doceri? Have you ever used any other videos (Khan, Youtube, etc.) on EDpuzzle?

    Sorry – I will have to think more deeply about the questions you asked for feedback (too excited seeing the EDpuzzle video!). One quick thought in response to your reply to the first comment – this year, I’ve been trying to emphasize both understanding as well as memorization of the unit circle values… So throughout the trig unit, I’ve been giving them short quizzes on quadrants of the unit circle (I basically cut the the circle like you have into four pieces and give it out randomly), but I’ve also been trying to ask a lot of comprehension questions. I’ve been going back and forth between the unit circle (+ special triangles) and the sinusoidal graphs as much as possible to continue to cement those connections, but the students that are continuing to draw and redraw the special triangles are definitely falling a bit behind.

    • Thanks! I think most of the students appreciate the flipped model, even though they’re doing more work (well, they weren’t doing hardly any HW before, so whatever).

      That’s a good question about “does flipping give students a deeper understanding?” I would say not *deeper* for my average student, but I think that *more* students are “getting it” than before. It also allows me to move faster through the material, so I suppose I could use the extra time for deeper investigation if I wanted to go that way. Right now I’m using the extra time for more 3-act lessons, which I suppose could be considered deeper.

      And no, I have not used EDpuzzle with other’s videos. I really want students to use the graphic organizer as they’re watching the video so it’s active, not passive, learning. It’s easiest for me to make the GO that I want them to use and then create a video for that. I also think that using Doceri speeds up the video considerably: primarily because I can write ahead of time and just click “play” on Doceri, and it looks like I’m writing inhumanly fast :). Khan and others write in real time on the video, which slows down the video and takes a lot more time. I think that’s why EDpuzzle first tries to emphasize you cutting the video down to “just the essentials”.

      We haven’t gotten to sinusoidal graphs yet, but that’s something I hope to make connections as you are. I can also appreciate trying to get understanding *and* memorization, though I’m wondering what the end goal of memorizing the Unit Circle is? Are there questions on the AP calculus test where it’s helpful to have the unit circle memorized? If so, I should definitely start having them memorize it for speed as well.

      Thanks for reading and commenting!!

      • I was so pumped up by watching your video that I made one the other day! I’m going to try it for a month to see how it works… My first period class last quarter went sooooooooooooooo slow because so many of them stopped doing homework at home (partly b/c they didn’t understand, partly b/c they were busy/stressed with college apps and partly b/c they just didn’t want to) so we had to do all of the lessons and classwork and homework in class. I’m hoping this starts rebuilding some of their sense of accountability.

        BTW, do you have an iPad pen that you recommend? Or do you just use your finger? I found one but the squishy round tip is throwing me off… are all of them like that?

        With regards to the AP Calc test – it’s not so much of knowing the “unit circle” per se than knowing the common trigonometric values. For example, they could be asked to find the derivative at a certain point of a trig graph or the definite integral of a trig function. I’m fairly sure (though not positive) that they have questions like that on the non-calculator section of the test, so knowing the values quickly comes in handy. (This is what I gather from the AP Calc teacher at my school who asked me to make sure the kids know the values.)

      • Yes, that’s exactly why I made the change this year–slow moving class because of not doing HW for various reasons.

        I just use my finger. I’ve been using Doceri as an in-class tool for about 3 years now, so I feel comfortable just using my finger. For the first year I had a stylus, but the rubber rubbed off cause I used it so much. I just make my fingers in the shape as if I was holding a pen and it seems to work. Doceri also has the “wrist guard” which can help, though I don’t use that anymore either.

        That’s good to know about the AP calc! I’ll certainly tackle memorizing it more this year!

        Thanks for sharing!

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