Reciprocal Trig Functions

Usually I introduce the reciprocal trig functions early on, but this year I decided to hold off on them, as they seemed to just make everything confusing, especially when we learned about graphing.

So this year, first we learned all sorts of things about sine, cosine, and tangent, including how to graph them and variations on them. Then after a test, I started on secant, cosecant, and cotangent.

We began with a do now that wasn’t about trig at all. I had taken pictures of the stuff on the board, but apparently I don’t properly know how to manage storage on a smart phone, so they’re gone now. Anyway, I’m going to do my best to recreate them.

Here were the instructions on the board when my students came in:recip parabola do now

So anyway they did that, and got this graph:x^2 and x^-2 points

They weren’t sure what was going on with the 1/x^2 graph, so I added some x-vales between -1 and 1, and then they got the idea:

x^2 and x^-2

After that I asked them to help me fill in a table comparing the two graphs. They came up with something like this:

Screen shot 2013-05-12 at 10.00.46 AM

All true things. So I gave them a sheet with 6 graphs on it: sine, cosine, tangent, cosecant, secant, and cotangent. Then I asked them to cut them out and match them into pairs — pairs in which each graph was the reciprocal of the other. Obviously they also had to explain how they made their matches.

I also made up an instruction sheet, but I didn’t end up using it, except for with a girl who did the activity the following day.

I thought it was interesting that many of the students’ first instincts told them to just pair up the graphs that looked alike — sine and cosine, tangent and cotangent, secant and cosecant. (I didn’t label the new graphs with their full names — just the shortcuts, sec and csc, so I don’t think that was part of it.) When I reminded them of the table we had created on the board, that seemed to help, and students sorted it out quickly enough.

Afterwards, I graphed the reciprocals together on the board (using desmos of course) so that we could double check that they followed the table’s guidelines that we had come up with earlier.

This is the first time that I’ve gotten the sense that my students understand why the secant and cosecant graphs look the way that they do. I will definitely be doing this again next time.

A few days of teaching scared kids

So, the first two days back at school in Watertown after the marathon bombing and terrorist search/killing/capture have gone okay. Apparently there was a pretty bad fight in the cafeteria at lunch yesterday, which I just found out about today. Certainly this was not a good experience for students who had just recently witnessed so much police activity and gunshots and explosions — now another bit of violence is right inside the school. I also imagine that stress from the week’s events was probably part of the reason that it occurred in the first place, although I don’t personally know the students involved and shouldn’t speculate.

My very first student interaction of the day was with a freshman girl — let’s call her Minnie — who usually comes to school early and chats with me a little bit. She is extremely sweet and hard-working, and I know she cares a lot about school and her family. It turns out that her father is a police officer who was part of the gunfight in which the older bombing suspect was killed. Again, this is the very first student conversation I had. Usually, I must admit, I shuffle around the room getting things ready, but of course I put my white-board marker down to listen to her. She told me about the insane hours that her father had spent not that far from home, and excitedly explained how the Bruins had signed a hat for him and how many people had been treating him like a celebrity. She was rattled but clearly saw her father as a hero.

In each class yesterday (aka the first day back), I started off by asking each student to share both a ‘brag’ and a ‘nag,’ which I have blatantly ripped off from some wonderful dancing friends of mine. I gave basic examples that didn’t reference the weeks event, because I wanted to wait to see whether they would, and if so, what they would say. I allowed students to say only one or the other, but nobody could totally pass, even if they just said ‘I got a lot of sleep’ or ‘my dad baked brownies.’ The second you let somebody pass, suddenly nobody wants to share anything.

In my very first class, a rather precocious freshman said that the thing ‘nagging’ him was that all these terrible things had lumped up in the same week — the bombing, gunfights in town, Texas explosions, some more miners were apparently trapped, Denver gunshots, et cetera. A few students in addition to Minnie shared that their parents were police officers who were called into action in Watertown. Many said that the thing that ‘nagged’ them was that the lock-down kept them from enjoying the nicest day on break or going to New Hampshire for the weekend. I was surprised by the number of students whose families had planned weekend trips to New Hampshire and Maine.

After the initial check-in, I opened the class up to a general discussion of what had happened. Again, I wanted the kids to talk to what they wanted/needed to talk about. I made a point of saying that, and also telling them that we didn’t need to talk about anything if they felt that they didn’t need to, or if it was too difficult.

My students were great. They expressed sympathy for the people who were hurt. Generally, they just swapped stories and asked each other questions about what they had seen. It seemed like they had been scared. Which makes a whole lot of sense. Many of them discussed hearing gunshots and explosions. One boy had heard the bombs explode at the marathons. One boy said that there was still a bullet in his house. Many live very close to where the second brother was found hiding in the boat, and some saw what was happening. Some had connections to people killed at the marathon and to the younger brother. Besides the most violent events, many clearly were very affected by the general situation. One girl told me she cried because she was worried about her grandmother who lived in the most closed off part of town. Another told me she had shut and locked her windows at night in worry.

In leading the conversation, I tried not to do too much besides add factual information and get students to answer each others’ questions. I said “thank you for sharing” about a thousand times.

One thing that is interesting about the whole situation is that people in the world are talking about how these terrorists happened to be Muslim. Some ignorant people have been doing terrible things with these facts. The really cool thing about Watertown is that it is incredibly diverse. We have a large number of students who have come from Armenia (lots of kids from Armenia), Lebanon, Russia, Jordan, Iran, and other students from Brazil, Guatemala, Honduras, and so forth. Many girls, some recent immigrants and some who in every other way fit the American teenage girl norm, cover their hair or wear more full traditional dress.

The religion of the bombers came up only three times. The first time, a student said that there were non-Watertown affiliated strangers who had said angry, cruel things to some Muslim Watertown students on their twitter pages. My kids didn’t understand this. In another class, I mentioned that I was proud of the fact that students at my school had been so cool about the issue, and one of my students was so confused that people wouldn’t be. He said “these are just two horrible people — why would anybody think that’s what all Muslims or all Russians are like?” What a freaking awesome kid.

The last Muslim-related comment was from a whip-smart Muslim girl, who covers her hair (but once let me see it!). I noted that she had been very quiet in class, which was unlike her, especially when political sorts of things come up. She told me that everyone at school was fine, but that the event had been difficult on the Muslim community at large. I hate that. Then again, she was giggling with her pals about some silly youtube video they had seen for quite some time. I think she’ll be okay, though I’m definitely going to check in on her.

Oh! I also gave my students the opportunity to write letters to the local police officers. I stressed that they didn’t need to, and that I wouldn’t think any less of them if they didn’t want to. I think about 2/3 of my students wrote letters. Minnie offered to give them to her dad the next day to bring into the station. I took her up on the offer — I think she was proud to be able to do that.

photo

Today, Tuesday, kids seemed better. They still talked about it. In the beginning of my classes, I made sure to remind them that they could still go see a guidance counselor if they needed to at any time, and if they felt that anything we were doing in class was beyond them right now, to let me know. When I was one on one with various students, I made sure to ask them how they were feeling. Again — they’re still skittish, but I think they’re nicely on their way to feeling settled again, even if this ‘settled’ feeling is different than before.

Watertown – teaching after trauma

I teach at the high school in Watertown, MA. So, this past Friday, my students were all instructed by police to stay in their homes while SWAT team members searched their homes for bombs. I believe most of them heard the hundreds of gunshots that sounded during the gunfight between the marathon terrorists and local police. I have no idea what will happen at school tomorrow.

It’s going to be a long, difficult day. I was fortunate to be out of town during the whole experience in Boston, as it occurred during my school’s spring break. But everything that happened was very physically close to my home (I live in Cambridge), work, previous school, etc. And everything was even closer to my students, inside of their actual homes at times. So I will have to be the reliable, strong adult to a hundred teenagers.

I’ve read a little about how to handle teenagers who have recently experienced trauma. Here are a few things that I have found useful:

http://www.nasponline.org/resources/crisis_safety/terror_general.aspx

http://www.nasponline.org/resources/crisis_safety/memorials_general.aspx

http://www.scribd.com/doc/137248164/Disaster-Trauma-Resource

http://www.scribd.com/doc/137247675/Talking-Violence

Overall, my focus will be on stressing the safety and power of our community, making sure to mention our safety at school, and steps we can take to heal now and to feel safe in the future.

The school will be sending each student to their advisory/homeroom with a particular lesson plan. This lesson plan has not been sent out yet, so I don’t know what we’ll be doing. In addition to that, I plan to begin my classes by discussing the tragic events of the weekend and then, given time, play some relaxed math games, rather than getting into the curriculum.

Discussion:

1. discuss why we’re safe in watertown and at school

2. what are ways we can keep safe?

    -what should school do?

    -what can students do?

3. raising money for the one fund

4. write thank you letters to cops, support letters to victims’ families

Math games:

Dan Meyer recently wrote a post asking people for ‘tiny math games,’ which is lucky, because my brain isn’t working too creatively. I scanned the comments for some and grabbed these, some of which I’d seen before and some of which are new to me:

    1. Pick a number. Say 25. Now break it up into as many pieces as you want. 10, 10, and 5, maybe. Or 2 and 23. Twenty-five ones would work. Now multiply all those pieces together. What’s the biggest product you can make?   –Malcolm Swan (from main body of article)
    2. take the car number (usually 4 or 5 digits) and add operations between the digits and an equals sign (somewhere) to make a true equation. Try to come up with as many different solutions as possible.  -Jonah
    3. First player picks the starting whole number (greater than 10). Second player decides who starts. Players take turns either subtracting 1 or dividing by 2, rounding down. Whoever reaches a result of one wins. -Yaacov Iland

Who knows? I plan to post at the end of the day about how it goes.

teach it to yourself, kids. (what happened?)

To recapitulate: I’m teaching the next unit in my honors algebra 2 class by not teaching it (well… ideally? kind of?) but rather having my students go through the book on their own with except for a guiding packet as a flotation device, along with literally every other resource that they might want except for me creating off-book lesson plans.

First of all, literally one student brought a book one time when I asked them all to bring their books to class. This is out of 36 possible times. Oh well.

Second of all, they were all very very overwhelmed with the idea of teaching the idea to themselves, but some took to the internet and some asked their parents, and many came in to let me know that it was hard, which I think is great.

I gave them a quiz after a weekend with a packet, which they would all have failed except that I made it not count. Instead, they swapped quizzes and graded each other’s afterwards, and I gotta say, they were pretty into it. Even some kids who tend not to do so well were engaged and asking tons of questions.

They are continuing to be frustrated by this 20-something year old book, but they’re definitely learning stuff. I’m also definitely doing more explanation than I wanted to do, but, hey, at least they’re spending time with the book, which they usually never do.

Text Book

The textbook that I was given to use with my algebra 2 students is older than I am.

When I told my students that they would be learning the next section out of the book with no help from me, they protested that it doesn’t explain anything.

But I’m making them do it anyway. I’m not obsessed with this idea.

When my department head met to discuss my precalc class recently, one thing that he mentioned was that at the end of class I asked ‘can I assign some homework tonight?’ Obviously that’s not ideal teaching. He then was asking why I don’t make them do homework every night or make them use the book more. He wanted to know what would happen when they took advanced math classes in college.

I really do need to show them that the book exists. A friend of mine runs a support-type class and said that one of my students was confused when she suggested he look in his textbook for help, claiming that we don’t use the book. Which is funny because he’s actually one of my freshman who uses the CME book, which is the only good textbook that I have, and the only one of my textbooks that I use to plan out of.

At first, I felt like the implication was that I was doing my students a disservice by teaching them in ways that made sense and made an effort to meet them where they are. As though my efforts at good instruction were spoiling them, babying them.

But it is true that at some point you do have to teach things to yourself using guides that are difficult. And I do spend a great deal of time trying to convince my students that even if something in school seems scary, they should try chipping away at it anyway. So I figured they could handle the book.

I’ve decided to start with my honors class, mostly because I think they are most likely to need the skill to learn math out of a book. And maybe a little because the next chapter I’m supposed to be teaching them is real boring. (These kids are so much better than simplifying radicals.)

Anyway, considering that I wanted them to develop the skill to read out of textbooks, I figured that it wouldn’t be enough to just throw them the book and say “k we have a quiz Monday.” Instead, I made up a little packet that leads them through the section and at the end mentions Monday’s quiz and what will be on it. After seeing how their eyes skipped over tons of text in their programming assignment, I wanted to walk them through how to read something that wasn’t written for the unfocused, speed-reading, distractable internet generation that they and I are a part of.

Here’s what I made:

I made an effort to make the kids read the important things that weren’t written in glittering ink. I also tried to have them connect examples to each other and to the section’s main ideas.

We’ll see if it works. I think they’ll do it at least … they are my honors kids. Mostly the different between honors and other kids is that these ones do their homework.

3 Days of Computers

At the end of every unit, my students take a test and complete a project. The project uses the ideas we’ve been studying, obviously. They’re usually rather challenging, but I think fairly engaging. But I recently came to the end of the rational functions unit, which I had never done before, and had no ideas about how to spin that into something more interesting.

So, screw math, let’s program.

Should I really take 3 days to do something completely unrelated to the curriculum when I’m already half a chapter behind the other teacher who does this course? YEAH DUDE.

Because most of these kids have no idea what computer programming is! Which is ridiculous! We spend hours and hours and hours using computers/smart phones/kindles/nanopets. And we use (most of) them to do really important things. I am a little confused that most schools don’t require any computer knowledge beyond how to use word. I don’t care whether my students are going to become master programmers; it’s important that they understand just a little bit about how machines process information. We make kids take all kinds of science classes in order to understand the world in which we live and figure out how to ask and answer questions about it. Considering that computers are becoming a humungous part of our daily life, I think we need to teach kids what they are. Also adults.

Also (and this is unrelated), I have read a whole bunch of people say that not everybody can program. That is SO DUMB. No offense. But I have a whole variety of kids, some who will never move beyond 7th grade math, in my programming club, and everybody can figure out a few things. Which is important when every single kid in a school system that can afford it is using/seeing computers daily.

Okay enough of that. Sorry.

This week I was doing programming with a bunch of real quick honors kids, so it was pretty easy. I put together a few blog posts with instructions and set them loose. I had them program in CoffeeScript, which is derived from JavaScript. I used this partially because my boyfriend had selected this for the programming club this year and, frankly, I’m currently used to it. It’s a rather forgiving language that doesn’t freak out about whether you put a space in front of an opening parentheses. Also, you can code and run programs right in their browser (www.coffeescript.org — click “TRY COFFEESCRIPT”), so my students would be easily able to work at home without worrying about downloading stuff.

I started off with this google slides presentation, which admittedly is extremely slow for an extremely dumb reason, which is just that I thought it would be cute to make a background that looked like code:

The reason that I included that, basic as it is, was that I don’t think my students have any sense of how their computers work. I wanted them to know how the little thing we did in a blue box on a web page was connected to it all.

In the two lessons, I kind of just threw my kids into the code without much explanation. Last year I had tried to start by explaining what things were and then asking them to start something from scratch, and it was a disaster. So I did the opposite here. Since we only had 3 days, I wanted to focus on how to make stuff happen on computers, rather than definitions.

Here are the two lessons I put together:

http://whsprogrammingclub.wordpress.com/2013/03/31/h-alg-2-basic-programming/

http://whsprogrammingclub.wordpress.com/2013/04/01/h-alg-2-programming-lesson-2/

The one thing that didn’t go so well was just that students kept skipping stuff. They just didn’t read it. I’m not sure what I could have done to fix that.

 

In any event, the kids did good work. They created programs that took user input values and then did the quadratic formula and the pythagorean theorem (in two separate programs, that is). And they were so PROUD of themselves. It was fun to see them buzzing about their successful work and being so satisfied when their programs worked.

Radians Intro

I started Trig in my precalculus classes last week and decided to do things very differently than I have before, seeing it hasn’t gone especially well ever before.

I must admit to largely stealing what the CME Algebra 2 and Precalculus books do. The Algebra 2 book does a nice introduction to using the unit circle, and the Precalculus book does a great job explaining how to use radians. Since my students haven’t seen anything about trig in a few years, and what they did see was very basic triangle stuff, the Precalculus material on its own was too advanced for them.

As my ‘velcro,’ as one of my grad school teachers referred to the idea of introducing new material as an extension of something that students already know, I asked kids to use the Pythagorean theorem to find the missing sides of some triangles.

Here was the really crucial question of the do now:

45-45-90 pythagorean

 

 

 

 

 

 

Crucial, I say, because it makes question #4 on the classwork handout much easier.

 

Speaking of, here’s the handout:

What I think went well is that it doesn’t even mention degrees until the second page. Rather than having my students constantly converting back to degrees, I’d like them to think fluently in radians. Of course conversion is an important ability.

The idea of a person walking around a circle seemed to work okay. Sometimes they had some goofy interpretations of what that could mean — though not in a ‘how cute are these teens and their creativity’ sort of way. More in a what-the-heck-are-you-thinking way. I can’t even remember what they were — everybody figured out the correct way of thinking about it by the time they handed in their work, so I don’t have any copies left.

Somehow it only occurs to me now that a good way of dealing with this would have been to have my students actually walk around a circle on the ground. Suddenly this seems very obvious. I was trying to think up something that involved wrapping hot pink rope around a paper plate. Silly.

I was pretty impressed with how quickly students did pick up conversions from radians to degrees. I didn’t give any more help than sometimes suggesting:

“So if pi is 180, what would that make pi/6?”

I expecting that converting back would be more difficult, but it seemed to go pretty well.

 

The next thing that I did was reintroduce SOHCAHTOA. Then we used this worksheet:

I did stick to multiples of pi/6, pi/4, and pi/6. I want them to get used to these angles. We’ll talk about multiples of pi/2 next.

Rather than just announce that the (x, y) value where an angle hits the unit circle is (cos(z), sin(z)), the hope is that they simply notice it. I haven’t had many kids finish this up, so I can’t be sure yet. One girl does keep thinking she’s making mistakes because the same numbers keep showing up. Little does she know, that’s exactly how I knew she was doing it correctly.

One thing that I have not done yet is make sure that students are using precise values, including square root notation. So far they’ve been giving approximate decimal answers. It’s important that they use the square root ones so that all sorts of things make sense in the future, but for now I like the decimals because I think they’re more meaningful to my students. I don’t think that answers like “square root 2 over 2” are very satisfying ending points for them. So I have to figure out how to introduce those.

Next up for the dear kids: filling in the whole unit circle.

Then after that: tangent. You may notice that I have completely neglected it. That’s on purpose.