Category Archives: trig

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

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[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?

Category: Precal, trig | Tags: , , ,

Life-Size Sine Curve

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Last weekend, there was some twitter chatter about making life-size graphs so students could explore points/transformations/what-have-you. That got my wheels turning, and after a quick trip to the dollar store and about an hour’s worth of work I ended up with:

trig graphSteps:

1) Buy four shower curtains at the dollar store. Bonus if they are prelined!

2) Tape together with packing tape. (hint: tape them down to the floor with washi tape to hold them down, then tape the seams on the front. When done with the whole thing, flip over and tape the back)

3) Use duct tape to mark the x-axis. Print labels (file here), cut out, measure your axis, and tape down with packing tape.

4) Apply colored masking tape for 1/2, root 2/2, root 2/3. (see next picture since the masking tape was at school)

Total cost depends on how much tape you have around the house. I used almost 3 full rolls of colored masking tape, but the good news is I bought it from naeir.org. Have you heard of this site? One of the teachers at school shared it with me. Basically companies donate overstock and you get to buy it for the cost of handling. You do have to spend at least $25 and shipping takes about 2-3 weeks, but holy cow, can you get a lot of stuff for $25!  My first shipment I got 8 rolls of patterned/colored masking tape, 2 packs of 12 small post-its pads, 8 post-it pop-up cubes, 12 correction tape thingies, a pack of sharpies, 3 sets of dividers, 2 packs of post-it labels, and I think some other things I’m forgetting. It’s crazy!

Anyway, in class, I handed out dry-erase pockets with a sheet that had an x value in it (0, pi/6, pi/4….2pi) (file here) and told the students to find sin x, 2 sin x, sin 2x, sin1/2x, and cos x. Holy moly. We could have easily spent the day doing that. No, if x = pi/3, sin2x does not equal 2pi/3. Once we got that sorted, we went out into the hall. I had all the students stand on their x-coordinate, then step to the y for the function I called out. It was very easy to find people who made wrong calculations! 🙂  Here’s what cosine looked like:

cosine curveAnd sin 1/2x (with an outlier!)

sin half x

By third time I ran through it, I had worked some of the kinks out:

  1. In my first class, I had more people than x-coordinates, so I gave coordinates that were more than 2pi. This did not go well. They were way far down and we couldn’t really see the pattern continuing. The next class I handed out 2-3 points per group and had them work together to find the values, then as we graphed we substituted people in who hadn’t graphed yet.  (The class shown had just one person extra.)
  2. I only did each graph once. We talked about max/mins, who didn’t move and why, how many cycles fit on the mat, etc. I think it would have been beneficial to do sin, then cos, then switch back and forth faster and faster. Then do the same for 2 sin x and sin x, sin 2x and sin 1/2x, etc. And also positive and negative. The last class we even tried sin x + 2 (“oh, that means we need to all step up 2!”)
  3. Have them write down noticing/wonderings as we are doing it, or a quick sketch of the graph (maybe have some axes printed on the back of their point card?) to help solidify the concepts.

The next day, when we went to graph, I asked them if it helped to visualize what we did yesterday. Only a few raised their hands. I told this to Mr Craig, wondering if I would do it again or if it would be more efficient to just jump into graphing then practice. He said, “Hey, you helped those 5 kids see it better! Plus sometimes it’s about the experience, not about being efficient.”  Sometimes that Mr Craig can be pretty smart. (Don’t tell him I said that, his head is big enough already.)

In other news, this happened on Twitter the other night:

8,006 tweetsDo you think I will hit 1,000 followers or 10,000 tweets first?

Category: Precal, trig | Tags: ,

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. 🙂

Category: Precal, trig | Tags: , , ,

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!)

Category: Precal, trig | Tags: , , , ,

If you teach trig, you need this post.

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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.

trig6

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: , ,