Tag Archives: unit circle

Trig Addendum: Modified Radian Fraction Cutouts

Read about the awesomeness that is Radian Fraction Cutouts in Shireen’s original post (and my implementation). After using them in Algebra II last year, I made one little modification that made them even better (one of last year’s students that I have again even said so!):

unit circle fraction cutouts dotted

File here.  The secret? Cut only on the dotted lines!!!  So easy to see the whole enchilada (or em”pi”nada as Shireen also says) divided up into sixths, thirds, fourths, etc, then seeing if you add one, take one away from one, or take one away from two.

Comments I heard from the first class I showed it to (the others will see it tomorrow):

“This is so easy!”


“Why did no one teach it this way earlier?”

“These are really great!”

[In response to another Ss question about 7pi/6]: “Use your cutouts!!”

Not one person wanted to convert radians to degrees!  Woot woot!!!!!

Timing wise: In one 53 minute class, we cut them out, made a pocket (fold a half sheet into thirds [like a letter] and tape down the sides to the back of the NTM, leaving the top flap untaped), talked about degrees, talked about radians, and finished the front of this notetakermaker. The back has more practice that they had for homework:

degree radian WSFile here

DO NOT LET THEM LABEL THE UNIT CIRCLE!!!  Otherwise they won’t have to work at any of the rest of the worksheet. 🙂

And since it is the first day all week that I’m finished will all my homework before 9:00, I’m going to go read a real book!  That’s not about math! Or teaching! Take that, SCHOOL!!!!!

Precal Files: Dude, I Could Trig All Day.

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tl;dr: Files for unit circle, graphing trig, and inverse trig functions.

So I’m going to post my precal files in the order that I taught them (see more of my precal files and FAQs here).  I met with a PreAP curriculum committee at the beginning of last school year, and they suggested that we do all the trig stuff in the fall, then go all the way from functions -> quads -> polys -> exponentials -> rational -> limits -> derivatives in the spring. It did work really well in the spring, but I need to do better at spiraling back to trig–I have a fear they won’t know what sin of pi is next August!

Ok, are you ready?  Here we go!

Starting with trig values at a point:

Trig files from megcraig.orgFile here.

Then angles review, but I think I like the worksheet from Algebra II better.

Trig files from megcraig.orgFile here.

Then the unit circle review:

Trig files from megcraig.org

File here. We also talk about the hand trick.  The hotmath at the bottom is for one of the better trig value flashcards website I’ve found.

The next day we expand past 0 and 360:

Trig files from megcraig.orgI use a worksheet from an Algebra II/Precal joke book for homework (which I just learned is frowned upon? I must say that these are usually well done and have some good questions that catch conceptual errors).

Then it’s time for one of my favorite group work worksheets, (that I already wrote about here):

Trig files from megcraig.orgFile here.

At this point we stopped, reviewed, and took a small quiz.

Trig files from megcraig.orgFile here.

Then it’s onto graphing. This is about the time I first learned about the windowpane method, so I taught some classes one way, some the other, and some both. This shows the window pane.

Trig files from megcraig.org

File here. This should have gone faster, but took over a day. The graphing from scratch at the top was like pulling teeth.

This is their practice/hw, which shows the old way of marking the graph into “exciting points”

Trig files from megcraig.orgFile here.

Then we did a real life sine problem from Math Teacher Mambo.

Trig files from megcraig.orgHere’s her post on it. Be prepared: it looks like a cosine graph so they all wrote cosine equations because who reads directions?  Then I had to tell them to actually read #7.  Next year, I may have them choose whichever function they want, then make the last question be “convert from sin to cos or cos to sin.”

Next, cosecant and secant:

Trig files from megcraig.orgFile here. I teach cosecant and secant graphs using a suggestion from a student: we sing “The Grand Old Duke of York,” since when you’re up, you’re up, when you’re down, you’re down, and when you’re only halfway up, you’re neither up nor down (asymptote!).

Ugh, tangent graphs.

Trig files from megcraig.orgFile here. This is another example of the “exciting point/pattern” method of graphing, which looking back, I think I like better. Or maybe I need to come up with some hybrid.

Then, because it ties in so well with graphing, we did inverse trig functions in this unit.

Trig files from megcraig.orgFile here

Trig files from megcraig.orgFile here. Even if you’re not a homework gal or guy, you may still want to use those last 3 problems as a lead-in for the next section

Trig files from megcraig.orgFile here. Although next year I want to spend more time on the even/odd/unit circle-ness of sin/cos to discuss, “ok, well, we can’t use 4p/3 in the allowable region for cosine, but what angle in the allowed region should have the same cosine value?”

Trig files from megcraig.orgFile here. *Note! The answer to #17 should be pi/3, not 2pi/3! It should be fixed in the file. Thanks to Chikae for spotting that!

Study guide time!

Trig files from megcraig.orgFile here.  And, yes, it comes complete with review powerpoints (that could also be used for whiteboard practice).  And they come in both exciting points and windowpane varieties–choose one or both!

Trig files from megcraig.org

Trig files from megcraig.orgExciting Points file here.  Windowpane graphing file here.

But wait there’s more!  If you act in the next 20 minutes (just like the real commercials, the 20 minutes starts whenever you read this 🙂  ), you can get a video of me working out some of the study guide problems!

#11-17 video here and #18-26 here.

I post these the night before the test and the students who watch them are very appreciative.

So, be honest: am I the only one who could Trig all day?  (Except for tangent graphs, obvs!)

If you teach trig, you need this post.

The reason you need this post is because Math Teacher Mambo has unlocked the secret to teaching radians so kids will understand.  YES IT IS TRUE. She posted this fabulous idea on cutting out radian pieces to use, like this:

Image from mathteachermambo.blogspot.com

Image from mathteachermambo.blogspot.com

How can I describe using them in class?  Well…

Exhibit A:  After two days of working with both, I informally polled all three classes about whether they prefer radians or degrees.  At least 25 – 50% raised their hands for radians, and many of them said it didn’t matter to them.  That’s right, a class where kids prefer or at least do not actively dislike radians.

Exhibit B: In one of my classes, after the poll, I told them I was so excited because this was a new way of teaching it and it actually worked.  One of the students asked, “so how in the world did you teach it before?” “We just thought about it.”  Their reaction:

Exhibit C: After that reaction, another student said that they were great to start with, but then after a while they didn’t even need them. Woohoo!

Since I use NoteTakerMakers® instead of INB, I modified my NTM from last year to accommodate Shireen’s circle files:

trig1.doc file

We started with degrees. By “bow-tie triangle” I mean reference angle (we had done trig values at a point the day before and practiced drawing our bowtie) and by “type” I tell them short, medium, or tall.  We did the first five together and then I sent them on their way.  Yes, with greater than 360 and negative angles, which was great because everyone had a different way of thinking about where they were.

Ok, just so I’m not kicked out of the #MTBoS, I would love to do a radians activity where we discover what they are, and that one radian is the same for every circle, and it takes 2pi of them to go around, lalala discovery learning, but I have 13 class periods to go from 0 to translating sine and cosine graphs, so I showed them a quick animation from Sam and pi, 2pi, pi/2 and 3pi/2 using Math Teacher Mambo’s empinadas analogy. (except with quesadillas because our Moe’s actually asks if you want them cut in half or fourths). We had just enough time at the end of class to cut out radians out, label them, and put them in a pocket (hint: Give each student a third of a piece of paper. Fold strip into thirds. Tape two sides and you got a pocket with flap.  You can even tape it down to the NTM between the two circles, but it does over lap the chart a bit.  I could probably make it pretty so it doesn’t, but I didn’t.)

The next day we talked about the radian examples at the bottom of the NTM.  After the examples I asked them about what type of triangle we will have with denominator 6, 4, 3, and 2. I think next year, I’ll have them cut out the pieces like this:

trig3Because lining up 7 or 11(!) of the wedges was time consuming and easy to land on the wrong space.  Plus hopefully this might help them think, “is this more than a whole quesadila?” when they are deciding which pieces to use.

After the examples, I let them loose on this page:

trig2.doc file

I stamped the first five and ten as they were working to make sure they were on the right track.  They didn’t even balk at the last few that were greater than 2pi!

Warning: the rest of the post is less essential than those radian cut-outs. 🙂

The next day was the big intro to the unit circle.  I’ve moved away from “these are the coordinates, let’s memorize the unit circle and draw it on everything really quickly” because I realized when I started teaching precal we need to know sin 5pi/6 without having to draw the whole thing. So instead we talked about short/medium/tall triangles and just remembering 3 numbers: 1, 2, 3 and which one is short/medium/tall (or skinny/wide). I have them draw the triangle for each question. I think maybe I should also have them highlight the part we care about?  Or now I’m thinking (and I’m going to try this tomorrow with reciprocal functions) of sacrificing one of our wedges and making it a triangle we can label and move around.  I’ll report back.


Doc file  and extra practice file

We spent the next day practicing and me stamping off correct answers, which I need to find an equitable way to do. I normally go around the room and stamp, but I always seem to miss tables.  I need one of those numbered ticket things like at the butcher or Joann’s.  Anyone have any great ideas on that?

They are definitely not where I want them to be after 5 days of this, but I think they are getting there.  We shall see.  Maybe I’ll just throw in the towel and start doing timed unit circle quizzes again.

Category: Alg II, trig | Tags: , ,