Christmas Sunshine

Having really warm weather for Christmas makes me feel like a kid again, as though I had never left California. (In SoCal where I grew up, it was not unusual to have 90+ degrees of sunshine on Christmas Day.)

Anyway, thought I'd take this ample free time (now that we're mostly set for going away on Friday) to put up some updates of what we've been up to in San Salvador. :)

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We recently climbed Santa Ana, which is the tallest (not steepest) volcano in El Salvador. It last erupted in 2005, which by nature's time frame is like a second ago! (So, I think that makes it tied for the second most active volcano we've climbed in Central America.)

Here is what the rim at the top looks like around the crater. Pretty steep when you try to look into the crater (left). Those little specks are people up ahead.


This particular crater has 3 strata, formed from 3 different eruptions over (a long period of) time. It's stunningly beautiful, and you can see all the layers even from afar. Down at the bottom there is a green boiling crater lake. Our friend Greg kicked a rock over to see how far down the crater is. For almost a full minute, you can still hear the rocks rolling/rumbling down below.



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Sometime in November, we went to the Marine Ball (again). It was fun to play dress up (putting my $20 dress to good use), but I think I've officially gotten it out of my system. This year, the good thing is that we got a deal. Dinner + dancing + open bar for $40 ain't bad at all.


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Lastly, some other time in November, the German embassy sponsored an Oktoberfest. It was pretty cool (especially because there were some social dancers there, tearing up the dance floor), but it was a teaser for the real thing we want to go to next year. :)

Here are pictures of us, with Will and Andrea, two of my favorite gringos:



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That's (really) it for now. We're super excited about going to Argentina!! :) We leave on Friday, and will be gone until after New Year's. On our agenda are glaciers!! -- and maybe some dancing and some wine tours. And definitely lots of eating. Argentinian steaks are supposed to actually come from happy (free-range) cows. :)

Jobs and How Funny Life is

I am always interested in finding out how people ended up where they are. I myself have always been surprised with the jobs I actually end up with.

After college, I thought I was for sure going to go work for Intuit (the guys that make your TurboTax software). I had had a successful internship with them the summer prior, and since I was pretty sure software was just a temporary gig until I figured out what I wanted to do with my life, it made sense to stay in California.

But, you know, all of my other engineering friends were going to on-campus job fairs the spring of my senior year and looking at interviewing with big companies, so I figured why not. I sent in some resumes and got a first round of interviews with Amazon.com, followed by a second round of all-day interviews on campus. It was sort of... exciting. I did it because I had nothing to lose (since I already had a job offer from Intuit), and I kind of liked writing code on the spot and discussing data structures with strangers. (In a totally geeky way, interviewing was really fun.)

Once Amazon decided to give me an offer, they flew me and some other people out to Seattle to visit their company site. As it turned out (as it always turns out), it was love-at-first-sight with me and Seattle. My boyfriend at the time was clearly non-committal about staying with me for the long term, so I decided that moving was clearly the right move for myself.

Two years later, I was a pretty happy programming monkey at Amazon (I had an amazing boss, a great boss's boss, and my best friend sat 3 feet from me), but I decided I had figured out what I wanted to do with my life -- teaching. I was so excited actually, that I couldn't sleep at night.

And so began my research into teaching options. I pretty much figured right away that some things, such as going back to school full-time, just weren't for me. In my research, I encountered some programs like TFA and the New York City Teaching Fellows that offered you the opportunity to teach while getting certified. That seemed to me like the way to go. So, I went ahead and applied. Long story short, I ended up in NYC, looking for jobs during the first summer immersion training.

My former principal liked to tell the story about how I basically stalked him. It's more or less true. I had interviewed with only a handful of schools, but nothing really was working out. (My demo lessons pretty much sucked.) Eventually, somewhere on insideschools.org, I came across a review of a new small 6th-through-12th-grade school called MS 241. The principal called me once, and then I called him about 10 times. Never got through, so I went and found him at a job fair. Sort of a fluke sort of thing, but I knew he was the person I wanted to work for as soon as I sat down with him. The school he described to me sounded like a dream. Kids learned by playing, kids learned math and science that were real and active. And, miraculously, after a demo lesson, this amazing school still wanted to hire me.

In the three years that followed, I learned more about teaching than I probably ever will in any stretch of 3 years in my career. I have little doubt that that school has some of the most incredible staffers anywhere in New York. (And trust me, I know there are a lot of good teachers in NYC.)

When I decided that I wanted to do the international teaching thing, Geoff and I agreed that for the first few years, we'd stay in relatively the same time zone. He needed a few more years to get his self-run company to a point where he wouldn't need to be available everyday from 9 to 5. So, I began contacting every Latin American school I could find. El Salvador was, funnily enough, the first school to show serious interest in hiring me, after receiving my cold email contact. They were willing to offer me a job even without having met me in person.

Like any sensible person who has already jumped through the hoops of registering for a recruitment fair, I postponed accepting their offer in order to check out other schools. In the end, however, after sifting through all the offers on the table and talking to each school administrator about the complication of having a non-teaching "spouse" who might bring about visa issues, the school in El Salvador still seemed to be the best fit. (Funny how life works.) :) And we're both very glad we had made that decision; this school has been very good to me, and more than accomodating about my situation with Geoff.

I can only hope that the next place will be a blessing the same way that every other employer has been. Job searching is so exciting! And so scary. I could be throwing my whole life away, or not.

Things useful to have in a job search

Since I've been looking for a job recently for 2011-2012, I thought I'd throw out there what I think is useful to have in a job search, at least if you are looking for job through cold email contacts. Some of this might be super common sense.

1. A nice cover letter with a photo. Everyone likes photos, and a compact, nicely written cover letter can immediately set you up as a purposeful applicant.

2. A CV instead of a resume. Why? To me, a CV is more comprehensive. My CV starts off with a one-paragraph summary of myself, to be followed up by my list of experiences and academic achievements underneath. Now that I am on my third job, I think a one-page resume isn't really sufficient to highlight my accomplishments, once you take away the space for my name and contact info. Besides, if a school can't even take the time to look through the second page of an already condensed CV, they're not likely going to make a great employer, right?

3. Open reference letters (with contact details). Nothing replaces confidential references, but open letters are nice to have, because if a school is not sure whether you are a viable candidate, these could tip the scale without them having to jump through the hoops of contacting each reference. Open letters are also nice to have because you get to keep and re-use them for later. So, if you get a couple of letters from each school you've worked at, you could easily have a bunch of letters on file after a while. Your reference letters should include some supervisor letters, obviously, but you may also include those of former professors, co-planning colleagues, and student parents. My portfolio contains all of the above, because they each highlight a very different strength.

4. A website. Mine is very simple (sorry, no links here because I don't want to mess up the stats I am tracking), but I think it's pretty straight-forward and effective. You link to your CV, provide an email link, provide a summary of yourself, a picture, and a list of "testimonials." It sounds mad cheesy, but it works. The point of a website is so that when you send unsolicited emails, the school heads aren't annoyed when you send unsolicited attachments as well. And the testimonials, as cheesy as they are, are supposed to make you sound more attractive at first glance.

Anyway, I've contacted a good number of schools, just sort of on a first-round basis. I've heard back from some of them, but few definitely-interested ones at this point. I'm not worried; I am going to that fair in London in January and I'm sure things will sort themselves out at the fair. But, my goal before that is to contact ~50 schools. (It seems like a very obtainable number, considering that I have amazingly already contacted about 30 schools.) So, here we go! :)

By the way, recruitment fairs are totally the way to go, if you ever want to go for an international teaching job. It's quite a bit of a hassle to collect all of the necessary confidential references by ~November, but you get invaluable face time with each school, and those schools can follow up on your file before and after the fair (if you choose the right recruitment agency to go with). A lot of schools won't even consider cold-email candidates, since it saves them a lot of work to go through a recruitment agency.

For me, I do the international teaching thing because I love to travel. And I love new languages. It doesn't preclude me from the possibility of settling down in the States in the future and/or working at a public school again, although these days there is definitely less tying me down to the States...

Salvadoran Customs and Math

There are two Salvadoran customs that amuse me greatly, and they make me think about the math involved in each case:

* Here in El Salvador, whenever people need to cut a round cake, they first cut a circle in the middle of the cake. And then they proceed to slice the outer rim of the cake into equal slices. When they serve, first they serve the slices on the rim, and then they serve the center piece last.


A natural math question that arises from this is: How big will the middle piece have to be, in order to ensure that all of the pieces served are of the same size?

The math is very easy, so I'll leave it to you to figure out. I'm going to give it to my Geometry kiddies at some point, as a warmup problem. (Of course, the kids will have to figure out for themselves that the answer will depend on N, the number of slices, and R, the radius of the cake.)

On a random note: I like how they cut it this way. It actually makes sense to me, because it's much easier to serve the slices when they're not all long and skinny.

* Also, another amusing custom here is that when Salvadorans have a really big raffle prize (ie. Taca's round-trip tickets at today's staff luncheon at our school), they often will announce something like, "We're going to pick 3 names. The first and the second people we pick out of the hat do not win any prize; their names get discarded on the side. The third name we pick out is the sole winner."

It's funny because it's clear that this does not change the probability of winning. Each person still has the probability of winning = 1/n, whether you compute it as a one-time pick of a single name OR as the consecutive-picking-of-three-names-without-replacement method as described above: (n-1)/n * (n-2)/(n-1) * 1/(n-2) = 1/n. But the Salvadoran way is much funnier. The first two names picked out of the hat are the sorest losers (or they go around and brag about how close they had come to winning). How funny!!

...I love how there is math everywhere! :)

Anyhow, I am on vacation now, and will very likely be a useless lump until January. See you then.

How I Handle Midterms

My school institutes a semesterly cumulative exam in all classes. This "midterm" exam counts as 33.33% of the student's semester grade, averaged against the two quarter grades the student has earned. (Curiously enough, Q2 doesn't end until January, even though the fall midterm is in December. I actually prefer this setup, because it makes it possible for me to give the kids some extra-credit work, should they be on the border of passing/failing for the semester once the midterm grade has been computed and included. But, I only do that for the first semester, as I then tell the kids that they will need to earn their second-semester passing average on their own. It's a bit of a carrot-and-stick approach.)

I am pretty sure there are various people out there lobbying against midterms, and admittedly, these cumulative exams might not be something that works equally well in every subject, but I personally find them to be a valuable opportunity to spiral back to what we've learned and to do a cumulative assessment to see what the students have really retained. So, it's important to me that both my review and my exam reflect that philosophy.

This year, I structured my review as such:

* A few weeks in advance, I posted a list of topics and associated practice problems on the web. The kids could start doing those problems whenever they have free time, and the problems I assigned were either extra practice problems from the textbook that we didn't have time to do during the regular instruction, or they were problems from old exams they've taken. Each problem is aligned with a specific skill in the list of key skills that I think the kids should retain from the first semester. The kids know that if they show excellent math work (which includes verifying their answers with the back of the book) and they complete all assigned problems, they can gain up to 5% extra credit toward the midterm exam.

* During the allotted 3 days of review time in class (this is standard across all of the school), each day I would give the kids a short sheet of practice problems to jog their memory on older, dustier topics and terms. If they are sharp on that stuff and/or are aggressively asking me for help, they can get through those practice problems each day in about 30 minutes, and then use the remaining 30 minutes in class to work on the extra-credit study packet. (See above.) Or, if they're unmotivated and only idly working, they are kept busy reviewing pertinent material for 45 minutes to an hour (since in that case, the worksheet would take them longer to get through), so they're not bothering other people in class. (If they do start misbehaving instead of choosing to work on the extra-credit study packet, obviously you assign them a consequence. Another carrot-and-stick approach.)

* In the end, the midterms I gave were always a mixture of: problems similar to those on old exams; problems similar to those in the extra-credit study guide; and problems similar to those we practiced in class during the review days. I wasn't looking to surprise the kids for the midterm; I don't feel that's quite so fair for a test with so much weight. Plus, I wanted to make the entry point of each problem accessible for every student, so that a kid feels like they have some hope for getting every problem correct, using what they've learned. But, they were definitely asked to show, explain, and apply concepts, so the tests were by no means easy. One kid said, "Wow! We learned a lot of stuff this semester, and every single thing was on the midterm!" And others told me they thought the tests were not difficult, as long as you did your part in reviewing.

The result? Each class had a pretty nice distribution of scores, and in all cases, it was evident to me whether a kid had put in the extra effort to get through the entire study guide of practice problems. :)

I know that sometimes (...many times??) other teachers complain that midterms are a waste of time, but I don't think they have to be, if you see them as an opportunity to further the kids' familiarity with those older topics.

Mandatory Christmas Bonus

Here in El Salvador, employers are required to give their employees a Christmas bonus. Geoff and I will be giving our maid an extra week's pay (plus she will get paid time off while we're on vacation in Argentina). But, for the full-time employees (such as myself), the mandatory Christmas bonus breaks down as follows:

Up to 1 year of employment: Proportional bonus of 10 days' pay. (ie. If you've been employed for 6 months, your bonus is equivalent to 5 days' pay.)

Over 1 year, up to 3 years of employment: 10 days' pay.

Over 3 years, up to 10 years of employment: 15 days' pay.

Over 10 years of employment: 18 days' pay.

That's a pretty real example of piecewise functions, besides progressive taxes and volume discounts on products. How do you feel about the government dictating when you get bonuses? I know (from having interviewed with schools in Brazil in the past) that Brazil has all kinds of funny salary-related laws. Among them is one that says that you must get paid 13 months of salary a year; half of your "extra" month gets distributed as a Christmas bonus, and the other half of it gets distributed at the beginning of summer, in time for your summer vacations. How funny!

Learning as I Teach

I was reflecting upon how you always learn while you teach. This is especially true the first time you teach something, but also true generally, if you keep working at it over the years.

During the last day of review week for midterms, a Precalc kid asked me for help with a problem from the textbook. The problem describes a worm that is 5 cm on Day 1 and grows 3 cm the next day, and grows 1.8 cm the next day, etc. Each subsequent day, the worm grows 60% from the day before. What's the total length after 2 weeks?

So, I showed the kid how to use the geometric series formula u1(1-r^k)/(1-r) to find the sum of this geometric series. Then, the kid asked me, "But why can't we just use 5(1.6)^(n-1) as our formula, since the worm is growing 60% each day?" Took me a few minutes to think through it, and in the end I felt really silly about it all, because the difference is obvious as daylight.

The worm is not growing 60% everyday. If it were, it'd be growing exponentially faster! Instead, its amount of daily/incremental growth is diminishing at a rate of 0.6 (losing 40%) each day.

It's always so fun teaching something for the first time, because when kids ask me questions, I still have to do a double take on some topics with which I am rusty. :) (And then in real time, try to think of the best way to break it down.) Does this happen to you??

Anyway, MIDTERMS START TODAY!!!

A Kid-Friendly Analogy for Recursion

A kid I had taught last year came by after school the other day for some help with the concept of recursion. After a few minutes of looking at some simple examples, she still was kind of confused about why some values would depend on other values, where as some values (ie. the base case) don't need other values in order to be computed. I tried explaining it using the simple examples in the textbook and by looking at the recursive formula of the Fibonacci sequence. Still, she only half-understood, so I decided to throw out an off-the-wall analogy:

Let's say I tell you and a bunch of other kids to make a straight line, facing forward. Then, I tell every kid to smack the head of the kid in front of them. So, you smack the head of some kid, he smacks the head of another kid, etc. Until we get to the front. The kid in the very front, what does he do? He can't do anything; he doesn't have someone in front of him to smack. That sucker's sort of like the base case. The base case is the value you're given at the beginning of the problem because that's the one that you can't calculate.

The funny thing is, the kid totally got the crazy analogy! She said, "So basically, every number in the sequence depends on the one that comes before it, except for the base case. That's the number at the beginning, so it doesn't have any other numbers in front of it in order for you to calculate it, and so its value needs to be given to you by the problem." And then she proceeded to ask me smart questions like what happens if you're trying to calculate the 103rd element -- are you going to need to find all 102 elements before it? (And we got into a pretty good discussion about tradeoffs between different formula types.) ...I guess the moral of the story is that I should make kids smack each other more often in class, for the sole purpose of improving their grasp of confusing mathematical concepts. ;)

Life Matters

Brain dump. For me, mostly.

* The whole tax business in El Salvador is pretty funny to me. To start, we have a sidewalk tax. It's about $3.95 a year, and (I think) it means that you are allowed to use the sidewalk for the next 12 months. Then, when April comes around, you have to go in person to turn in your income tax forms. They don't accept it by mail or by internet. (But they do accept other people turning it in on your behalf. Strange.) And then, about 6 months later, you go pick up your refund -- in cash!! -- at any branch of Banco Agricola. At that point, the bank teller will tell you that they cannot issue you a check. So, if you're like me, you lug hundreds of dollars in cash while walking back from the bank -- an extra exciting experience in this country. It's strangeness all around.

* New York State Department of Ed is also pretty funny. A while ago, I was trying to get a duplicate copy of my teaching certificate, for job-searching purposes and also in case my current school gets audited. I went online, logged into the TEACH system, paid my $25 dollars, and then only afterwards saw a fine print somewhere on a totally separate FAQ page that they have, in fact, stopped the service of printing paper certificates for "time-limited teaching certificates!" In fact, they have completely eliminated the job of the person who used to print the paper certificates! Well, at this point, my options were to A.) pay another 50 bucks to upgrade my certification, still 3 years before the current one is going to expire, or B.) forget the paper copy. My potential employers and the Salvadorean Ministry of Education are going to just have to make do with my print-screen version of the "teaching certificate."

I was pretty mad (and almost equally amused). But, I am pretty stingy as well, so I decided to wait it out. --What do you know? Two or three weeks later, I get my duplicate certificate in the mail. :)

* I am about 85% sure I will be going to London in January for a job fair. I was waiting for days on a confirmation from my recruiters that my application and recommendations and payment all checked out, before I made travel arrangements. I had to follow up with them, because I noticed that the airfare had dropped $140 over the weekend (from $1100 to about $950). So, finally, I heard back from them this morning, and I rushed to log in to Kayak to buy the tickets. --Guess what? They're back up to $1100. And that's not including paying the recruiters, or hotel or food. So, that's all very expensive, and I'm back to being in a limbo about whether this is the right move. There is a good chance I won't get a job at this fair (for various reasons, timing and my lack of IB experience being the key ones), but if I don't go, I know I will regret it when I am stuck still looking for a job in April........

* Incredibly (as though I don't already have enough to do in the middle of job-searching and preparing kids for midterms), I am also working on applying to a summer program. I've already written my personal statement, put in orders for my transcripts (both undergrad and grad), and given the recommendation forms to my supervisors. I am feeling like this is not going to all pan out, but I feel OK about it. In case you can't tell, I am practicing being more of a go-getter, and my backup plan for the summer is to go to Herrang (in Sweden) during their month-long swing dance camp, and to dance until my legs break into pieces.

So, that's it. All of the things floating around in my head!

How the Allies Used Math Against German Tanks

A friend passed this article along. The story is super interesting, and might be relevant for those of you teaching statistics! (I tried to look up how they got all the formulas, but it ended up giving me a headache, because each link just led me to more formulas. Next summer, I really need to pick up a good book on statistics.)

About integrity

I have to say that it drives me absolutely nuts when a kid copies work off of another kid. That, and it breaks my heart that I have to give the kid a zero on the assignment, when the kid is already struggling (usually). Surprisingly (and thankfully), it doesn't happen much at my school.

Recently, after the resistors lab I did with my kids, I let the kids take the last problem home to finish up the trial-and-error calculation portion. Later, I found out while grading the labs that a handful of the boys copied the last problem off of this girl in the class. They actually didn't even use their own collected data values for the problem, but blindly copied the other group's values. So, obviously, I gave them all zeroes. But, I was feeling pretty crappy about this, because I supervised the groups doing the rest of the lab and I knew that all problems except for that one were actually their own work.

Well, before returning their lab, I thought for a long time about what to say to the kids, and every time I thought about it I just felt more upset about the whole thing. In the end, I didn't want to give them a drawn out speech and I just said two things when I returned the labs:

1.) You guys are a second away from college. In college, they're not going to be this generous when you get caught; you're not going to just get a zero on the assignment, but they will kick you out of the school.

2.) You're also about to be adults. If you're trying to grow up to be a lying and cheating adult, this is how you do it -- you cheat. But if that's not whom you want to be, then you need to start making different choices, because integrity isn't something that you just magically gain as an adult. You get it by practicing integrity everyday, now.

One of the boys came up to me after class and pleaded with me to give him some points back, if he would re-do the assignment. He said that he has never cheated before ("and Ms. Yang, you saw me do the rest of this myself!"), and the only reason why he copied off of someone else was because he really wanted to get full credit on the entire assignment, after all that work he had put in during class. It broke my heart, but I told him firmly that this is one issue I don't budge from; you can't cheat and gain back those lost points. It's a hard consequence, and 20 points lost on a lab is a very cheap lesson for learning that.

I've been in this situation before, and it never gets easier. When kids make mistakes, they need to learn the lesson. But it turns it into a me-versus-them sort of situation, instead of me-helping-them-do-the-best-they-can. After school, that same kid came back to me to get help to prepare for tomorrow's quiz. He wants to really do well on the next assignment, to make up for this one mistake. Did he learn his lesson? I hope so.

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.

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.

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.)

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.

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.

Choices

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.

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.

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! :)

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...

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.

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!

My Lovely Challenge

I am undertaking an awesome personal endeavor: reading my first Spanish novel! :) I am about 10 chapters (~60 pages) in, and the story is great! I had bought the book at a mall about a year ago, thinking that I would learn Spanish by looking up every word I didn't know (which was pretty much how I had learned English as a kid). But, very soon I realized that, as an adult, I am now much busier and have much less patience for looking up every single word in the dictionary. So, I struggled through about one chapter and gave up promptly. The book had since sat on my shelf, collecting dust.

Randomly, last weekend, on my way out to catch a flight to Austin for my friends' wedding, I grabbed the book since I didn't have any other handy reading material for the plane. I didn't bring a dictionary (it seemed like a hassle), so I tried to read the story using only context clues. Amazingly, it's now entirely do-able for me! Of course, there are still words I don't know, and still some verb tenses that I'm not familiar with, but as a whole, the novel is very enjoyable in all of its banter and irony even though I'm just reading it straight up without a dictionary. --HOW EXCITING!! (A year ago, I had started to write down the list of words I didn't know, that I was encountering in the first chapter of the book. Now looking back at that same list, those words seem really easy, so my Spanish has made a lot of progress! yay.)

Anyway, the book is really good so far; it's a translated American novel called La Loteria, and it's about a man who's mentally handicapped, who lived with his grandma until she passed away. After she passed away, the rest of the family swooped down to divy up her few worldly possessions, but didn't want to take care of him... until he wins the lottery. The really charming part about the book is that he would always state something as it appears, and then state it again in his own blunt understanding of the situation (without all the smoke and mirrors), which is extra cool for me as a language learner, because I get to see the same situation described with and without ill-intentioned euphemisms.

Yesterday, at school, we had a "Drop everything and read!" half-period to celebrate National Reading Day. I told some of my kids that I was slowly reading a Spanish book, and I think they were genuinely impressed! I too often forget that we're supposed to model for our kids that we, too, spend time learning things that are not easy for us. What is it that you do to model that mentality for your kids?

Why I Love Math Teacher Blogs

I did a really fun half-lesson today involving Dan's cup stacking idea, and I really let the kids try to struggle with it for a bit to figure out what they would measure, and how. It worked brilliantly! Especially having come after already several other exercises (see previous posts) where we had discussed the meanings of slope and y-intercept, I really thought that this one tied it all nicely together.

(I didn't use my own height; I picked a kid from each class, which was pretty funny for them, and I got to be the judge of when the stack of cups reached the top of their head.)

We also discussed why it's not as good an idea to measure only one cup, even though you could visually see roughly where the "stacking rim" starts and where the "extra part" ends on that cup. (If you're off by even just 1 mm in measuring the rim, that's really easy to do, but when you account for the fact that you're stacking tens or maybe a hundred cups all together, that margin of error will really add up!) We also discussed again that x is the cause, and y is the effect between the two quantities. In one class, their predictions were right on -- several groups were off by 1 cup only. That's pretty amazing, considering that each additional cup only contributes a couple of centimeters in height.

What a lovely activity!

And, in our precalculus class, we finally did the final testing for the catapult launches. They went really well, and most groups had M&M's that landed right around their targets (using the catapults that they themselves had built earlier)! :) How super cool. There was actually one group whose 3 launches out of 4 had landed right in the (pretty small) landing pad that they had placed on the floor. AWESOME consistency!

I also recycled those M&M's for Kate's estimation activity that introduced absolute values. Now my kids are spoiled because they got to eat M&M's for three days in a row. :)

This is why I love math teacher blogs!!

Tilted Parallelogram, Triangles, and Good Ol' Slope

Last year, when I taught parallel and perpendicular lines in Geometry, I took the kids to the computer lab and did some investigation via GeoGebra so that they can observe for themselves that the perpendicular lines indeed have reciprocal slopes, etc. This year, I'll probably do that again with the regular Geometry kids, but with the honors kids (who can already recall from last year how to write parallel and perpendicular slopes), I am skipping over that and giving them a more challenging GeoGebra investigation right away. This new task will ask them to complete parallelograms and tilted right triangles in the coordinate plane, thereby making them apply their knowledge of parallel and perpendicular slopes in order to find the missing vertices. (ie. If you are given three vertices in a parallelogram, where does the fourth vertex have to be? Are there multiple possibilities?) They can use a simple built in polygon tool in GeoGebra to verify that certain sides in their parallelograms (and/or in their isosceles right triangles) are indeed congruent.

Example problems from the activity (slightly rephrased):

• If A(1, 1), B(2, -2), C(4, 3) are three vertices inside a parallelogram, where is the fourth vertex, D? (Are there multiple possibilities?)


• If I(-1, 3) and J(0, 1) are two vertices in an isosceles right triangle, where is the third vertex, K? (Are there multiple possibilities?)


• Do the vertices M(-2, 5), J(0, 1), and K(the answer to the previous problem) make an acute, obtuse, or right triangle?


• Do the vertices N(-1, 4), J(0, 1), and K(as before) make an acute, obtuse, or right triangle?

I will also give the kids some already-drawn triangles at the end of the investigation, for them to find the slopes in order to determine whether those are acute, obtuse, or actually right triangles. (Because on my last test, some kids drew some triangles that looked sort of like right triangles but were actually not, when you take a closer look at their slopes. I took only a couple of points off then for their "right" triangles, since we hadn't explicitly talked about right triangles in connection with the slopes of their edges. In the future, the kids should be able to catch those mistakes on their own.)

I am excited about this, because it is asking kids to geometrically apply/extend their understanding of slopes, and it also paves the way to coordinate proofs, which are just around the corner.

Orthocenter Curiosities

I had been reading about orthocenter properties on the web one day when I thought that you might be able to show some of its properties using a tactile activity. I tried it out during my prep period, and it worked as I imagined! Pretty neat.

Here are some pictures, taking you through the steps.

Step 1: Draw a circle.

Step 2: Draw any triangle inscribed inside the circle. (Now is a good time to introduce vocabulary words like "inscribed" and "circumscribed"...)

Step 3: Cut out the circle, and fold it inwards along the edges of the triangle.

Step 4: Mark the point where the three arcs coincide. This is the orthocenter!


Step 5: Verify that your orthocenter is indeed the intersection of the three altitudes by connecting each vertex with the orthocenter, and making sure the result looks perpendicular to the opposite edge.

Step 6: To explain why it works, we label the sides that are congruent with tickmarks!

It doesn't help make orthocenters useful, but it is a fun/easy tactile activity (very 9th-grade appropriate, methinks) that shows visually some of the deeper mathematical properties about orthocenters.

Addendum: Oops - sorry, I was being sloppy with the vocabulary earlier. Can you tell I have circumcenter on my mind? :)

Fun in the Sun

THE SUN IS OUT! Has been for two weeks now. It's super lovely; I think we might be finally easing into the dry season. :)

Geoff and I have spent two beautiful weekends in a row at the beach, in good company. Last night, there was a music festival in El Tunco, so we (and apparently everyone we knew) decided to stay at the beach for the night. :) One of the bands was a rock cover band, and played such amazing old hits as "You Gotta Fight for Your Right." It was a great night... I won't divulge many details, but there was some spontaneous Charleston going on, with cheering Salvadorans. I almost had an asthma attack when we got back to our hotel, from trying to keep up with the crazyfast latino drum beats. Good times!!

Next week, we head off to Austin for our friends' beautiful wedding, and after that we will be in Tikal over the first (long) weekend of November! I LOVE this pre-holiday time of the year!! :)

Activities to Help Kids Understand Meanings of Slope and Y-intercept

I've been introducing linearity to my 9th-graders. I have some introductory linear activities that have always worked very well for me that I'd like to share with you. They work particularly well for regular 8th- and 9th- graders, but you can also adapt them to your struggling students in higher grades.

This first one is a one-day activity, spanning about 60 minutes. Kids go right into it without any teaching, and I only preface it by stating that there are multiple ways of representing changing quantities, and that they would have to reconcile the different representations against each other in order to compare the changing savings amounts throughout the year.





What I like about this first activity is that by the end of it, kids have a pretty intuitive understanding of what slope and y-intercept mean in an equation. They also get a preview of graphical systems of equations and "break-even points", if that is what your curriculum beckons next.

Then, I follow it up with this activity the next day, which again the kids try to do on their own without assistance from me.

For the most part, I observe that kids naturally will want to use either a proportional reasoning or a unit rate to solve the rate comparison problems, which is great! We go over why the unit rate is a direct connection with the idea of slope (once again), and why their process of making the predictions (while taking into account both the rate and the starting value) is very similar to using a linear equation. We also review visual connections with the graphs they made.

At the end, as a quick check in, we do a quick equations Mad Libbs, which takes 5 minutes and reinforces the idea of the meanings of slope and y-intercept. I explain while going over it that slope is "something per something else", and that this is illustrated through some of those Mad Libbs problems.

Again, my pet peeve is that kids often arrive in Algebra 2 without knowing what slope and y-intercept mean in the context of a problem, even if they know how to graph equations and how to write equations. --What is the point of having a bunch of algebraic skills if you don't know what they mean??

As a culmination for my higher-level kids (ie. Algebra 2 or Precalc), I think they should always be able to tell me given a regression equation, what x stands for, what y stands for, what the slope represents, and what the y-intercept represents. It's much harder for kids than you'd think, but I think it's super important for tying everything together.