Monday, November 29, 2010

Measurement Unit! A Sneak Preview

I am starting the Measurement Unit! yay. :) (Starting this week, to be finished in January, after the midterms.)

I did this unit with my kids last year, and it was where my whole outlook on the Geometry curriculum sort of started to turn around. (I had really dreaded my first few months of teaching Geometry last year...)

I'll write more about the revised lessons as they unfold, but the order goes more or less like this (modelled after last year's sequence):

* Day 1: Intro to measurement. What quantities can we measure? How do we convert between units? (Including move-around activity for measuring quantities around the classroom.)

* Day 2: Indirect measurement of height. How can we measure an object's height using proportions and a mirror, or proportions and shadows? (Including outdoors activity portion)

* Day 3: Direct measurement of height. How can we accurately measure the height of a balcony using a string, a water bottle, and a meter stick? (Including group competition of results!)

* Day 4: Measurement of volumes. How can we measure/calculate the volume of a container shaped like a cylinder or prism? How can we predict how high water will rise once transfered to another container? (Including hands-on measurements and water transfer activity)

* Day 5: Conversions between liters and cm^3. How big is a liter? A cubic meter? How many liters of water will it take to fill up our classroom? (Including demo for transfering water from a one-liter bottle to a cube of 1000 cm^3 volume, plus move-around measurement of the classroom)

* Day 6: Measurement of irregular volumes, density, liquid density. (Stations activity - kids rotate around to measure: volumes of irregular containers via transfer of water; mass and volume of rocks via triple-beam balance and displacement of water; weight / volume / density of liquids. Also included discussion of net weight and "Gee whiz!" demo of how mixed liquids will separate based on liquid density.)

* Day 7: Reading about how the emperor measured the weight of the elephant, plus a bunch of conversion practice.

The big difference is that last year, we ended up doing a bunch of practice at the end of the unit, and also had little focus on estimation. This year, I will be pushing more estimation throughout the measurement processes, and also sprinkling / spacing out more of the conversion practice throughout the unit. :)

On Grouping by Levels

When I was a kid, I moved with my parents to a country (ie. the States) whose language I didn't speak. In the few years that followed, I experienced being grouped with other language-learning kids in remedial classes, where the teacher taught less material every week simply because the teacher didn't have faith in our collective ability to learn. Even as a kid, I had decided that no one else was going to determine for me what my limits were. I went home and studied, on my own, so that I wouldn't lag behind other kids in other classes. In the end, I think I turned out doing OK, even though I definitely took a bit of time to get there.

As a teacher, I occasionally come across kids who really, really struggle with basic instructions/material on a daily basis. (As in, 20 to 30 minutes into the problem set, and they're barely starting #2.) Sometimes I find myself feeling very frustrated, wondering if such a kid is really placed in the right class. And then, a part of me always asks quietly, "Who has the right to say what any one kid can achieve? If I hold them back, if I ask them to change into a more remedial class*, then I am fundamentally doubting the kid's ability to achieve. Maybe all this kid needs is a little more time -- and a little more patience/a different approach from me."

The truth is, in the end I don't know if I have the ability to serve every child in my classroom the way they need to be helped. But I can do my best without giving up on that kid. (And, very occasionally, that mentality translates to seeing a 60 going up to an 80 by the end of the year.)

*I find this type of mentality to be surprisingly common amongst honors-level teachers. Often it's very easy to say that a kid doesn't belong in our honors class, because they aren't quite as "quick" as the others. To me, my favorite kids -- honors or not -- are those who give me 110% daily. If a kid is willing to REALLY try their best in an honors class, who are we to say that they don't belong? For that, I really like my school's open-enrollment policy for honors (and AP) classes.

Sunday, November 28, 2010

Thanksgiving with the Coxes

Geoff and I spent the last four days playing host to his parents. It was AWESOME - I actually never imagined that it would be possible to jam pack so many things into four days, in El Salvador. We saw a beautiful beach (and did a whole all-inclusive thing) that had a stunning salt-water pool, a jacuzzi, and many very large and luxurious pools. We had an oiled massage (here they are $15 per hour... very affordable!). We took the Coxes up to a beautiful (and very delicious/intimate) fusion restaurant at the top of the mountain, overlooking many mountains and the city while the sun was setting. (The owners came out and talked to us, and one of them told us the story about how the restaurant came to be, and also played and sang some tunes on his guitar. Geoff's parents LOVED that!) We visited an old Spanish colonial town, saw its church, had a drink by the lake, and even hiked down a bit to check out the awesome hexagonal-prism shaped rock columns that are completely natural. And on the last night, we went to a beautiful restaurant that's already all decked out in Christmas spirit, on top of Torre de la Futura. (Geoff's mom LOVES Christmas, so it was a special treat for her to see the whole place decked out already.)

All in all, it was an absolutely lovely weekend. Happy Thanksgiving, and may we all be thankful for family and for love.

Monday, November 22, 2010

Resistors Lab!

I loved doing the rational functions activities with my Precalc kids!

1. The laser lab was FANTASTIC. I have to say that in El Salvador, it's not easy finding laser pointers. Even though my school has awesome resources and even a full-time driver to whom I can make requests to run school-related errands for me during the day, he was only able to find two $16 punteros laser (muy costosos!!). In the end, I had to run around and borrow make-shift laser pointers that were originally part of some fancy schmancy USB powerpoint clicker device. The science teachers were awesome and let me borrow 4 of their fancy clickers, plus our awesome head librarian had two in her presentation technology collection, plus I had my two very-expensive $16 pointers. Made just enough for a class set! Yay.

But, all of the hassle was super worth it, because in the end, the kids collected beautiful data that fell neatly into a rational function pattern, and we discussed the conceptual linking between where the domain breaks, the amount of vertical shift, and our laser setup. (The domain, which represents the laser source's horizontal distance away from the wall, is x > 30 or so, because the activity instructions indicate that the mirror needs to be placed horizontally 25cm away from the wall, and realistically it's hard to stand/place the laser source right on top of the mirror when you're doing the laser measurements, since the mirror rests on top of a platform made of textbooks, that juts out another few centimeters horizontally. The range of the function, which represents the height of the reflected laser beam on the wall, is y > 15 or so, because the activity instructions indicate that the mirror platform be about 10cm high. Our kids estimated that the lowest point the reflected laser beam could reach on the wall is just above that, or ~15cm. This gave them the partial equation y = a/(x-30) + 15, and all they had to still do was to plug in a point to solve for a.)

It was super!

2. And then today, I ran another lab with my Precalc kiddies on resistors. (See below.) Previously, I had used Megan Golding's resistors questions as intro to solving for equivalent resistances. We did that intro on Friday. Today, for the actual lab portion, my kids first took color-coded resistors and used ohmmeters to find out the resistances across each individual resistor. (Since not all resistors that are color-coded the same actually have the exact same resistances AND since the class needed to share a small stack of resistors, I made the kids grab two of each color to find their average resistance to use in the calculations/predictions. That way, it didn't matter much later on if they grabbed another resistor of that color; its resistance value would be roughly the same as the average resistance they had found earlier.)

I then gave them some series and parallel situations, and they had to make a prediction for the equivalent resistance, and then use the ohmmeter to verify their prediction. It was super cool; the math on paper really came alive for them! They got to see their calculation results match what was popping up in their ohmmeters.

Another cool (but tricky) part of the lab was teaching kids to read ohmmeters. I'm not sure if all ohmmeters do this, but the ones I had borrowed from the physics teachers have a dial of different settings. Depending on the resistance value, you have to change the position of the dial to measure a different maximum resistance, and the result actually takes on different decimal places / different units. After about 10 to 15 minutes, kids were starting to get the hang of reading the different settings for the correct units, but in the beginning it was quite a bit tricky! (The physics teacher was really pleased when I told him this afterwards; he said it's good practice for the kids to read/interpret the outputs from a machine such as an ohmmeter.)

Sunday, November 21, 2010

Paradoxes in my Students

So, first I should preface this story by saying that we have some really great kids at my school. Their parents are some of the richest and most influential people in a third-world country (some, possibly in all of Central America), and many of them are expected to take over the family business regardless of how they do in school. Some of their parents are away from home all the time because of work, and they are raised by maids and drivers. Yet, despite all of this, about 90% of the kids are really kind. In various circumstances, I've seen the way they treat the kids who are less fortunate (ranging from orphans to poor kids to disabled kids), and their kindness is always genuine.

So, yesterday Geoff and I went to chaperon a volunteer trip to build houses in San Vincente for the victims of Hurricane Ida last November, who are still displaced from their homes. We took a group of 6 juniors who voluntarily met us at school at 7am and who helped to carry rocks and to paint houses in a remote village/work site from 8am to 3pm on a Saturday (getting back to school ~4:30pm). Geoff and I worked on the metal foundation of a house for that time, to improve its earthquake-preparedness. All in all, the kids were fabulous. They really enjoyed the experience, especially because they got to meet the families whose houses they were helping to re-build. The families said some really powerful things, like they've had the strength to go on (after losing everything in Hurricane Ida) only because they have seen the help that God had sent them via all of the international and local volunteers. (I'm not religious, but my kids certainly are, having grown up in a conservative Catholic country. So, I'm sure hearing this is even more moving for them.)

But, there were strange things I observed that were characteristic of even our best kids that I wish were not. For instance, during lunch, our kids went into the school van, turned on the air conditioning, and slept in the AC while every other Habitat for Humanity volunteer sat in the dirt and hung out. Or, at 3pm, they came to me and asked if we can stop at a gas station on the way back, so that they could use the bathroom, because they couldn't stand using the outhouse. I was pretty embarrassed for them; I told the kids (because it was what I was feeling and I've taught more than half of those kids) that they were re-affirming the impression that the American School kids were too good to use the same bathroom as everyone else. I also told them that in some countries/places, those kinds of bathrooms are all people have, all the time. After I said that, a couple of the kids chuckled in embarrassment and half of the group went to use the outhouse. I took the rest up to a "nicer" bathroom up the hill, because I figured it was better for them to use another bathroom than to hold their need in the car ride. But, I couldn't help being awed by the irony of it all.

There they were, volunteering their entire Saturday to sweat under a ridiculously warm sun in order to help out people who had lost everything a year ago in a flood. Yet they couldn't bring themselves to use an outhouse. Amazing.

Friday, November 19, 2010

The Next Big Move

It's official: I am looking for a new job! Geoff and I had decided a few months ago that we wanted to move to Europe after this school year, the reason being that in a few years, we might be married with kids and will not have the same freedom we have now to travel and look around.

It took me a while to tell all three of my supervisors (mostly because they're each insanely busy, and it's not one of those things you want to say during the passing period), but now the deed is done. Next up: Looking for a job!! Scary. I don't have IB experience, which is a biggie when looking for European jobs, so Geoff and I will have to be extra flexible. But, we're hopeful that since I'm starting relatively early (now), that I'll find a job by June 2011. :) (Geoff's working on getting his British passport in the meanwhile.) The exciting part is that we get to go to somewhere different, that hopefully will also allow me to teach something different (ie. AP Calculus or IB)!

So, keep your fingers crossed for me that I won't be jobless (and homeless) by June.

Thursday, November 18, 2010


This is the part of the year when I start to emphasize to students that every day, they're making choices towards their learning. During Quarter 1, I pretty much hand-held the freshmen through all quiz corrections. Every time a kid did poorly on a quiz, I emailed home and convinced their parents to talk them into staying after school for some remediation. Last year, there was a change sometime during the latter part of Q3 where kids started to be proactive on their own about their learning. I want that to happen sooner this year. Like this time, I told kids specifically if I thought they needed extra review time with me after school before the test. Most came, although a couple of the kids didn't come because of sports commitments or other things. I told those kids sternly that they're making a choice, and they have to understand that consequences follow their every choice. That way, if they don't end up doing too hot on the exam, it'll be a learning experience for them about making positive choices.

Sunday, November 14, 2010

Miscellaneous Geometry Projects

We're more or less through with a few tedious, very algebraic weeks in Geometry! yay. Next big unit will be super hands-on (Methods of Measurement), so it'll be a nice break from all of the heavy-duty algebra. In the interim, I've taken some projects from the wonderful Nancy Powell and modified them a bit. Check them out!

For the mini-golf project, I took Nancy's project and added a couple of scaffolding questions. I also added a section where the kids would design their own golf course (which I think she does make the kids do on the computer, in GSP, but it wasn't in this version of her project).

For the string art project, since I don't actually want to spend a lot of class time making the artistic portion of the project, I made the whole sewing-with-strings thing to be optional (extra credit). Instead, the focus of the project is on identifying symmetries and constructing regular shapes using a compass. (The kids will need to be able to construct these same shapes later, when we begin to build nets of 3-D solids.)

That should take us to almost Thanksgiving. After Thanksgiving, we will have only a short week or so of instruction before we have to start reviewing for midterms (given twice a year)! Wildness.

Thursday, November 11, 2010

Thinking Aloud

I noticed on a recent quiz that my kids have trouble identifying angle relationships once there is a network of more than 3 lines. I have an idea for making kids construct parallel lines using the angle concepts, that will hopefully help to further their ability to visualize angle relationships.

In my mind, the exercise looks like this:

1. I'll first let the kids draw a scalene triangle and label it ABC.

2. I'll ask the kids to use a protractor to construct "a line parallel to AC at point B, using the concept of alternate interior angles." (The kids should be able to do this quickly, since that's the same angle relationship we used during our tessellations project a while back to create parallel lines. But, in my experience, kids need some help interpreting things like "a line parallel to AC at B." It's surprisingly difficult for them to decode what that means!)

3. Then, I'll ask the kids to construct "a line parallel to BC at point A, using the concept of same-side interior angles."

4. Finally, the kids will construct "a line parallel to AB at point C, using the concept of corresponding angles."

5. Depending on if the kids feel like this has been a difficult exercise or not, at this point I might optionally insert requirements for written explanations next to each newly constructed line, explaining which angle pairs are which type of angles. (Good practice with naming angles with 3 letters.)

Just thinking aloud.

Rational Functions Fun!

I found a GREAT rational functions activity over at NCTM, that really focuses on helping kids understand the meaning of basic rational function equations. I'm in the process of doing some review / test with my Precalc kids, but I have already given them the packet and we're going to be doing a good chunk of it (including the reflection activity!). yay! So excited. That, along with Megan Golding's resistors lab and Kate Nowak's intro to rational expressions (which my kids have already seen), is going to make a neat mini-unit on rational functions! :)

Sunday, November 7, 2010

Constructions with Compass and Straight Edge

I had my honors kids do a series of constructions in class with compass and a straight edge. To help them overcome the temptation to "cheat", I gave them popsicle sticks as straight edges. Most kids figured out right away how to do an isosceles triangle (all on their own), and about half of the class figured out on their own how to make a kite. (I figured it was a small hop from being able to do isosceles triangles.) Some kids fumbled their way to an equilateral triangle while trying to get the isosceles one. Then, everyone struggled with constructing a pair of parallel lines, so after they struggled for a while, I let them open up to a part of the textbook that describes the "rhombus method" for constructing parallel lines. Except, the way that the book describes does not allow them to construct 5 equally spaced parallel lines as I had requested. So, either the kids had to fiddle and figure out on their own a modified "rhombus method" (which a handful of kids did manage to do), or they had to get a hint from me.

All in all, the kids liked the activity so much that I decided to turn it into a project. So, the next day I gave them a list of specs, and they brought me clean final drafts with explanations for justifying why the sides are indeed congruent.

I tried to scan in the best piece of student work, but the scanner doesn't pick up on the arc marks all too well. For a kite, he started with two intersecting circles of different radii, and connected the circles' centers to the points of intersection in their arcs. He constructed equilateral triangle using two intersecting circles of the same radius, and his parallel lines are formed from a network of congruent circles.

Neat, eh? Lots of math with no numbers.

We'll be seeing construction again, very soon...

Saturday, November 6, 2010

All About Angles

Admittedly, parallel lines and transversal problems are very contrived in most cases and are not very real-world relevant. But, I still like those problems because they give the kids some basic algebra practice, while forcing them to think about the meanings of their equations.

In the last two years, I have consistently introduced the series of parallel-lines-and-transversal theorems using the same worksheets, and they have worked very well for my kids. My theory behind these worksheets is that: A.) In order to learn the angles vocabulary, kids need to be actively engaged in visualizing the relationships. So, why not start with making them visualize the relationships before introducing the terms? B.) Once they learn the terms, kids can discover all angle relationships via a protractor. C.) At the end, you give them some memory tools or some color-coding shortcuts to quickly figure out angle relationships for setting up angle equations. That way, even if a kid can't remember the name of an angle relationship, they can still have enough geometric knowledge to move through the algebra part. D.) In the end, as a quick check-in, kids should be able to quickly pick out the correct equations corresponding to different diagrams.

So, here we go.

1. Getting kids to visualize angle location relationships before introducing the terms.

2. Getting kids to conjecture about angle measurement relationships on their own.

3. Using colors to re-inforce angle relationships. (And I give them the memory tool that only "Same-side interior" and "Same-side exterior" angles are "Supplementary." Every other pair of recognizable relationship is congruent.)

4. Final check-in. Kids should all be able to pick out the right equations.

These are worksheets and not activities, but they are very effective in teaching the basics of angle relationships. Follow it up with a day of textbook algebra problems practice, and your kids are golden on this often-tested concept! (My Holt Geometry textbook also has an interesting angles word problems activity that I have adopted the last couple of years, which works well as an extension to ask kids to look at the application of angles in a slightly more realistic situation.)

And of course, you can always defer to Erastothenes to show kids some gee-whiz angles math.

By the way, stay tuned for my kids' straight-edge and compass construction projects. Neatest pure-geometry thing we've done in a while.

Wednesday, November 3, 2010

Austin and Panama

We have spent the last two weekends away, first in Austin for a wedding, then in Panama over the four-day weekend (para Dia de los Muertos). Some observations:

* Downtown Austin is just as fun as rumor has it! But, you have to be ready for bars to smell like 18-year-olds (ie. throwup). ...And for the bars to close at 2am (demaciado temprano...).

* Taxis in Austin cost something like $2 a minute! It's pretty insane. Geoff almost asked one of our drivers whether the meter was broken. I think every single time we left our hotel to go anywhere, it was about a $15 ride -- even if the ride takes only 5 minutes! Thank goodness we were kindly given a ride by some of Allen's non-drinking friends to and from the wedding, so that we could party it up without having to be the DD.

* Best restaurant we found in Austin (recommended by the locals), hands down: "Moonshine." It has various Southern food, but with a unique twist. It also feels like you're sitting in someone's back yard, having a nice brunch.

* San Antonio, TX, is also a nice town. Downtown San Antonio has a man-made river-front that's really nice, and with very friendly bartenders.

* Panama Canal is just a bunch of locks. It has got a cool history bit, obviously, but isn't actually much to look at. If you go, you should definitely pay the extra $3 to watch the introductory movie, because you can get a sense of the past, present, and future of the canal and its continuing importance to the country and the world.

* Panama is diverse! I love that. Panama City is probably the most metropolitan city I've seen in Central America. They also have ethnic foods (including Indian, even though our taxi driver messed up and took us to a Lebanese restaurant instead), which is really exciting. The area around the casinos is verrrry "working girl" friendly, which we discovered by accident.

* By law, only the native tribes can own the land on the islands scattered around the 365 beautiful San Blas islas. We stayed with a native family overnight, and went on some island-hopping during the day. The village was very rustic! (For example, various households, if not the entire village, share two "toilets", which are merely two holes that hover above the ocean. They don't have a sewage system. The villagers live in grass huts and throw their trash directly into the ocean.) It was a really neat/unique experience!

That's it for now. Progress reports are due this Friday, so things are busy with work, obviously. Soon, Geoff's parents will be here visiting us (over Thanksgiving weekend), and we'll be busy on that end with the preparations, as well. :) I can't wait. The year is just flying by!