Tag Archives: trig

Evolution of a Theorem (or How to Save All the Animals)

[Edited 7/31/2018 with further information and credit about the origins of the theorem]

Greg (@sarcasymptote) is responsible for the first known instance of puppies in peril. Chris Lusto formalized it into a theorem.   It all started when Bowman Dickson posted The Dead Puppy Theorem and Its Corollaries. Others joined in on the effort, most notably the additional corollaries developed by Math Curmudgeon   I also joined the cause, making a worksheet that covered the do’s and don’ts of exponents.   Lisa Winer at eatplaymath took that idea and made an a full-page warning worksheet for her Algebra II students.

Lisa posted her worksheet at just the right time as I was about to start one of the more dangerous chapters: Trig Identities and Equations!  EEEEK!  So I made this and we spent about 15-20 minutes discussing the various problems that come up, why they’re illegal and what to do instead.Save the Trig Kittens

(file here) We also added more to the back as we needed them (e.g. can’t cancel sin x in 1/sin x + sin x.)  I really think it worked as the identities section of their quiz was really quite pretty!  Only 4 kittens were harmed in 75 tests!  It really made them think about each step and I got a lot of “Mrs Craig, is this hurting an animal?” or “Mrs Craig, is this legal?” questions when we were working on them, whereas before I think they just did magic and didn’t care.

What are some constant mistakes that you would want to warn your students about?

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 told you I love Trig.

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tl;dr: Notetakermakers, homework, and study guides for trig sum/difference/double/half angles, trig identities, and solving trig equations.  Part of my ongoing series of posting all of my files; see more and FAQs on this page.  Plus I tell you about an awesome book at the end!

Yes, I love trig. I love that I there’s always new ways to think about and teach it. I love that it’s so elegant. And I love that it’s one of those topics that looks scary and is scary and new but eventually most kids get it and feel so smart about it.

Trig files from megcraig.orgFile here.

Now, check out the middle box of the “three fraction hints” above. If it’s the first time you’ve seen this multiply-by-the-common-denominator-of-the-small-denominators, then be sure to read this post about it.  It’s totally awesome and is so handy in Precal and Calculus!

Now, don’t worry, we don’t do all of those in identities in one day!  We do the first six together:

Trig files from megcraig.org

This is also the first time I talk about Q.E.D and I tell them they could use any symbol to show “YES! I DID IT!” such as a check, smiley face, corgi, or unicorn.

Then I have them work on the rest of the first column for homework with the rule: if you’ve been on a problem for more than 5 minutes without getting anywhere, stop and move on. Since I teach honors, I know some of them would get trapped in a problem for 20 minutes and then just get frustrated with the whole thing. Then we work on the others in class on group whiteboards for a day, and finish them up whenever we have a few minutes throughout the week.

Another reason I like trig is because there’s ACTING! involved. Sure, you could just show the powerpoint of Sinbad and Cosette when you teach the sum and difference formulas. But why tell it, when you can get four chairs at the front of the room, make some nametags (write the names really big, put them in page protectors, then tie some string through the holes of the page protector), and then act out the whole thing with 3 volunteers?  I even bring in a tie and a scarf for when I’m playing each driver. And yes, as Cosette, I wear sunglasses so I can do this move:

and say, “we do not have the same sign.”  Although, confession: I have no idea how the story is supposed to help memorize the tangent sum/difference formula–please let me in on the secret if you know it!  Another confession: Crazy Stupid Love is one of my favorite movies of all time.

FOCUS!  Back to sum and difference:

Trig files from megcraig.orgFile here. We also decided this year not to do the problems like 7 and 8 so I will allow you to skip those as well.  You’re welcome.

Trig files from megcraig.orgFile here.  If you do skip 7 and 8, also skip 12, 13, 18-20 on the homework.

Some double/half angles:

Trig files from megcraig.orgFile here. Fun tip: have them derive the double angle formula of sin and cos from the sum formulas and then everybody gets to feel smart.

Trig files from megcraig.orgFile here.  Now’s a good time to consolidate all our knowledge:

Trig files from megcraig.orgFile here.  (omit #19 if you’ve been omitting stuff) And then begin solving trig equations!

Trig files from megcraig.org

Check out that awesomeness about sin 2x having twice as many answers, but 1/2x could have the same number of answers or even no answers between 0 and 2pi.

Trig files from megcraig.org

Yeah, I was really clip-art happy when I was making all these.

File here.

And then some quadratic and mixed equations!

Trig files from megcraig.org

Cute and cuddly, boys, cute and cuddly….

File here.

It’s only one section, but worthy of its own study guide and test.

Trig files from megcraig.orgFile here.  There’s even a couple showme videos for the study guide: #1-9 and #10-16

My Thing

My thing this week is Simon vs. The Homo Sapiens Agenda. I read about 50 pages of it the night before last, then spent all afternoon yesterday finishing it because I HAD TO KNOW WHAT WAS GOING TO HAPPEN TELL ME TELL ME TELL ME.  And it’s obvious that the author works with teenagers because the dialogue is spot-on.  And they’re normal teenagers doing normal teenager-y things which is a rarity in YA. And it’s just a nice pleasant story where no one dies, not even the dog. 🙂

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: , ,

Graph(s) Paper

So the Kickstarter for CoordiMate was discussed in the MTBoS recently.  It’s a self-inking stamp of a coordinate grid aaaaand I kind of want one.  Because I am the laziest/non-scaliest grapher in the world. “No, trust me kids, this would look like a line if I had the ability to space my tick marks.”

But until then, at least I have really pretty graphs I can print out and/or cut and paste into worksheets, tests, etc.  And now you can too!  Let’s check out the rewards for my own personal mathstarter campaign:

Pledge at least $0

Get a full page of -8 to 8 graphs that you or students can print out for homework and practice.

page of graphs  Claim your reward here in doc or pdf!

Pledge at least $00

Get a whole page of -10 to 10/ -11 to 11 blank dotted graphs. My preferred graph for conics.

blank dotted graphsClaim your reward here in doc or pdf!

Pledge at least $000

You want trig graphs?  Do ya, punk?  Ok then, here’s a page of blank trig graphs from -2pi to 4pi.

blank trig graphsCurrently available as PDF only.

Pledge at least $000000

For the ultra backers at this level, we have the ultimate rewards package.  This collection of ten different graphs comes as a one-page download so you can quickly copy and paste single graphs into your document.  All of them are made of grouped drawing objects which means you can modify the size but keep the proportions by dragging the corner handle while holding done the ctrl key (but if you’re an ultrabacker, I’m sure you already knew that trick).

variety graphsClaim your reward here as a doc file!

So there’s my mathstarter project. I think it’s next Exploding Kittens, don’t you?

Sunday Summary

3 things to share

1. Man, don’t you hate it when you figure out a better way to teach something the day after you teach it?  Although trig equations went pretty well this year (well, we’ll see tomorrow on their quiz), I think next year I will structure it differently. Here’s where we started:

CaptureCapture2Next year, I’m going to start at the end with the calculator/desmos, with a -4π to 4π window to discuss the general form.  Then do some examples with θ between 0 and 2π, then non-calculator examples.  Then on day 2, graph 2θ (or θ/2) on the same graph to discuss getting more/less answers.  If you’d like to modify this for me (doesn’t hurt to ask, right?), or if you want to use it as is, here is the .doc file (with bonus homework at the bottom!)

2. Now, day 2’s note-taker-maker, I’m kind of in love with. (ok, technically this was day 3 because I ended up teaching day 1 slowly)

Capture3The only thing I might change is super-reinforcing the ZERO product property rule is not the “ZERO or sometimes 2 or 3 or -7 product property rule” because some of my students are still having issues with that. (btw, Snoops is dancing because it’s already factored.  Also, it is super fun to have a kid ask later, “How can we solve this?” and you reply, “Cute and cuddly, boys!”  File link.

3. I finally figured out about 3 years ago how to make linear programming less painful…get the mechanics of it out the way first!

Capture4Let them spend a day finding the feasible region, vertices, and max/mins.  Then the next day you can focus on the finding the equations from those long scary problems and the rest is the same as these notes.  Much less stress than trying to introduce all of it the same day.  File here.

2 good books I’ve been reading

We took a quick weekend trip to Chattanooga last week where I started David Benioff’s City of Thieves and could not put it down!  It’s like a buddy cop movie set in the absurdity of World War II.  It reminded me of Anthony Marra’s Constellation of Vital Phenomona, one of my favorite books of the last few years.



Hector Tobar’s Deep Down Dark is the reason I haven’t gotten anything done today!  How did he make this so compelling when I already know the outcome?  I’m about to go draw a bath and try to finish it tonight; it’s such a page turner!  And don’t worry, he does a really good job making sure the reader doesn’t get lost with all the people in the story (I’m horrible at remembering names in both book and real life.)


1 thing I’m meh about

I have to write a unit plan about complex, polar, and parametrics using some premade lessons (and adding others as I see fit). The premade lessons are really expecting a lot from our students and I’m not sure if I will get the outcomes desired by using them (unless the desired outcomes are tears and frustration).  But I’m having trouble finding a lot of great stuff that I can easily replace them with (I have a few things thanks to @mrdardy and @crstn85).  So if you have some cool stuff to share, please do so!  The state of Alabama will thank you!

MTBoS Challenge Sunday Summary

3 Favorite Questions From My Favorite Activity This Week:

We did these extension questions in PreAP Precal for 2 days: (.doc file here)

These came from a variety of places–textbook, MTBoS, a resource with free-response-questions for Precal level, my own mind. My three favorite questions:

1) Take a look at #6 there.  With 5-7 groups in each class, we got 5-7 different orderings.  What rich discussions we had!  What beautiful diagrams they drew (big whiteboards, I want to marry you)!  MIND-BLOWING FACT FOR ME: The absolute value of tangent of an angle is always bigger than the absolute value of its sine value. 

Good topics to bring up: how does tangent relate to slope?  what does “approaching undefined” mean?  what is a good way to discuss numbers whose absolute value is bigger, but since we’re in negative land, it means they are smaller?  what is a good way to organize these?

2) Ah, the fabulous #7. We had just discussed odd and even functions in the previous chapter so this was a great opportunity to bring up the algebraic definition and how it relates to the graph of these guys!  (Not going to lie to you, this is first time I realized that the graphs of sine/tangent are odd and cosine is even)

3) Here are the answers they will tell you for #10: (1, 1) (1, -1) (-1, -1) (-1, 1).  Then they will not believe you when you tell them that it is incorrect.  Then they will draw a picture to prove it and tell you the square root of one squared plus one squared is 1.

2 Awesome Things That Happened This Week

1) My version of Math Maintenance bellringers paid off!  Since bellringers are “non-negotiable” this year, I thought I’d take advantage of the fact that they don’t have to be ACT review questions until the 2nd nine weeks.  So I’ve been throwing review questions up, not from the day before, but from the previous week.  My kids did somewhat better on those sections than the teacher who did just random review, and I know I’m not a better teacher than her!  So I’m going to have to say that it was the math maintenance bellringers that made the difference!

2) I made it through the entire week without crying, which is a first for this school year.  Winning!

1 Thing I’m Looking Forward To

Trying my first Mathalicious lesson in Algebra II!  A little worried, but hey, gotta grow sometime!

Category: Precal | Tags: ,