I was playing around with Prezi tonight, and this is what I came up with for organizing middle-school math topics:

Does it work? Can you see the embedded Prezi applet? Suggestions welcome on either the format of the presentation or (more importantly) the organization of math ideas!! Are there important topics or questions that I am missing?? I am definitely teaching middle school next year (with I think a high-school class thrown in there somewhere, but I'm not so sure about the latter.) A part of me is fully nervous about this going-back-to-middle-school goodness. Another part of me is very excited about all of the poss!b!l!t!es!!! :)

PS. You might have noticed that I didn't bother with adding flashy effects to the presentation. That's intentional... It's kind of my pet peeve when the tech takes more of the spotlight than the content. If I do decide to incorporate some Prezi stuff next year to help my kids draw mental connections between big ideas, I'm going to keep it as vanilla and non-distracting as possible.

## Monday, June 27, 2011

## Friday, June 24, 2011

### On Being a Multi-Faceted Educator

I've noticed that as time passes at the Klingenstein Summer Institute, each day I more fully develop a sense of how much I suck as a teacher.

Fortunately, unlike the grad classes I took before which pointed out some gaps in our teaching practice but did not offer any solutions, at KSI we actually spend a significant amount of time brainstorming small things we

One thing I think I can immediately implement at my new school is to initiate a professional development community consisting of teachers who are seeking to actively examine their own teaching practice. I've been talking with other teachers here who have reading groups at their schools, where teachers read and discuss educational articles. In KSI, we've also been practicing a lesson study protocol for allowing teachers to voluntarily share the weaknesses in their teaching, in order to ask for feedback and advice. I think that sounds wonderful, and it is something that is very manageable that I believe I can tackle. (I know that there are teachers who do some of this on Twitter, but I'm more motivated by local communities.)

I've also gone to some great tech sessions that discussed various uses of technology inside/outside the classroom. I don't know if all of those things will be available, or if the use of some of them (such as Edmodo, a social networking site) will fit with the school's technology philosophy. But, I feel like the least I can do is to really investigate those options and to talk to the right people when I arrive at the new school in July. If I can integrate some new modes of tech use into my class, they could enrich my teaching style greatly without being too much additional work for me or the kids and (for these particular methods I am thinking about) without distracting from the content.

Teaching about diversity and bringing about positive changes to the school culture is the hardest piece of all of the Klingenstein lessons to implement. I think that is because as educators, we cannot all necessarily agree that teaching about those topics is 1. pertinent and necessary, 2. feasible given the community culture that surrounds the school. (I personally believe those discussions are necessary, but I believe it would be difficult to convince others on my staff of the same.) I am going to start by doing things like putting up pink triangles around the perimeter of my classroom, to signify to GLBT kids that it is a safe space if they need an adult to talk to. And then, once I build some rapport with the administrators and gain their confidence, I can try to tackle bigger tasks. But, that'd have to wait until I get over the hump of being a brand new, young-looking teacher at the school.

Aiyayay. Teaching is hard, and it seems to get harder by the day.

Fortunately, unlike the grad classes I took before which pointed out some gaps in our teaching practice but did not offer any solutions, at KSI we actually spend a significant amount of time brainstorming small things we

*can*do to address various learning and diversity issues in our classroom and at the school.One thing I think I can immediately implement at my new school is to initiate a professional development community consisting of teachers who are seeking to actively examine their own teaching practice. I've been talking with other teachers here who have reading groups at their schools, where teachers read and discuss educational articles. In KSI, we've also been practicing a lesson study protocol for allowing teachers to voluntarily share the weaknesses in their teaching, in order to ask for feedback and advice. I think that sounds wonderful, and it is something that is very manageable that I believe I can tackle. (I know that there are teachers who do some of this on Twitter, but I'm more motivated by local communities.)

I've also gone to some great tech sessions that discussed various uses of technology inside/outside the classroom. I don't know if all of those things will be available, or if the use of some of them (such as Edmodo, a social networking site) will fit with the school's technology philosophy. But, I feel like the least I can do is to really investigate those options and to talk to the right people when I arrive at the new school in July. If I can integrate some new modes of tech use into my class, they could enrich my teaching style greatly without being too much additional work for me or the kids and (for these particular methods I am thinking about) without distracting from the content.

Teaching about diversity and bringing about positive changes to the school culture is the hardest piece of all of the Klingenstein lessons to implement. I think that is because as educators, we cannot all necessarily agree that teaching about those topics is 1. pertinent and necessary, 2. feasible given the community culture that surrounds the school. (I personally believe those discussions are necessary, but I believe it would be difficult to convince others on my staff of the same.) I am going to start by doing things like putting up pink triangles around the perimeter of my classroom, to signify to GLBT kids that it is a safe space if they need an adult to talk to. And then, once I build some rapport with the administrators and gain their confidence, I can try to tackle bigger tasks. But, that'd have to wait until I get over the hump of being a brand new, young-looking teacher at the school.

Aiyayay. Teaching is hard, and it seems to get harder by the day.

## Wednesday, June 22, 2011

### Klingenstein Summer Institute

A while ago, through Sam Shah's blog I learned about a program called Klingenstein Summer Institute, which is a professional development workshop that runs for a couple of weeks each summer, and is geared towards teachers in independent schools who are still within their first five years of teaching. I researched the program briefly on the web and decided that it was probably unrealistic for me to apply and be accepted, but I went ahead and did it anyway. (I am trying to practice being a more aggressive version of myself -- if that makes any sense.)

So, long story short, here I am sitting in Lawrenceville, NJ, which houses the Klingenstein program. The first few days were pretty hectic, but I think it's calming down. Thought I'd take a few minutes to jot down some of the things I've been learning/thinking about since my arrival at the institute.

1. Math-specific stuff:

Thoughts? That's my first stab at re-organizing the topics of a course based on this idea.

2.

3.

Ok - dinner time. :) Ciao!

So, long story short, here I am sitting in Lawrenceville, NJ, which houses the Klingenstein program. The first few days were pretty hectic, but I think it's calming down. Thought I'd take a few minutes to jot down some of the things I've been learning/thinking about since my arrival at the institute.

1. Math-specific stuff:

**The old way of organizing topics around content chapters needs to be replaced by organization around themes**, to facilitate the development of effective schema in the students' minds (which increases retention and all that good stuff). I am obviously a newbie at this, but I'll try to organize Geometry units to illustrate these. I'm not quite fleshing out everything.- BIG IDEA: Measurement

--> Dimensions and quantities

--> Conversions

--> Measurement methods

--> Indirect measurements such as Pyth Theorem and similarity ratios

--> Trigonometry - BIG IDEA: Conjecturing

--> Induction

--> Deduction

--> Layered math formalism (definitions --> postulates --> theorems)

--> Making conjectures about circles and triangles and angles inside parallel lines

--> Multi-step solving of diagrams using various known pieces of info - BIG IDEA: Multiple Representations

--> Coordinate plane concepts and skills - BIG IDEA: Transformations

--> Both inside/outside of coordinate plane

--> On objects as well as "functions" - BIG IDEA: Proofs

--> Basic postulates

--> How do we prioritize info?

--> Formal Geometric proofs (both algebraic and geometric)

--> Meta-cognition: How do we assess a proof/argument? - BIG IDEA: Spatial Visualization

--> Proportional reasoning

--> 3-D to 2-D, or 2-D to 3-D

--> irregular area/perimeter/volume/surface area

Thoughts? That's my first stab at re-organizing the topics of a course based on this idea.

2.

**Small changes in the classroom can make big impacts on student learning.**Things I am going to try this fall: slowing it down (wait time/questioning) and being positive/encouraging/empathetic. I'm going to ask a colleague to observe/evaluate me on my goals and to hold me accountable.3.

**We do (individually) have the power to substantially affect school culture.**One Klingenstein alum once started the first Gay-Straight Alliance (I forget the name) as his year-long project. Now there are GSAs all over the country, in private and public schools, and they help to foster awareness and compassion. I need to believe in the power I have to change the things that I think are problematic in the school.Ok - dinner time. :) Ciao!

## Monday, June 13, 2011

### End-of-Year Mini Reflection

To be frank, I did not enjoy the end of this school year. For some reason, towards the end of May a bunch of stuff happened that left a sour taste in my mouth. Like, plagiarism type of stuff. It made me feel sick in my stomach to know that cheating is so rampant at the school -- at all levels, honors and regular, and relatively few kids seem immune to it. It seems like lots of kids care overly much about their grades and think it's harmless to cheat occasionally, and lots of other kids who do value the actual learning still think that it's harmless to help their friends cheat once in a while.

This realization turned the year around for me, in a negative way. Instead of looking forward to my classes everyday, I walked in daily and felt like I was in the middle of strangers whom I couldn't trust, and I was just counting the days until it was over. Not a good feeling to end the year on, especially because I had felt all along that we were doing great projects and the kids were putting in a lot of effort. It just seemed like something had happened during the last two weeks of the school year, because all of a sudden my high-effort kids just dropped off and stopped studying, around the same time as the series of distressing and disappointing plagiarism incidents.

Thinking back, last year (2009-2010) at around the same time in late May, my bottom kids were staying with me once a week for extra help, and the rest of the class envied them as their test grades jumped up 20% from the beginning of the quarter to the end -- a reflection of their improved understanding, as a direct result of their efforts and perseverance. I remember feeling euphoric, like I knew the kids were ready to be sophomores and to take on additional responsibilities and challenges. This year? I am not so sure. What an awful feeling to end the year on.

...So, there I was, ready to go into the summer wondering about my own worth as a teacher. ("Maybe if I were a better teacher, my kids would not have ended the year with these negative choices. Clearly I messed up big somewhere to cause this..." type of thinking.) And yet, all of a sudden, kids started trickling in to thank me for having taught them and to say goodbye before I leave for Germany. Some of the kids I have not taught in a year. One girl whom I have taught now for two years wrote me a thank-you card that said, in short, that I have changed her relationship with math.

Then I was busy with packing and cleaning and closing accounts, so I did not have time to sort through my feelings about all of this. Until today.

Today was our last faculty meeting. During the meeting, the principal wanted us to go around and share what were the key uplifting moments for us this year. Mired still in my (I guess you could call it) anger and hurt about the plagiarism incidents, I had trouble recalling the highlights that made my year. And then two things came into mind. One of them was when my juniors in Precalculus started showing up to see me after school for help, sometime in Q3. Some weeks they would come once in a big group; other weeks they'd come one or two at a time, everyday. And how, a couple of months later, when the intercom announced that it was Teacher Appreciation Week, this class burst into spontaneous cheering and applause. For me! When I was not even looking at them. (I was writing something on the board at the time to take advantage of the morning announcement.) The other outstanding memory was the thank-you card that I had described above. I can't do it justice, so I took a picture of it to share with you why a single card had such an emotional impact on me.

Thinking -- I mean,

So,

This realization turned the year around for me, in a negative way. Instead of looking forward to my classes everyday, I walked in daily and felt like I was in the middle of strangers whom I couldn't trust, and I was just counting the days until it was over. Not a good feeling to end the year on, especially because I had felt all along that we were doing great projects and the kids were putting in a lot of effort. It just seemed like something had happened during the last two weeks of the school year, because all of a sudden my high-effort kids just dropped off and stopped studying, around the same time as the series of distressing and disappointing plagiarism incidents.

Thinking back, last year (2009-2010) at around the same time in late May, my bottom kids were staying with me once a week for extra help, and the rest of the class envied them as their test grades jumped up 20% from the beginning of the quarter to the end -- a reflection of their improved understanding, as a direct result of their efforts and perseverance. I remember feeling euphoric, like I knew the kids were ready to be sophomores and to take on additional responsibilities and challenges. This year? I am not so sure. What an awful feeling to end the year on.

...So, there I was, ready to go into the summer wondering about my own worth as a teacher. ("Maybe if I were a better teacher, my kids would not have ended the year with these negative choices. Clearly I messed up big somewhere to cause this..." type of thinking.) And yet, all of a sudden, kids started trickling in to thank me for having taught them and to say goodbye before I leave for Germany. Some of the kids I have not taught in a year. One girl whom I have taught now for two years wrote me a thank-you card that said, in short, that I have changed her relationship with math.

Then I was busy with packing and cleaning and closing accounts, so I did not have time to sort through my feelings about all of this. Until today.

Today was our last faculty meeting. During the meeting, the principal wanted us to go around and share what were the key uplifting moments for us this year. Mired still in my (I guess you could call it) anger and hurt about the plagiarism incidents, I had trouble recalling the highlights that made my year. And then two things came into mind. One of them was when my juniors in Precalculus started showing up to see me after school for help, sometime in Q3. Some weeks they would come once in a big group; other weeks they'd come one or two at a time, everyday. And how, a couple of months later, when the intercom announced that it was Teacher Appreciation Week, this class burst into spontaneous cheering and applause. For me! When I was not even looking at them. (I was writing something on the board at the time to take advantage of the morning announcement.) The other outstanding memory was the thank-you card that I had described above. I can't do it justice, so I took a picture of it to share with you why a single card had such an emotional impact on me.

Thinking -- I mean,

*really*thinking and letting my emotions sink in -- about these two particular incidents during the meeting finally brought me out of the semi-depressed rut that I had been in for the past few weeks. I still have so much to learn and I want to be SO much of a better teacher in Germany than I was here, but maybe, in the end, this year itself wasn't all bad.So,

*Prost!*to a new beginning! This is the true blessing of a teacher: each year, we each get a fresh chance at getting it right. Or, at least, getting closer to getting it right.## Sunday, June 12, 2011

### A Little Shoutout to Our Favorite Cafe

Apparently the best barista in the world is Salvadoran and works at the local coffee chain that we frequent. I guess I shouldn't have made fun of Geoff all of those times when he claimed emphatically that Viva Espresso has the best cappuccinos he has ever tasted in his entire life. Turns out that his taste buds are quite sophisticated.

My boyfriend has sophisticated taste buds and is willing to eat anything that has not been rotting for too long. That's dangerous.

My boyfriend has sophisticated taste buds and is willing to eat anything that has not been rotting for too long. That's dangerous.

## Thursday, June 9, 2011

### 3-D Calculation Work Samples

A reader asked me to post some samples of student calculations for the 3-D project. I had been meaning to all along, just didn't get around to doing it earlier because things have been so busy.

Anyway, here they are, on the eve of giving our printer/scanner away. :) I had full intention of scanning in an array of student work until I remembered halfway through the process that scanning is very tedious.

So here it is, what I managed to get done before my impatience took over. These are from regular Geometry kids.

Student #1:

Student #2 (in the same group as previous kid, but doing his calculations independently... they share only the grades for their constructed project and their 2-D nets):

Student #3 (totally separate group):

By the way, did I mention that it's trickier than I had thought for kids to build cones from scratch? Here's a sample 2-D net that one group used. They had to calculate the central angle using some proportional reasoning, in order to find the amount of arc length they want on the bigger circle, in order to create a lateral surface that perfectly fits around the base of the cone.

Not bad, eh? Truthfully, across the board the groups did very well. One group had an awesome hexagonal pyramid design/calculation (but I'm not posting it here because one partner copied calculations off of the other, so I ended up punishing both severely for plagiarism).

PS. I feel that I should take this opportunity to give a shoutout to my honors kids' beautiful 3-D projects. I could not bring myself to throw some of these projects out today while packing up my classroom, so I'm leaving some of them behind on the shelf in hopes that the next Geometry teacher will adopt them as classroom displays.

Anyway, here they are, on the eve of giving our printer/scanner away. :) I had full intention of scanning in an array of student work until I remembered halfway through the process that scanning is very tedious.

So here it is, what I managed to get done before my impatience took over. These are from regular Geometry kids.

Student #1:

Student #2 (in the same group as previous kid, but doing his calculations independently... they share only the grades for their constructed project and their 2-D nets):

Student #3 (totally separate group):

By the way, did I mention that it's trickier than I had thought for kids to build cones from scratch? Here's a sample 2-D net that one group used. They had to calculate the central angle using some proportional reasoning, in order to find the amount of arc length they want on the bigger circle, in order to create a lateral surface that perfectly fits around the base of the cone.

Not bad, eh? Truthfully, across the board the groups did very well. One group had an awesome hexagonal pyramid design/calculation (but I'm not posting it here because one partner copied calculations off of the other, so I ended up punishing both severely for plagiarism).

PS. I feel that I should take this opportunity to give a shoutout to my honors kids' beautiful 3-D projects. I could not bring myself to throw some of these projects out today while packing up my classroom, so I'm leaving some of them behind on the shelf in hopes that the next Geometry teacher will adopt them as classroom displays.

## Tuesday, June 7, 2011

### Money Matters Associated with International Move

I wanted to write down some moving logistics for people who might find themselves in similar shoes as us later. Moving

...Anyway, all things considered, I'd say that our moving process is going swimmingly. My only regret is that I am an idiot and didn't go in time to a Salvadoran doctor to get my wisdom teeth removed while it's still dirt cheap. (The doc refused to do it since I'm flying out in under 2 weeks.) But, I am hopeful that perhaps it won't be too bad under the German insurance to take care of this little

*after*having lived somewhere for two years is entirely different from moving abroad the first time, because...* Now you have a bank account that you get your salary directly deposited into, that you have to close before you leave the country (without butchering any direct deposits).

* If you want to wire any money to your Stateside accounts, you have to allot about a week to do that BEFORE you close your foreign accounts.

* You have presumably been contributing to a foreign social security system, presumably with matched contribution from your school and presumably it's some significant amount of money that you wish to get back at some point. It's not totally straight forward how this is done, because if you want the full amount back (including your school's full contribution up to your length of contract), I believe the standard operation is to start the refund process AFTER you leave the country, which means someone else than you will have to pick up that refund for you...

* In order to take care of some of these things smoothly, you may have to transfer your power of attorney onto someone else who will remain in the residence country after you leave. Obviously, you should choose someone that you can trust, but also the paperwork process can be a bit of a hassle and take up to a week.

* If, like me, you are moving to another foreign country afterwards, you should contact the school well in advance to figure out if there are temporary places to store your stuff. In my case, I've had to wire over some advance payment to my relocation agency in order to get a storage unit started for me (which, again, takes time and some follow-up energy).

* Did I mention that you cannot assume that you will be able to afford the moving costs?? Good thing Geoff and I don't have much stuff, because the estimates we got from two different moving companies for our ridiculously few items were both around $2000. I guess that depends on where you're shipping from and to, but the point is that you want to look into this stuff early so that you will have options. In our case, the only things we're shipping are my teaching supplies, in small boxes. Everything else we're going to squeeze into our two suitcases per person as when we arrived here.

...Anyway, all things considered, I'd say that our moving process is going swimmingly. My only regret is that I am an idiot and didn't go in time to a Salvadoran doctor to get my wisdom teeth removed while it's still dirt cheap. (The doc refused to do it since I'm flying out in under 2 weeks.) But, I am hopeful that perhaps it won't be too bad under the German insurance to take care of this little

*weisheitszahn*problem. (My first German word!)## Sunday, June 5, 2011

### Last Days in El Salvador

Now that our time in El Salvador is coming to an end, I wanted to count our blessings a bit before we move on to our next stop.

Things I will miss about El Salvador:

* yummy, cheesy pupusas

* incredibly warm people

* tropical climate

* how time seems to be suspended every time we go to the beach

* the loveliness of Spanish

* how El Salvador is almost untouched by tourists and English

* $15/hour massages and our $13/week maid

* living within walking distance to the school

* freedom from any test-prep pressure

* free round-trip annual tickets to the States

* having incredible teaching resources at the school and various support staff helping me set up labs for my classes

* the kids I taught last year, who still came back to me all the time this year for help and advice...

* The Head of our school is a great boss and someone whom I feel comfortable walking into the office of without appointments.

Things I will not miss:

* unreliable water, power, and sometimes internet

* giant rain-filled pot holes that they just don't bother fixing

* Salvadoran banks, which tend to prolong every process and make everything difficult to accomplish

* feeling unsafe at night or during the day in the boonies

* hearing people complain about stupid things while many other people in this country have literally nothing and NO opportunities!

Top 13 memories during our 2-year stay (I couldn't fit it into a neat list of 10):

* Pacaya - live-lava volcano in Guatemala where we roasted marshmellows on the lava stream

* Machu Picchu and Wainapicchu - sky-high Inca ruins in Peru

* Calafate - still growing glacier mass in Argentina

* Rio Celeste - natural bright blue river in Costa Rica, in the midst of a rain forest

* El Tabacon - labyrinth of luxurious hot springs in Costa Rica, channelled from natural volcanic springs

* Cerra Negra - "volcano boarding" in Nicaragua

* amazing snorkeling in Belize

* Tikal - Mayan ruins in the jungle of northern Guatemala

* Santa Ana, Izalco, Coatepeque - Salvadoran volcanos: one with a smoking green crater lake, one with bare rocky top, and one with a beautiful volcanic lake.

* Atitlan - giant and beautiful volcanic lake in Guatemala, complete with cliff-jumping!

* staying with a local family on an island in Panama (in a village with only huts and no sewage system) and then island hopping between untouched paradise islands

* a bunch of teachers going to the lake house of one of my students and doing lake sports and BBQing with their truly awesome family

* Almost getting robbed by a masked bandit in Guatemala at machete-point and driving off while screaming. (This is not a good memory, obviously, but it taught us a helluva lot about being safe. So, it's a very memorable experience.)

As the year wraps up (Geoff leaves in 7 days and I leave El Salvador 4 days after him), I am endlessly thankful for the experiences that we have had. Geoff and I always say this, but it's true -- we are some of the luckiest people, because we have jobs that we love, we are saving money and traveling at the same time, and we have each other to share this amazing experience with.

Summer (and Germany), here we come! :)

PS. I think the more Spanish I have learned, the worse an effect it has had on my ability to spell in English. Take "glacier" for example, it didn't immediately occur to me that I was spelling it "glaciar" like it is in Spanish, since when I visually checked the word, it looked very familar to me. And for the same reason, my natural inclination was to type "masage" (similar to

Things I will miss about El Salvador:

* yummy, cheesy pupusas

* incredibly warm people

* tropical climate

* how time seems to be suspended every time we go to the beach

* the loveliness of Spanish

* how El Salvador is almost untouched by tourists and English

* $15/hour massages and our $13/week maid

* living within walking distance to the school

* freedom from any test-prep pressure

* free round-trip annual tickets to the States

* having incredible teaching resources at the school and various support staff helping me set up labs for my classes

* the kids I taught last year, who still came back to me all the time this year for help and advice...

* The Head of our school is a great boss and someone whom I feel comfortable walking into the office of without appointments.

Things I will not miss:

* unreliable water, power, and sometimes internet

* giant rain-filled pot holes that they just don't bother fixing

* Salvadoran banks, which tend to prolong every process and make everything difficult to accomplish

* feeling unsafe at night or during the day in the boonies

* hearing people complain about stupid things while many other people in this country have literally nothing and NO opportunities!

Top 13 memories during our 2-year stay (I couldn't fit it into a neat list of 10):

* Pacaya - live-lava volcano in Guatemala where we roasted marshmellows on the lava stream

* Machu Picchu and Wainapicchu - sky-high Inca ruins in Peru

* Calafate - still growing glacier mass in Argentina

* Rio Celeste - natural bright blue river in Costa Rica, in the midst of a rain forest

* El Tabacon - labyrinth of luxurious hot springs in Costa Rica, channelled from natural volcanic springs

* Cerra Negra - "volcano boarding" in Nicaragua

* amazing snorkeling in Belize

* Tikal - Mayan ruins in the jungle of northern Guatemala

* Santa Ana, Izalco, Coatepeque - Salvadoran volcanos: one with a smoking green crater lake, one with bare rocky top, and one with a beautiful volcanic lake.

* Atitlan - giant and beautiful volcanic lake in Guatemala, complete with cliff-jumping!

* staying with a local family on an island in Panama (in a village with only huts and no sewage system) and then island hopping between untouched paradise islands

* a bunch of teachers going to the lake house of one of my students and doing lake sports and BBQing with their truly awesome family

* Almost getting robbed by a masked bandit in Guatemala at machete-point and driving off while screaming. (This is not a good memory, obviously, but it taught us a helluva lot about being safe. So, it's a very memorable experience.)

As the year wraps up (Geoff leaves in 7 days and I leave El Salvador 4 days after him), I am endlessly thankful for the experiences that we have had. Geoff and I always say this, but it's true -- we are some of the luckiest people, because we have jobs that we love, we are saving money and traveling at the same time, and we have each other to share this amazing experience with.

Summer (and Germany), here we come! :)

PS. I think the more Spanish I have learned, the worse an effect it has had on my ability to spell in English. Take "glacier" for example, it didn't immediately occur to me that I was spelling it "glaciar" like it is in Spanish, since when I visually checked the word, it looked very familar to me. And for the same reason, my natural inclination was to type "masage" (similar to

*masaje*) instead of "massage"... Dangerous!## Friday, June 3, 2011

### Unit Circle and Wave Functions Project - Part 3

Just to round it out a bit, I thought I'd finish writing about how I leveraged my students' understanding of basic sine and cosine functions and their relationships to circles, in order to get at some basic solving of simple trig equations. It bothers me that in the textbook we'd see exercises like:

1 + 3*sin(5x) = 2 and kids would be expected to solve the equations without ever knowing what it means.

It's simple:

A kid who understands how sine and cosine functions relate to circles should be able to write the function f(t) = 3*sin(5t) + 1 based on the above description. And then, they should be able to say, "I want to solve for time given the height is 2, so I'm going to plug 2 into the left hand side of the equation. Thus, 2 = 3*sin(5t) + 1." And then and

We can also illustrate the use of other simple trig equations such as [sin(x)]^2 + [cos(x)]^2 = 1 the same way. Instead of giving our kids cos(x) = -0.9 and asking them to solve algebraically for all other 5 trig values without knowing what it all means, why don't we instead say...

What kids should be able to do (with a bit of practice) is to go from something like f(x) = -2cos(0.5x) + 8 and g(x) = -2sin(0.5x) - 3 to drawing a circle of radius 2, centered at (8, -3), with its rotation starting at (6, -3) and moving at 0.5 radians/sec counterclockwise. From there, given a description of a certain height OR horizontal location, kids should be able to choose the appropriate equation, substitute in the appropriate value in the appropriate place, solve, and explain in a simple sentence what the numbers mean. And then, they should be able to take those waves and graph the wave functions while explaining the transformations that occurred to the waves as a result of the transformations inside the circle... thereby showing a full and flexible understanding of sines and cosines.

...I'm a newbie Precalc teacher, but I feel strongly we cannot be teaching kids how to solve trig equations -- even the basic ones! -- without re-focusing on the meaning behind those sinusoidal functions. I'd love to see how other teachers can take this idea and apply it to more complex trig operations!!

PS. You know what's cool about kids playing around in GeoGebra? Some of them discovered for themselves that if you do

Addendum June 4, 2011: Here is a sample worksheet I used in my class that gets at some of these concepts in an integrated manner.

If you want the full file, you can grab it here and let me know how it goes. :)

1 + 3*sin(5x) = 2 and kids would be expected to solve the equations without ever knowing what it means.

It's simple:

*Let's say you've got a point C rotating around a circle of radius 3. The center of the circle is at (0, 1) and the point starts rotating at the normal "3 o'clock" position, (3, 1) in this case because the radius is 3 units. The rate of rotation is 5 radians/sec and the point rotates counterclockwise. At what times during the first cycle will this point reach a height of exactly 2 units above the x-axis?*A kid who understands how sine and cosine functions relate to circles should be able to write the function f(t) = 3*sin(5t) + 1 based on the above description. And then, they should be able to say, "I want to solve for time given the height is 2, so I'm going to plug 2 into the left hand side of the equation. Thus, 2 = 3*sin(5t) + 1." And then and

*only then*should this kid find the two time values t that correspond to this y value! My kids can sketch a unit circle and articulate that we have two solutions (within the first full rotation) because there are two points along the circle with that same height. And then they can justify using a sketched circle why you find one radian solution, and then the other one is just pi minus that first solution (each of which you then divide by 5 to go from radians traveled to time it takes, if you are moving 5 radians/sec).We can also illustrate the use of other simple trig equations such as [sin(x)]^2 + [cos(x)]^2 = 1 the same way. Instead of giving our kids cos(x) = -0.9 and asking them to solve algebraically for all other 5 trig values without knowing what it all means, why don't we instead say...

*You have a circle whose radius is 1 and its center is (0, 0). Point C starts rotating around this circle counterclockwise at (1, 0) and moves at the regular speed of 1 radian/sec. Where (x, y) will Point C be after 5 seconds? When point C reaches a horizontal location of 0.9 units to the left of the y-axis, how high or low will it be? Find all 6 trig ratios at those points.*What kids should be able to do (with a bit of practice) is to go from something like f(x) = -2cos(0.5x) + 8 and g(x) = -2sin(0.5x) - 3 to drawing a circle of radius 2, centered at (8, -3), with its rotation starting at (6, -3) and moving at 0.5 radians/sec counterclockwise. From there, given a description of a certain height OR horizontal location, kids should be able to choose the appropriate equation, substitute in the appropriate value in the appropriate place, solve, and explain in a simple sentence what the numbers mean. And then, they should be able to take those waves and graph the wave functions while explaining the transformations that occurred to the waves as a result of the transformations inside the circle... thereby showing a full and flexible understanding of sines and cosines.

...I'm a newbie Precalc teacher, but I feel strongly we cannot be teaching kids how to solve trig equations -- even the basic ones! -- without re-focusing on the meaning behind those sinusoidal functions. I'd love to see how other teachers can take this idea and apply it to more complex trig operations!!

PS. You know what's cool about kids playing around in GeoGebra? Some of them discovered for themselves that if you do

**C = (3cos(t), sin(t))**, you end up with the point rotating around an ellipse (because the amplitudes are imbalanced horizontally and vertically). How are textbooks supposed to replace*that*type of discovery??Addendum June 4, 2011: Here is a sample worksheet I used in my class that gets at some of these concepts in an integrated manner.

If you want the full file, you can grab it here and let me know how it goes. :)

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