## Tuesday, August 31, 2010

### Geometric Series Fun!

I'm excited about an upcoming lesson about applying geometric series. In fact, I am excited about it enough to be writing about it in way advance. :) (I'm not getting around to it until late next week. But, anyway, it's making me feel very OCD to be pretty much finished with planning for all 3 classes all the way through next week! I am kind of really in love with my job.)

Here it is. I love problem #1, because it brings up the idea of a summation approaching a particular value! If you keep constructing new rectangles at half size (adding counterclockwise each time), you'll see that they just fill in the middle portions and the overall geometric series approachs 2*(original area). YEAH! Chivisimo.

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In other news, Geoff's and my car broke down on the way back from the beach on Sunday. :( --OUR FIRST BREAKDOWN!! The mechanic has been sending us emails with estimated costs, and I just want to know which parts of the car are not broken, because it sounds like he's going to have to replace everything that I either did or did not know existed in a car!! (Fun fact: Our car was spilling oil all over the engine area after having spontaneously broken down on the side of the road, an hour away from our house, when it was already getting dark! It was a bit nerve-wracking, and it even started to rain just as we were waiting on the side of the road. Our friend Greg got out of the car, peeked under the hood, and said seriously, "You guys got a fire extinguisher?" ...Thank goodness we've got awesome friends who live near the beach, who arrived within 15 minutes with a tow truck to drag us all the way back to the city!!)

So, that's gonna be fun... I learned some key words recently from dealing with all the miraculous breakdowns in our lives: grua means tow truck, and calentador is the water heater at our house (whose thermostat broke because of electric current fluctuations that are a side effect of the heavy rains that we had over the summer). Really fun stuff.

## Monday, August 30, 2010

### Geoboard and Area!

I've been itching to do some neat Geoboard exploration with my Geometry classes, ever since our shipment of math goodies arrived. So, today during my prep period, I randomly pulled together a Geoboard activity for my honors kids to start on during our overflow time. We started it near the end of class, as a change of pace after doing some practice for an upcoming quiz. The kids got through only the first three problems, but it was a really fun / productive time for them! Check it out. I liked it a lot, so I am now thinking about using this activity for my regular kids next week, as warm up before their next big project. The honors kids will probably have some extra time this week to finish off the whole assignment, so I can preview how it would go for my regular classes. :)

(It was fun to observe that kids couldn't immediately build a 5x6 rectangle, because their instinct is to count points/lines instead of # of spaces. Fixing their misconception via adjusting the rubberbands is surely a lot more fun than making them re-draw a shape on a grid!!)

## Saturday, August 28, 2010

### Tangrams Sample Projects

This year, I am going to try to be good about scanning in stuff that my kids have made, so you can see how a project is intended to look / work. :) I love it when other teachers post pictures or videos of what goes on inside the classroom. It makes their lessons seem so much more real!

Here is one. The tangrams project was super fun, for both me and the kids. In the process, we learned how to find angles using adjacent angles in a diagram, how to find angles inside an isosceles angle, how to use Pythagorean Theorem to find diagonal lengths, how to find lengths of sides that "stick out" in a picture, and how to combine / simplify square roots and whole numbers! Wow. So much math in 2 or so days of class time. :) :) And the kids loved the puzzly parts of the project, piecing the tangram pieces together to make the silhouettes. Some kids also took the initiative to create their own designs and to find additional perimeters, which was really fun for me to grade!

Here are the scanned images of a couple of example projects. They look lovely in person, but the math work is in pencil (and some of the colors are also light), so the scanner unfortunately doesn't pick it up all too well. If you zoom in, you can see the math that they wrote next to the diagrams.

Entire project from Kid #1:

Parts of project from Kid #2:

## Thursday, August 26, 2010

### Lovely Week!

I am having an amazing teaching week. :) Highlights:

• I started doing MATH origami with my Honors Geometry kids! It's fantastic!! The whole idea is that I would give them diagrammed instructions (with no words on there whatsoever), and they would have to try to figure out how to make the end product by reading the diagrams. On Day 1 of trying this (ie. today), I gave them something that I had thought would be really easy (I had tried it myself and it was relatively easy for me to figure out), and both honors classes were completely stumped (and incredibly engaged)! Yessss. This module is going to be great.

This first one was supposed to show them how to fold / cut out a regular pentagon out of a square sheet of paper. In the end, after their struggling with it for 15 - 20 minutes and having had varied levels of success / frustration, the bell rang and I announced that we'd have to re-do this particular exercise next week (ie. making pentagons again). The difference is, next time, I'll guide them through reading the diagrams step-by-step (paying careful attention to where the creases are supposed to be and which points should meet up), and we will work on precision as a class. My hopes are that their regular pentagons from next week will be precise enough to allow us to introduce some good geometric vocabulary, and to measure congruent interior angles. :) After that, I have plans for them to make a regular hexagon, followed by a tetrahedron that requires no glue!! (Even other teachers are loving the tetrahedron. I went around Open House yesterday to show off my new origami toy in between parents talking to me, and all the teachers wanted to run off with it! haha. That's how you know it's kid-ready, I guess!)

By the way, all this good stuff came from 3-D Geometric Origami, which is a super fun booklet of foldable geometric shapes (2-D and 3-D). In doing these constructions, we can introduce naturally a slew of geometric terms about faces, edges, angles, etc. I have plans to get the Honors kids stronger at independently building stuff throughout the first quarter, so that when we move on to heavy-duty compass/ruler constructions of polyhedron nets in the 2nd and 3rd quarters, they'll be kinesthetically (and emotionally) ready.

(The regular Geometry kids will benefit from the origami lessons as well, but because they will need more time getting through the regular curriculum, I'm not going to give them the serious polyhedron net construction projects later in the year. They'll still do some good compass constructions though, because I plan on doing some simple line designs with all my Geometry classes, and those require the construction of regular polygons.) :)

• In other news, all of my Geometry kids have been working on setting up algebraic equations with info embedded in geometric diagrams. They seem to be doing better this year with distinguishing the different scenarios for setting up different types of equations. I'm not sure if it's just because it's my 2nd time teaching these exact topics, or what, because I don't feel like my approach has changed drastically from last year, and the kids are having significantly less trouble. There's only been micro-changes, like this year I have emphasized the idea, "Do we have enough info to set up an equation? Do we now? Do we now??" as I put up algebraic lengths piecewise on the board. Does that really make a huge difference in their comprehension??

• Precalc has been smooth-sailing as well. This week (thus far), we did some good graph-reading, reviewed what a function is, and introduced how to identify its domain and range given a graph. I have a lovely activity for introducing functions, which I didn't come up with but I feel like has been smoothed out over the years every time I present it. Now, it's gotten down to me preparing (in advance) a series of ordered pairs representing a function, and putting them on post-its. (Input goes on one post-it, output goes on another. You stick the two post-its together, front and back.) I draw a "function magic" box on the board with an arrow going in and one coming out, and I start to reveal ordered pairs as I stick the input post-its in and return with the output value on the other side. The reason why I use post-its is simple: as I reveal the ordered pairs, I stick them onto the board in function diagrams (the ones that look like two big ovals, one for domain and one for range), and kids copy them down from the board, one ordered pair at a time. For this intro demo, I always choose some numbers with patterns, so that kids can get excited about making predictions -- and I also use a function that has shapes/symbols as inputs and where the output is consistent, ie. always 5. This way, I can show them that even when one output is shared by different inputs, the function still behaves "predictably" (ie. if it returns 5 always, that's pretty damn predictable) and is still a function. And it also shows them that functions are not limited to numeric domains and ranges.

After this, I have the kids pair up and I give each pair a set of flash cards, showing functions and non-functions in graphs, tables, diagrams, and word descriptions (ie. "a function that maps age to height"). The pairs of kids have to separate them into two piles, one for function and one for non-function. It's not fancy, but it works really well to quickly go through MANY examples and the kids are really actively engaged. In the end, we quickly go over the answers as a class.

As for teaching about domain/range of graphs, I totally stole Sam Shah's idea of domain / range meters, and the kids thought it was funny (and probably thought I was "not cool"), but it worked like a charm!! I was thrilled. Kids were beeping (umm, quietly... these kids are waaaay docile...) when they were doing their own classwork problems. heehee

I love teaching! I love the world. Off to Day 2 of Open House (this time, with freshmen parents)!! :) :)

## Sunday, August 22, 2010

### GeoGebra... times 3!

I made a GeoGebra spirals applet -- for use in my Precalc class! Yay. I'll be introducing geometric and arithmetic sequences sometime during the next week, and even though we'll spend most of that period working on finding numeric trends embedded in composite patterns and such, I think that at the end of class, I am going to use the applet to help them appreciate the connection between nature and geometric sequences. :)

Anyway, that's enough preamble; check it out for yourself! You can make a spiral outwards (if geometricRatio > 1) or inwards (if geometricRatio < 1), and I'll ask the kids to make predictions about what they will see with those ratios before we toggle the numbers. You can also turn on Trace for point B in order to see the actual spirally points.

Amazingly, that's three times in one week that I'll be using GeoGebra. (First time I'll be showing my Geometry kids briefly what Mr. H created per my request, so that they can better visualize why 3 points will define a plane in space. The second time, I'll be taking my Geometry kids down to the computer lab to do a full period of GeoGebra exploration in pairs about Segment Addition Postulate and the midpoint of a segment. See worksheet below if you are interested in seeing just how explicitly I structure an activity like this for 9th-graders who've never seen any geometry software before. If you want the actual file, drop me a note. I did all the way up through step #15 with my 9th-graders from last year and they had all loved it, so I am interested to see if the more challenging parts I tagged on this year will end up "breaking" it. --I guess we will see!)

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Also this week, I'm pulling out an old middle-school activity that I liked a lot, in introducing angles and the estimation of angles. I found out on Friday (during the beginning of our tangrams project) that a handful of my kids didn't know how to recognize right angles in a diagram. Some of the same kids also didn't know that when one angle is 90 degrees (at what looks like a perpendicular intersection), its adjacent angles would also be 90 degrees. ...I have to say, that's pretty alarming. So, umm, we're starting from the basics.

The activity is one that I've done with middle-schoolers many times. You give kids each a piece of patty paper, and you get them to fold it several times and to cut an arc so that when they open it back up, they have a circle that has creases across every 22.5 degrees. Then you just start calling out angles that are multiples of 22.5 degrees, and kids have to figure out (without a protractor) how to use their patty paper to show 90 degrees, 45 degrees, 180 degrees, 135 degrees (at which point, they would have to put it together with someone else's angle), 67.5 degrees, etc. Essentially, the kids learn to estimate angle sizes by comparing them against benchmark angles and thinking about fractions of 90 degrees. (You could also use this manipulative to discuss why when you compare two angles, the sizes of their rays don't matter.)

Let's go, Full Week #2! :)

## Saturday, August 21, 2010

### Activity of Reading Instructions

I mentioned in the previous post that one of my goals is to encourage my kids to parse instructions and to translate them into mechanical procedures or a diagram. Here is a good (fun) example from Neufeld Math, which Ms. Cookie had linked to; I think I'll use it when I introduce compass constructions. The idea is that I'll let the group struggle with it for 10 or so minutes and not help them out much. (I'll force them to go back to the written directions.) Then, we'll go over the correct results; I'll let a kid present the solution if they are able to parse all the instructions correctly.

In the space provided, draw a circle and then carry out these steps:

• IMPORTANT: Keep the same radius used to draw the original circle, for this entire exercise.

• Select a point on the circle’s circumference. Label it A.

• Center your compass on A. Use the compass to draw an arc that passes through the center of the original circle and intersects the circumference in two places. Label those points B and C.

• Use the new intersection point C as your center to draw a second arc that passes through the edges and the center of the original circle.

• Continue around the original circle in clockwise to construct more arcs that pass through the original circle's center and its edges, in order to complete the pattern and to return to point C.

• Erase all the marks outside of the original circle, leaving only the arcs that are inside. At this point, you should have what looks like a flower with 6 petals inside a circle.

• Shade in every second region with your pencil.

• Can you figure out how to construct a regular hexagon that is inscribed in this circle?

What I really like about this activity (besides the rigors of reading and parsing instructions) is that it introduces, in a natural way, some important mathematical terms. Instead of me lecturing on the board, I can simply write or project the pertinent definitions onto the board for the kids to refer to as they delve right into the activity, and I can merely circulate and clarify the math vocabulary (arc, inscribed, circumference, regular hexagon) if they still have questions.

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Addendum to the graphs powerpoint: a more complicated (and INTERESTING) graph about pay disparity and gender. I always think this issue is so fascinating, because like my friend observed once, we can't really fairly compare the pay between men and women until women begin to periodically re-negotiate their pay the way many of their male counterparts do. I know that I had never re-negogiated my pay while I worked as an engineer, even though at least one of my guy friends did/does it every 6 months. (He actually goes out and interviews with other companies every 6 months to re-assess his market value. Then, he takes their offers back to his HR Department and requests a raise even when it's not annual review time. That's so smart / aggressive.)

### Process Goals and Fun with Graphs!

It is timely that I came across this nice post today about goals, because I had been thinking yesterday about listing out some goals on my wall that are not topic-specific, so that I can refer to them throughout the year as I work with my kids on different assignments intended to address one of those life (or processing) skills.

Examples for Geometry:
• Visualization of parts vs. whole (Isolating info from one part of a diagram; combining info across multiple layers / perspectives.)

• Mechanical precision (Measuring and constructing lengths within 1 mm error and angles within 1 degree error; building solid models that won't fall apart.)

• Comprehension of diagrammed instructions (ie. for building anything)

• Attention to details in written instructions

• Judgment of reasonableness (of any physical or visual quantity)

• Effective written and oral communication

• Perseverence and resourcefulness

• Team work

Examples for Precalculus:
• Interpretation of data (Understanding trends / real-world significance / impact.)

• Fluidity with technology (Using graphing calcs fluently and flexibly.)

• Flexibility / creativity of problem-solving approaches

• Risk-taking and reasoning about the unfamiliar

• Precision and clear step-by-step organization of thought process / math work

• Increased self-management of progress, frustration level

• Effective written and oral communication

• Perseverence and resourcefulness

• Team work

Maybe it's just me being cheesy, but I think both the students and I might benefit from my posting these goals on the wall -- especially if I do actually highlight different skills throughout the year, as is appropriate to a given lesson, so that the kids would feel like they're not just learning content for content's sake and we are working towards a bigger picture.

My lists of goals are pretty generic (not really even math-specific, for the most part... Most are just good habits of mind...), but they are reminders of what is most important to me. Naturally, between the freshmen (Geometry) and the juniors (Precalculus), there is quite a bit of a "life experiences" gap. So, my goals for the two groups are pretty different as well. My juniors don't need as much for me to hold their hands on following directions, but they might need a nudge to be more creative and/or persevering. Versus the freshmen, whose first task of this year is to learn to consistently read and follow instructions. :) They each are at a different developmental stage with their meta-learning.

We have an opportunity each day to impact kids, mathemagically or otherwise.

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By the way, I found some nice graphs for my graph-reading lesson that is coming up. They're neat, even though they are not terribly complex in a mathematical sense. (The textbook's got a couple of nice complex graphs to use for rigorous classwork exercises, so I feel like I could afford to spend the rest of the period looking at interesting -- albeit simpler -- graphs with the kids.) They have to do with the mobile market, digital textbooks, and digital music. I plan to have fairly open-ended discussions about trends in those graphs, who might be interested in reading them, and what their implications might be for those people. And then maybe end with why these kids should finish college.* It could end up being a total flop of a lesson hook, but I am curious to see what these juniors can bring to the table.

If I have time, I am also thinking about adopting a middle-school post-it bar graph activity to composite bar graphs and scatter plots. I would give every boy a yellow post-it and every girl a pink post-it, and we would start a bar graph template on the board -- for example, models or brands of cell phones. They would go put their post-its on the board, in the appropriate column, and we would re-arrange the post-its to stack the pink post-its on top of the yellow post-its, in order to create a composite bar graph showing how many boys, girls, and total # of students have each type of phone. You can also do this exercise with scatterplots (for example, height vs. foot length) and see at a glance 1. what the overall trend is within the class, and 2. what the gender-specific trend is within the class. Using only colored post-its! And requiring maybe only about 10 minutes, discussion included. ...I'll try to squeeze it into my Precalc lessons this week and tell you lurkers how it goes.

*From what I hear, many Salvadorean private-school kids end up dropping out of college in the States, because they either lack the academic skills or -- more commonly -- can't deal with the fact that they no longer have a live-in maid and a designated chauffeur. This waste of an opportunity is terribly sad, considering that their parents can afford them to obtain a U.S. college education, while the majority of people in this country are living in poverty and many are starving. :(

But, can you really blame the kids for this injustice?? They are a product of the system (of huge disparity in wealth). sigh.

## Thursday, August 19, 2010

### Vectors and Wait Time

I've been working with my (regular) Precalculus kids for a full week on vectors. Since I didn't want to teach them just rote algebra procedures, I made them do everything graphically. Adding and subtracting vectors graphically, scaling graphically, etc. As predicted, some kids figured out for themselves that <a, b> + <c, d> = <a + c, b + d>, and I told them that it's all good to use this algebraic method to double-check their graphing work. But, I'm pretty pleased that...
1. Kids were able to answer this question using only graphical addition: List 5 vectors whose sum is <-3, 4>.

2. When I wanted to introduce the graphical subtraction of vectors, I first gave the kids some diagrams of pairs of vectors to look at and I asked them to choose a pair that represented a vector and its "negative." Almost every kid got it right! They figured out for themselves that vectors of equal magnitudes and pointing in opposite directions represent "negatives" of one another. And we discussed how V + (-V) = <0, 0>, just like how 5 + (-5) = 0. That was pretty neat, because it was a tiny reference to set theory. (Or some such goodness in higher math.)

And I am truly amazed by how great these juniors have been in class. We spent a good chunk of class today working on using protractors and rulers to carefully duplicate vectors, and I was fully expecting them to throw frustration tantrums because of the precision I required. But, they were super! I am not holding my breath that this will remain the case, but I'm certainly doing my best to keep the class extremely smooth-running to avoid the 3-week-in rebellion. (I won't lie; it also helps to have a handful of kids from my honors class last year now in regular Precalc with me. I still think it's extremely silly of them to drop out of honors, but given that they're taking many other honors classes, I guess I can sympathize. MAYBE.)

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On the Geometry side, I had a really great experience with the shapes puzzle I had posted a link to. Kids were talking to each other using geometric vocabulary; they wanted more time on the assignment before discussing solutions as a class... I actually felt bad when I cut them off at 15-20 minutes of individual efforts struggling with the puzzle, because I could tell that the kids were really trying to figure it out on their own.

It was awesome. It made me think again about the issue of wait time. Giving kids just 5 more minutes on a thoughtful assignment sometimes makes such a huge difference.*

*Note: I think the issue of wait time is intrinsically tied with the timing and type of hints you give. For example, my first regular Geometry period felt significantly more frustrated on the assignment, because I gave them no hints while working. My second period (also regular Geometry), started to make visible breakthrough when I said after a good 15 minutes of individual efforts: "By the way, here is a hint: everytime you tag on a triangle to a shape you have already found, you end up changing the number of sides in that polygon."

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Geoff comes back this weekend. :) I'm so excited!!!

## Monday, August 16, 2010

### Common Knowledge?

I guess the moral of the story is Don't assume common knowledge:

Me: (addressing my honors 9th-graders...) Does anyone want to tell us how you estimated the distance from home to school in kilometers?
Class: [silence]
Me: Okay, that's a hard one. Let's see... (trying to figure out how to guide kids' thinking...) Well, what do you think is your average driving speed on your way to school? Like, if you drove for an hour, how far would you go?
Class: (rustling a bit, but still silent)
Me: Well, have you guys ever seen a posted speed limit around here? What does it say? 'Cause those are measured in kilometers per hour, aren't they?
Kid A: 30!
Kid B: I think 80?
Me: --Wow. Okay. So you guys have never seen a posted speed limit sign before.

## Sunday, August 15, 2010

### Lesson plans for Week #1

I am excited about Full Week #1! I'll probably write more about this later as the week unfolds, but basically, this week in Geometry I am going to finish up the metrics conversion/dimensional analysis stuff and go into some basics of Geometry. Here are some links (really good ones) that I dug up to help me plan those lessons. I'm posting them beforehand because if you are a Geometry teacher, it might be too late for you to wait for me to do the lessons and give a play-by-play review of how it went. (...PLUS, giving my own play-by-play analysis is awkward. Versus you can just read about where I'm finding inspiration -- much less awkward!)

So, here we go:
* As a basic review of some geometric terms they might already know from middle school, I'm giving kids a warmup that looks like this: a visual puzzle involving geometric vocabulary and recognition. It is surprisingly challenging! Took me a bit of energy to find those heptagons and octagons and whatnot. I envision it taking a good while if we want to have a good discussion about the results. (The rest of the period will be overflow/extra practice time for earlier content. I'm pretty sure the kids will need this extra time, since dimensional analysis problems are difficult.)

* The next day, we go into basic geometric postulates and naming conventions. I super-duper like Dan's take on why we name things the way we do, so I took that and adopted it to show a bit more vocabulary -- like collinear, coplanar, interior/exterior to an angle, and whatnot. The idea is for kids to look at something visual and to come up with a natural way of grouping/describing objects, and then all I need to do is introduce the math term associated with those concepts.

* Last year, I did an activity that I liked, which was giving two kids a string, and then giving two other kids another string. I have them intersect their strings and start to walk towards/away from each other, to see whether the number of intersection points ever changes. Obviously, it doesn't, and it helps to show in a physical way that two taut lines, if they don't overlap / aren't parallel, will always intersect at exactly one point. I'm going to re-do that demo this year, because the physical representation seemed to really help out last year.

* Lastly, I plan on wrapping up the week with some cool tangram activities adopted from this link: using tangrams to build shapes and to find their perimeters. As part of the lesson, kids will calculate (not measure!! They can measure at the end to verify if they want..) the angles and perimeters of each individual tangram piece, and also build tangram shapes (one assigned by me and one that they design individually) and find the perimeters.

I think that sounds pretty fun / productive for the first week, no? :) The following week, we will move into some more heavy-duty algebra stuff, and the kids will hopefully get a chance to get on the computers as well, to get acquainted with GeoGebra.

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On the Precalculus end, I will spend the entire week on vectors. I decided that I am going to build it up piecewise and culminate with a mini-project on graphically constructing vectors, using ruler and protractor (lengths and reference angles) only. And then, we'll do some physics-like word problems as time allows. That's going to be pretty tough, for sure, but the physics teachers say that they need this level of understanding. So, here we go! I hope they are troopers and are already good at measuring with rulers and protractors! (...Fat chance.)

### Bacon JalapeĆ±o Brownies

A while ago, my friend Amy had blogged about baking bacon brownies, and so Geoff and I had tried it out. We put the chopped-up, somewhat-cooked bacon into the brownie mix, blah blah, and it was pretty darn good. So yesterday, for my birthday potluck (at my friends Colleen's and Eric's house), I thought I would make some more -- and this time with jalapeĆ±os!

To my great delight, they were quite popular with all who had a chance to try them. Bacon + jalapeĆ±os + Ghirardelli brownie mix = super duper yumminess! (They actually ran out almost immediately. I only got a chance to nibble on the leftovers in the tray.)

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Being Asian sucks. I always get hungover while drinking, because my body can't metabolize the alcohol very well. My dear Asian friends, does this happen to you guys?? I also get red patches on my stomach whenever I have more than 1 drink, and my eyes become bloodshot as though I have pink eye. I am probably allergic to alcohol (but I don't really care, because I drink fairly minimally, and it keeps me a cheap date). I have read somewhere that our allergy to alcohol is what keeps most Asians from becoming alcoholics. It's just so toxic to our bodies that we end up consuming only minimal amounts of alcohol, not enough for an addiction.

## Saturday, August 14, 2010

### Birthday Thoughts

I turn 28 today. :)

It feels brilliant to be this old, because I am doing exactly what I want to do (and I had thought that it would take years more to feel this way). Geoff and I are so blessed; we travel, we love our jobs, and at the end of the day, we are still saving money! I also have no doubt that Geoff is the person I want to spend a really long time with, and when we look back on the four years that we have already shared and look ahead at the life that we hope to still build together, it astounds me just how blessed we are.

So, I am turning 28, and I feel absolutely, utterly fantastic.

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...On a not-so-fabulous tangent, yesterday I stepped through mud on the way to school, and by the time I had realized this, it was way too late. (Why was there mud right outside of the school, even though it hadn't been raining??) So, the entire back of my long WHITE skirt was covered in mud stains, along with my legs. Yikes for a first impression upon the kids.

That was non-fab. Fortunately, my skirt had a lot of folds (it's a convertible skirt/dress, so there's definitely extra fabric), so I just folded some of the worst stains (quarter-sized) under other cleaner parts, to minimize the visible damage. But, yikes!

## Friday, August 13, 2010

### Teaching Estimation!

I had decided sometime last year that as one of my focuses (foci?) for Geometry, I would push their estimation abilities this year. We would estimate before measuring everything.

Today was Content Day 1 of Geometry, and I gave them a list of questions to answer as the Do-Now. Kids had NO IDEA how to answer some of these questions, which was my intention. We ended up with a really great discussion about how you would go about estimating some of these. (We ran out of time today, but I plan on returning to these very quantities on Monday and have them actively measure some of these, to check their own accuracies in estimation. Estimate-measure-estimate-measure-estimate-measure!)

• What is the height of this classroom in centimeters? In meters?

• What is the length of your foot in centimeters? In millimeters?

• What is the length of your hair in meters?

• What is the distance of your house from the school in kilometers? In meters?

• How many hours are you in school a week? How many minutes?

• How many kilograms does your textbook weigh? How many grams?

• How fast can you run in m/s?

It was great! (And funny that even when they know how big a meter is, they still can't estimate how fast they can run in m/s.) Some kids said really awesome things like, "I am about 160cm tall, and the classroom is about twice my height," or "Our new principal* is about 200 centimeters, and he is about a meter away from the ceiling." I followed this Do-Now/discussion by teaching them the mnemonic device for remembering how to convert between metric units. (Kangaroos Hopping Down Stairs Drinking Chocolate Milk).

The big concept (often overlooked) that I always try to teach for unit conversions is that when the unit gets larger in size, the number needs to get smaller. ie. 1 kg = 1000 grams. Otherwise the right-hand side of the equation either shrinks too much (if both the unit and number are getting smaller) or expands too large (if both the unit and number are getting larger) to remain equal to your original quantity! I say this even during metrics conversion, because if the kids get this concept down, it's a lot easier for them to figure out that 78 cm = ______ inches should have an answer that is smaller than 78, even though the numbers are messier.)

We used the Kangaroo Stairs to figure out how many decimals you move. (ie. Number of steps you take on the stairs = number of places you move the decimal.) Again, once the kids can reason through whether the number needs to get bigger or smaller, they can usually figure out the rest with a small bit of practice. Moving the decimal isn't the hard part; knowing which way to move it is the crux of those problems.

It was lovely.

*Yes, our new principal is a giant. :)

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My Precalculus class was also awesome today! We had to teach vectors as part of the support for their Physics classes, so I adopted Kate's activity of pushing desks around to show how vectors add differently when they point in same/different/perpendicular directions, and I also drew arrows (vectors) on transparencies and moved them around a large projected grid, to demonstrate how you would add vectors graphically. The kids figured out quickly as a class that you can find the magnitude of a vector with simple Pythagorean Theorem. Yesss!

It was really cool. I think I've got some difficult kids in there (or so I hear), but my plan is to kill them with kindness for now, so that three weeks in, they'll feel bad to act up in my class. So far, so good.

## Thursday, August 12, 2010

### What is this?

Bonus points for whoever can figure out what this picture is about. :)

Hint: Mnemonic device.

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By the way, here is another cute SAT problem from David Marain, in his words, "solvable by 7th-graders" (as long as they understand how to visualize fractions and how to perform simple inductive thinking):
If 1 - (1/2 + 1/4 + 1/8 + 1/16 + ... 1/(2^N)) = 1/(2^2010), then N = ?

## Tuesday, August 10, 2010

### First Day Details

A small but neat trick that I picked up from another teacher last year is that you can write the kids' names on post-its, and then put the post-its on their desks before they walk in on Day 1. That way, they will already be quietly looking for their assigned seats on their way in, you can take roll easily on Day 1 (by seeing if there are post-its that remain unoccupied), and they can also hand you back the post-it with their names corrected if they prefer to be called something else.

I really liked that idea. Especially because last year, the school was scrambling their schedules until the very last minute, we didn't receive the updated rosters until the morning of Day 1, and I especially wanted the semblance of order in my classroom to set the tone for the rest of the year.

This year, I am taking that idea a bit further and adding on a layer of getting-to-know-you. I'll write their names on a survey beforehand and stick the named surveys on their assigned desks. (...Yes, I do realize it's quite a bit more work this way than having them fill out their own names, but for some reason I like it better than posting a seating chart on the wall!) That would give them a semi-thoughtful do-now in addition to filing in quietly. Afterwards, I'll have a short chat with them about my expectations. (I have re-written my syllabus as a letter to the students this year, to emphasize the idea that we are building an important relationship right from the start.)

None of this is original, but as with all disciplinary measures at the beginning of a year, the devil's in the details.

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I am heading into my fifth year, and generally I am feeling really upbeat about it. I'm up to two-and-a-half preps this year. (To make it a bit more manageable, I am going to try to keep my honors and regular Geometry classes paced similarly, except going into greater depths and throwing in extra projects that are sequence-independent for the honors classes.) After a bit of paperwork chaos, it looks like I'll also be teaching Precalculus as I was hoping for. Yay!

I've pretty much got the whole year outlined, in both Geometry and Precalculus. Since it's the first time I'll be doing Precalculus, I can't yet gauge how ambitious those units are, so I'll reveal them piecewise as I figure it out. I'm feeling pretty great about Geometry, because I have revamped my entire curriculum from last year (keeping bits here and there that I liked, but re-doing the whole sequence and adding a whole bunch of new activities), and my two Geometry partners for this year seem to be on board with the plan.

Onwards! I think it's gonna be a very busy, but good, year. :)

## Monday, August 9, 2010

### KenKen and Other Goodness

Another couple of cute geometry SAT problems from David Marain:
Problem 1: Chords PQ and PR are each of length 6 and form an inscribed angle of 60°. If the area of the circle is k*pi, what is the value of k?

Problem 2: From point X, the 2 tangent segments to a circle have lengths 6 and form a 60° angle. Find the square of the least distance from X to the circle.
As a visual person, I love geometry word problems, because the first thing I do when I read problems like these is that I sketch them out quickly. (And then I tend to "see" information that I wouldn't otherwise observe from the flat phrasing of the problem.) That's a good habit I would like to instill in all of my geometry students.

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By the way, I've been incredibly addicted to the KenKen puzzles in the free daily newspaper during my latest visit to NYC. Since my friends could only meet either for lunch or dinner (or sometimes happy hour), I found myself spending a lot of quality time by myself in the city, and each time I would skim through all the gossipy news and go directly to the crossword and KenKen puzzles.

(I'm terrible with crossword puzzles, but the ones in trashy newspapers are usually very do-able for me. I guess I'm their target audience. By the way, I find it hysterical that Wyclef is running for president in Haiti; it's like South Park in real life, or something.)

Here is one KenKen puzzle I kept from the papers. I know these things have been around for a while, but I never really tried them until these last couple of weeks. I find them much, much more enjoyable (read: less repetitive -- and actually somewhat mathematical!) than Sudokus.

The rules are simple:
1. Each row and each column must contain the numbers 1 through 6 without repeating.

2. The numbers within the heavily outlined boxes, called cages, must combine using the given operation (in any order) to produce the target numbers in the top-left corners.

3. Freebies: Fill in single-box cages with the number in the top-left corner.

Try it. And tell me if you find it as addicting as I do. :)

## Sunday, August 8, 2010

### Holes in the U.S. Education

Oh, so here's a funny thing that happened in Peru that I forgot to relay: At some point, we were hanging out at Machu Picchu and the guides wanted us to list all the things we knew about the famous ruins or about the Incas. Out of a group of 13 of us, guess how much we knew? ...Next to nothing.

Those guys were shocked. They said, "What do you guys learn in the U.S. schools?" ...Yeah, if only I had a dollar for every time I heard that.

Being a teacher, that's a pretty sad thing to hear. My Taiwanese friend (who is really smug about a lot of things, whether rightfully so or not) is quick to point out that Americans don't get geography jokes about other countries, because given a random non-Western-European country name, we can't really tell you anything specific about that country's geography, besides roughly where it is located in the world.

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So, I'm back in El Salvador. I had thought I would be relieved to be home after four weeks of consecutive traveling. And, I am. But I miss Geoff already, and it's only Day 1. :(

Work starts tomorrow for me. I feel a fair amount of anxiety about returning to work (for reasons not to be discussed here -- but suffices to say that this is a rare occurence for me) and about the fact that Geoff will remain in the States for at least another 2 weeks. (Maybe longer, if he needs to close a deal on a house.) So, I am home, yes, but calm the way I normally feel at home? That's still a work in progress...

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In other news, recently I've been telling people stories about how Geoff and I went about planning out the next few years of our lives together. Everyone's been choking on laughter when I say matter-of-factly that Geoff wants to make babies at some point, and that I have told him that that shouldn't happen when we're, say, 40.

It's funny how men have nesting instincts but are clueless about fertility rates. How come people always talk about women having mothering instincts? :) I think Geoff's fathering instincts far exceed my own nurturing instincts. I can't even hold babies! (I am scared of breaking them.)

Anyway, I thought I'd put it out there, so you can all tell him that you now know he has a sensitive side. ;)

## Saturday, August 7, 2010

### Affirmative.

Coming back to the States only once or twice a year is pretty funny, because it's like flying into the future. I am constantly amazed by how wired everyone is in the States. In El Salvador, I literally don't carry a phone with me half the time. I have only rough estimates of what time it is, at any given moment.

It's pretty ridiculous how technologically advanced everything is. It's pretty awesome, in a way, but in a way it also creeps me out a bit just how reliant we are on technology. Is it just me? Are we going to start saying "affirmative" instead of "yes"? ZerozerozerozerozerozeroOne.

## Friday, August 6, 2010

### Thoughts about Mothering from a Non-Mother

I want you to know that I've been talking to/reading about mommies and mommies-to-be and I have decided that once you become a mom, you don't have to just give up the things that you love to do!

Case in point: Last night, I went swing-dancing. It was an utterly lovely time, with a wonderful deejay and a crowd that wouldn't sit through any of the songs (the songs were literally awesome one after another), even though we were each dripping with sweat. Amidst all of this, I saw my dance friend walk through the door, who had recently become a mommy (about 6 months ago). I started talking to her, to find out to my surprise that she has been trekking from Jersey to New York once every two weeks since the birth of her baby, to swing dance! YEAH. (Her husband watches the baby on those nights.)

And recently, I've been reading our friend Katy's blog. (See your right-hand side for links.) Katy used to be -- wait for this -- a rugby player. She semi-recently had a baby and a C-section, but she has been getting back to the rugby field. ...DOUBLE YEAH!

One of my NYC friends is getting married soon (with plans to start a family), and she is a marathon runner. :) She has been talking with our mutual friend Annie about planning marathons into her pregnancy schedule! haha. And how once she gives birth, she can go back to running with the stroller, but she just can't be training for a marathon while pregnant.

--I love, LOVE how these mommies and mommies-to-be still try to maintain a life of their own outside of their kids. That makes motherhood seem so much more do-able. Because we, as women, are diverse (read above), strong, and beautifully independent. How can all of that go away with motherhood??

Thank you, women everywhere, for being my inspiration. (And good luck on motherhood.)

## Tuesday, August 3, 2010

### Math in Science

Before I decided to try this teaching-abroad thing, I used to work at a really great 6th-through-12th-grade public school that believes in teaching applied mathematics and sciences. They try to make everything hands-on and real for the kids. And I mean, everything! You would walk through the halls and see giant pulley setups, or see kids practicing pitching tents in preparation for their upcoming camping trip. In math classes, you see kids doing debates about tax policies and hosting probability game carnivals. (In fact, they are doubtlessly one of the best schools in NYC, and possibly one of the best schools around the country. They take in regular Bronx kids -- some of who can't read and have never done any math and many who come from broken homes -- and do magic to them; this year, 2010-2011, AMS will be having their first graduating class of seniors, and of the entire class of about 90 or so kids, only a handful is at risk of not graduating with their peers. The rest of them have passed their Regents and class requirements with flying colors.)

So, that's a long-winded way of explaining why I am always on the prawl for ways to integrate science into my math curriculum. (The science teachers at my current school sometimes make fun of me, because I am always going in and out of their classrooms and borrowing scales, beakers, thermometers, and whatnot. Some days they let me borrow their entire lab space, and those days are extra awesome!) One of the things I did somewhat recently is an RC circuit lab, to help the kids see that exponential data does occur naturally in physical space. They collected a bunch of data about charging and discharging capacitors and then did regression on their calculators to see that the exponential curve fits the data almost perfectly! Then, we ripped the equation apart to discuss every bit of it and how it related to the physical thing they saw.

Today, I spent some time looking at the skeletal lesson ideas at a NASA-sponsored site called PUMAS. Silly name, I guess, but they had some neat stuff. In particular, what I liked were:
...Anyway, these are just skeletal lesson ideas, so maybe they won't work well in the end, but I like them as raw ideas!

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On an unrelated note, I finally went to the restaurant from Seinfeld yesterday! It's so funny that I had never been inside, even though I used to live four little blocks away on Broadway and 108th! The inside doesn't look like the show set, which was a bit disappointing, but the food was pretty decent for a diner! (Geoff was disappointed that I didn't order a Big Salad. heehee)

...And, in between looking at apartments, contacting realtors, following up on building regulations, and hanging out with family/friends, my boyfriend hasn't properly rested in days. :'( I can't wait until all of this madness is over and we get to finally, Finally be back at home, just the two of us. (That won't likely happen until end of August, if even.)

## Sunday, August 1, 2010

### Media-rich Data

Best quote ever: "I've just spent the last 5 minutes trying to piece you together with the 8 other Asian girls I know, and it turns out that you're not any of them! I've never met you before!"

...It happens more often than you'd think. (Not the act of confusing me with other Asians -- which I don't mind so much -- but the verbalization of such. Tact seems to be a lost art these days.)

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Funniest story told to us by the realtor lady we had met yesterday: One time, the contractor she worked with went to a place in Newark, NJ, while there were still tenants living there. Afterwards, he had to call the cops, because he had seen a fresh body in the crawl space of the house. WHAT?! That's insane!

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Some interesting things I dug up while going through the archives at FlowingData.com:

Some of it is teachable material; others are just cool to check out. It's really interesting how media can enhance your interpretation of numeric data. The guy has some other really neat stuff, but I figured I would link to the short list that appealed to me the most.