Monday, February 28, 2011

Crazy Bus Trip to Tikal

This past long weekend was a whirlwind of a trip, starting with Geoff picking me up right after school on Thursday (after our day of professional development at school), and then rushing to the bus station. We took a 5-hour bus from San Salvador to Guatemala City, another overnight bus from Guatemala City to Flores, and then finally took a cab ride from Flores to Tikal. Altogether, we were pretty much commuting non-stop between 3:30pm Thursday and 7am the next day. Sooooo, totally hectic!!

I am a sleeping machine, so I slept pretty much the entire way to Tikal. Geoff, on the other hand, slept almost none at all on the way to Tikal. (As it turns out, the overnight bus drivers were insane, and they were going at breakneck speed for pretty much 8 straight hours. Every time I would wake up from my nap, I would feel the entire bus swaying from side to side on a fairly narrow road, because of the crazy high speed we were going at. As a result of feeling nervous, Geoff had trouble falling asleep.)

But, anyway, according to plan, we arrived in Tikal on Friday at around 7am. Which was fabulous! It gave us almost two full days in Tikal itself. We wandered for the most part by ourselves, using the map that we had bought for 20 Quetzales. (The exchange rate is about 8 Quetzales:1 dollar.) WE LOVED TIKAL!!!

Tikal is an impressive show of Mayan culture, obviously, but for me the personal highlight was the fact that it is located in the middle of a jungle and was all kinds of rustic glory. We saw some amazing flora and fauna (including tons of monkeys!!), and heard the howling monkeys roar for quite a while. It made me feel like I was in Jurassic Park. (The park even has jaguars running wild, but thank goodness we didn't run into any.)

Here is Geoff's favorite animal in the jungle (besides the scary howling monkeys). We call this one the "monkey anteater" (not to be confused with their real names).

We also saw an incredible sunrise at the top of one of the temples. For me, it was a really unique experience to drive through the jungle at 4:30am, while everything was pitch dark. (We weren't silly enough to go driving by ourselves, but even sitting in the back of a pick-up truck was pretty scary. I kept looking back at the pitch darkness and thinking about the jaguars that we were waking up...)

We were on top of the tallest temple!! We sat there for a good couple of hours, through all the phases of the sunrise. We also listened in admiration while the alpha howling monkeys did their jurassic roar, which lasted a good 30 minutes. (They aren't morning people, I see.)

Then, another personal highlight of mine was when I un-wussed myself into climbing a 7-story high temple, where the only way up was this really steep set of ladder-stairs!! See left side of the first picture below; the original Mayan steps are deteriorated and cannot be used. (I'm deathly afraid of heights. Actually, Geoff had to talk me into going up, even when I was already half-way up the ladder-stairs.) Real-life application of rise/run?? YIKES.

On our last day in Tikal, we left early and went to hang out in the "nearby" town Flores. It was a beautiful tourist town on a lake. Super chill! There, we caught another amazing sunset -- this time with blue streaks in the sky?? (Maybe that's common, but I had never actually seen that before! We thought it was stunningly beautiful, so we took a picture.)

All in all, what an amazing trip! And half a miracle it was that everything worked out exactly the way we had planned. I'm back and feeling good about being back, but it's going to be hard to be without Geoff for a few weeks.

And then, it's crazy, but as soon as Geoff comes back to El Salvador from his "business trip", we'll have to be tying all of the loose ends for moving and making summer plans.

Wednesday, February 23, 2011

Absences; Peer Observations

In my Precalculus class, there are 15 students. (I know, it's really nice.) In the past month, this was their absence pattern (the ones that are "extraordinary", anyway... I'm not counting the daily illnesses):

* Students A, B, C were gone to a Model U.N. conference for close to a week.

* Students B and D (who also happens to be a new student transfered into my class at the semester) were absent two days later for an all-day leadership conference.

* Student E was absent for 1.5 weeks for a health checkup in the States (that experienced extra delays because she needed to run extra tests and to wait for their results).

* Student C then broke her collar bone(!) playing soccer, and was gone for a few days at first, then later on for another few days for a follow-up visit in the States.

* Students D, F, G then missed 2 days the week before a test in order to attend a soccer tournament. (The soccer tournament was away and lasted all weekend and so it affected their performance the following week as well. On Monday they could barely stay awake.)

* Starting today, student H is going to be out for a few days for a knee surgery.

ALL of this for a class of 15 kids -- in the span of a month?! Is this normal?? It's AWFUL to hear about their various health problems, and to be sure, helping kids make up their work after school and (in lack of a better word) "competing" with their other teachers for their time / energy outside of class now feels like the norm rather than the exception. :( I am sure it diminishes their desire to learn, when they come back from a long, often unavoidable absence and they realize that they have so much catch-up work to do in EVERY class. Truly, the long weekend cannot come soon enough, for them OR for me!!


In other news, I have arrived at the hypothesis recently that how you observe someone else teach is immediately related to how you teach. I love watching how a teacher does group work. I like listening to them talk in front of class, but after about 15 to 20 minutes, if the teacher is still talking, I start to shift in my seat when I observe their kids starting to shift in their seats and starting to create that barely audible hum of impatience in the room. When they finally break into group/individual work, I like to watch the kids to see who's on task and who's not, and whether the teacher picks up on it and does something about it.*

Other teachers who have occasionally stopped by my room, in contrast, only like to stay for the 5 to 10 minutes when I am talking in front of the room. As soon as my kids move into groups/pairs, the teacher usually stands up and leaves, because they don't value that part of my class and/or don't find it interesting. They think that the "teaching" is done as soon as I assign work, when in reality that's when my real teaching starts. It's not quite the inverted classroom model (because I still deliver material during class), but the group/individual work they do definitely bears on their learning much more than anything I could/would say in front of class.

I wonder: how do you observe other teachers?** What is it that you pay attention to and that you value?

*Other things I pay attention to: If the teacher assigns a short "Just try it!" guided exercise, I like to see how much time they wait before going over the answer. (If they wait too long, the kids are bored. If they wait too short an amount of time, it removes the thinking for the kids and renders the exercise ineffective.) If they pick kids to present a problem, I look around the room to see whether other kids are invested. If they involve multiple senses in the lesson, that's my favorite! But only if the activities are still immediately relevant to the learning objective. If the teacher is writing on the board, I pay attention to whether they write out complete big ideas and how they organize/label their work on the board. (If a kid looks up after a couple of minutes of being "tuned out", can the kid still immediately follow what's going on?)

**Every time I walk into another teacher's classroom -- whether it's math or otherwise -- I always think, "Wow! This is great! I should do this more often!" When I was a super-newbie teacher, observing other people helped me learn their styles; now it helps me re-affirm some of the choices that I make inside my own classroom, and turns those details into conscious decisions rather than some de facto sort of thing. It seems so obvious that I should be regularly watching others teach, but it's always the most obvious things that we do the least of!! ...Anyway, how often do you observe your peers?

Sunday, February 20, 2011

Structure of Lessons

I've finished reading this book called Made to Stick, which talks about how to make your important messages memorable to people. The book mentions a variety of strategies, but the single all-encompassing strategy they offer is to tell stories, because stories have various qualities that make them naturally "sticky."

For example, near the end of the book they cite a study done on some Stanford students. A small group of students took turns each giving an impromptu speech arguing for or against some topic on crime. The students were asked afterwards to rate each other's performance, and the charismatic speakers naturally scored the highest. As a red herring, the professor then played a short clip of Monty Python to get the students' minds off of the speeches. After 10 minutes, he turned off the video and asked the students to write down everything they could remember about the speeches they had heard. Shockingly, these presumably sharp Stanfordites remembered few details from any of the speeches -- charismatic or otherwise. They remembered almost no statistics or supporting ideas. The only parts of the speeches that were memorable to them were the few personal stories that were given to illustrate an idea.

Obviously, the lesson here is two-folds:
1. Stories are sticky, because they illustrate an idea with cohesive details and emotional impact.
2. A message that seems immediately effective does not necessarily have the lasting impact that you think it will have.

That got me thinking about the format of effective lessons. In my classes, I rarely tell stories. I could only remember one recent example where I had started the class hooking kids with a question, "What makes metal boats float, when we all know that a piece of metal would sink when we throw it into water?" I didn't reveal the answer until the end of class, after we had worked for a full class on measuring and calculating densities of objects. After all the hard learning had been done, we came back to solve this mystery of the floating metal boat, and I further tied in the story of the Titanic and why it sunk. The lesson was formatted like a mystery that unfolded piecewise, and hooked kids to be curious at every step. That curiosity was not satisfied until the end of class. At the time, I thought that the lesson went well. Now looking back through the lens of this book, I have a better framework for analyzing this lesson. It was essentially formatted like a story.

It got me thinking about my other lessons. Are they crappy because they don't follow this format? (I thought so immediately, but Geoff convinced me that I can't be so narrow-minded.) I think it would be helpful to come up with some general lesson formats that have been successful in the past, so that in the future when I plan lessons, I can follow a blueprint structure that has worked well for me in the past. (That was another theme in this book: Creativity thrives, ironically, when you follow a general structure that has proven to be successful in the past.) It would also give me a better framework for analyzing my own lessons in the future.

Are you aware of any great teaching resources that de-focus on content and focus instead on the format of a lesson? What are some lesson formats that have worked particularly well for you in the past?

Friday, February 18, 2011

Learning Laws of Sines and Cosines the Hard Way

Yesterday, I gave the honors kids a warm-up problem on the board (the hardest part was for them to read and follow all directions, apparently):
1. Draw an obtuse scalene triangle ABC, where A is the obtuse angle. (I announced out loud that the triangle should be "about two fistfuls" in size. "Don't be stingy!")
2. Measure sides AB and BC only. Label their lengths in the triangle.
3. Measure angle B only.
4. Draw a dashed line down from A to create height.

(I drew this on the board to show them how to line up their protractor to make sure the "height" is perpendicular to the base.)

At this point, I waited to make sure that everyone had an obtuse triangle with two sides and their included angle known.

Then, I gave oral instructions for everyone to fold the triangle back along the dashed line, so that only the right triangle with the known "included" angle is showing. They then proceeded to individually find all sides (no rounding!) and all angles (no rounding!) using basic trig. When they were done with that first right triangle, I had them open the fold back up to find all sides and angles on the other side, paying close attention to which pieces of information "carried over" to the other side...

In the end, this activity was GREAT because it was self-checking. After they did all of the calculations, verifying their results with a ruler and protractor was easy to do and very rewarding! And it built their confidence that their knowledge of right triangles could now be extended to analyze all types of triangles. (A precursor to Law of Sines.)

The rest of the class, they worked in groups on something that was kind of difficult for some of them (but very manageable for others).

Notice that all of these problems could easily be solved using Law of Sines, which I will introduce to them next week. But, I wanted them to do it the "long" way first -- cutting scalene triangles up into right triangles -- so that they can gain a real appreciation for A.) that they can do it on their own without some fancy formula, and B.) it's a lot faster with the fancy formula.

(The "fancy formula" Law of Sines comes from #2A on the second page, actually. It's just a matter of re-arranging those terms once you have the equation sin(A)*b = sin(B)*a, so I don't anticipate them feeling surprised when they see it next week.)

Today, my intention was to return to this and finish it in class with a discussion, but instead I gave the kids another Do Now similar to the one they did yesterday, but this time WAY HARDER (ie. it's the "hard way" of doing Law of Cosines.)

1. Draw an obtuse scalene triangle ABC, where A is the obtuse angle.
2. Measure all three sides of your triangle. Label their lengths in the triangle.
3. Draw a dashed line down from A to create height. Label the other endpoint D (directly below A).
4. Label BD as "x". You may NOT measure this length.

I had to give them heavy hints to setting up Pythagorean Theorem equations for both right triangles in order to solve for h^2 and to set both h^2 equations equal. In the end, they were able to get through the whole problem -- finding all sides and all angles -- and to verify their own answers with a ruler and a protractor, but I felt much less good about this. (The concept was just a step too complex for them to do all on their own, unfacilitated, the first time.) Even though I did little talking and they did most of the work, it still took about 35 minutes in each class -- a ridiculous amount of time for a Do Now!

But, I am hopeful! Next week will start off with a discussion of generally how to approach triangle analysis, and then a mixture of doing things the hard way (ie. still chopping scalene triangles up into right triangles) and then verifying their results using the Laws of Sines/Cosines "shortcuts." I am feeling really good about where they're at, because I can hear the gears turning in their minds as they push themselves to consider, "What do I know? What do I want to find? And how can I be strategic about getting there??" --which, to me, is the GREATEST part of math. :)

Wednesday, February 16, 2011

Multi-Step Trig Problems

Since they're really acing the introductory trig material, I am ramping my honors kids up to laws of sines and cosines. To do that, I am going to make them divide* scalene triangles into smaller right triangles, in order to really understand side and angle relationships within general scalene triangles. The longer-term vision is to build them up to Kristen Fouss's wonderful collection of complex trig problems. (Not to be overly confident in their abilities, but I really think that my honors 9th-graders may be able to handle dividing up quadrilaterals after a few days of practice dividing up triangles into smaller right triangles. --In groups, of course! Kristen's problems are just difficult enough where they will be forced to work together in order to get through them -- which would be absolutely perfect for these mathematically fearless warriors.)

Anyway, before we build up to all this fancy-schmancy laws and quadrilateral stuff, I thought it'd be good to just take a day and work out all of the kinks in their basic trigonometry application. Make sure they can fling sine, cosine, tangent at people in a hurry, that sort of thing. To that end, my Holt Geometry textbook (which I have a love-hate relationship with) has a nice collection of multi-step word problems. My honors kids were engrossed in them today, and feeling really good once they got through them. Check them out!

Intermediate problems (the book gave them the diagram for the first two of these... but I would expect the kids to be able to draw their own diagrams regardless):
From the top of a canyon, the angle of depression to the far side of the river is 58 degrees, and the angle of depression to the near side of the river is 74 degrees. The depth of the canyon is 191 m. What is the width of the river at the bottom of the canyon? Round to the nearest tenth of a meter.

Marion is observing the launch of a space shuttle from the command center. When she first sees the shuttle, the angle of elevation to it is 16 degrees. Later, the angle of elevation is 74 degrees. If the command center is 1 mile from the launch pad, how far did the shuttle travel while Marion was watching? Round to the nearest tenth of a mile.

Katie and Kim are attending a theater performance. Katie's seat is at floor level. She looks down at an angle of 18 degrees to see the orchestra pit. Kim's seat is in the balcony directly above Katie. Kim looks down at an angle of 42 degrees to see the pit. The horizontal distance from Katie's seat to the pit is 46 ft. What is the vertical distance between Katie's seat and Kim's seat? Round to the nearest inch.


A plane is flying at a constant altitude of 14,000 ft and a constant speed of 500 mi/h. The angle of depression from the plane to a lake is 6 degrees. To the nearest minute, how much time will pass before the plane is directly over the lake?

Really hard (differentiation for my super smarties):

Susan and Jorge stand 38 m apart, both to the west of Big Ben. From Susan's position, the angle of elevation to the top of Big Ben is 65 degrees. From Jorge's position, the angle of elevation to the top of Big Ben is 49.5 degrees. To the nearest meter, how tall is Big Ben?

A skyscraper stands between two school buildings. The two schools are 10 mi apart. From school A, the angle of elevation to the top of the skyscraper is 5 degrees. From school B, the angle of elevation is 2 degrees. What is the height of the skyscraper to the nearest foot?

I LOVED these! You have to assume for these problems that the heights of the observers are negligible, I guess. (Which makes it seem silly to round to the nearest foot or inch.) But, otherwise, I love these!! I went around and guided their thinking through a rigorous process of picking out what you're given, eliminating what you don't care about (ie. usually the hypotenuse, therefore we discard the cosine and sine), and then figuring out what to do with the rest of the info. After the first problem, they were all on their own and doing amazingly!!

*Would it be helpful to cut up the scalene triangles physically using scissors? I think so. I think I've found a hands-on Do Now task! :)


On a different note, I was trying to explain to my regular kiddies today why sin(x) = cos(90-x), and even though I drew diagrams on the board and showed them that they come from the same 2 sides in the same right triangle, I still wasn't convinced that they understood. So, I drew an analogy! (I am reading a book about how to make ideas "stick", and one of the tips they give is that analogies help people connect to ideas in a concrete/almost visceral way.) I think it worked well. What I said was, "Let's say I say that Lourdes is sitting across from me, and Sofi disagrees, 'No! She's in front of me.' But in reality, we're both correct, because we're referring to the same position, just using different descriptions because Sofi and I are looking at her from different perspectives." After that, EVERY SINGLE KID understood! Wow to the power of analogies.

(Obvious, I know. But I think I need to consciously draw more analogies on abstract concepts, so for me it was a really good reminder of a little trick I can use to make ideas immediately more accessible to more kids.)

Monday, February 14, 2011

Pringles Cannon!

Happy Valentine's Day! :)

Unrelated to St. Valentine, here are some pictures from an awesome cannon-shooting outing today.

I had to build a little launching pad in order for it to stand straight up (to try to get it to travel directly upwards).

Here's me, setting up (wearing goggles because people told me that the cannon might blow up randomly upon multiple uses). I told the kids to stand way back.

Here my kids are measuring their horizontal distances away from the launch pad. (Making the rough assumption that the cannon's going to shoot straight up.)

The girls are getting ready with their inclinometers to catch the action! (Notice that there was quite a bit of wind today. That might have contributed to the cannon shooting a bit crookedly.)

And... (drum roll, please) Here is a video. (Notice that I was no longer lighting the cannon by this round. I took a volunteer but I told her that there were some risks involved.) This wasn't our best attempt from a logistical standpoint, since the cannon shot the ball out crookedly and made it hard for the kids to get an accurate angle. But, afterwards I estimated the hang time using the video, and found that it was ~5 seconds -- which, if you use the equation H(t) = -4.9(t - 0)(t - 5), you get a max height of ~30 meters!! WOW.

(The highest straight launches they were able to measure were about 20 meters, so this one went quite a bit further. It was also a bit later in the day, so more of the ethanol must have vaporized into the cannon before the launch. But, anyway, according to the kids, it definitely went "mas lejos!" than the previous attempts.)

PS. Did you know that Obama is coming to El Salvador, and that we get gratuitous days off because they're blocking off parts of streets around the school? I can't believe it; never heard of anything like this before.

Sunday, February 13, 2011

Struggling Along

I have been writing a lot about Geometry, since it's my second time teaching Geometry and everything feels so amazingly better than last year.

But, in reality, (it being my first year teaching Precalculus,) I am struggling with making Precalculus accessible for every kid. The good thing is, they're learning at the level I would expect -- a little slowly, and usually it takes us several days to get through one dense section in the textbook, but the kids at the top of the class are understanding everything on the exams, and the kids at the bottom are working steadily in class in groups, through all of the problems, with some help from me. No one is drowning. Yet.

But they're moving more slowly than I need them to, even just through the "review" (Algebra 2-ish) material, and at the rate we're going, we're not going to have much of a chance to do much of trig identities this year! ugh. Everything we do, it feels like we're doing from scratch with more or less zero retention from years prior. The kids whom I had taught last year are picking up more or less right from where we had left off, and sprinting forward with minimal discomfort. The rest of them are struggling with basic things, like on Friday during a quiz a kid was struggling with what a rectangular prism looks like, even though we had done various volume-maximizing and surface area-minimizing practice problems in class (and also area-maximizing and perimeter-minimizing problems with 2-D shapes). Another new kid who recently transfered to my class had to be taught how to look at a function equation to figure out what quantities the x and y represent. His past teachers had NEVER explicitly asked him what x and y stood for before. EVER! (So, when he found the maximum point (5, 368), for example, he had no idea what those values meant.)

I feel somewhat discouraged. I think they are learning, because they're working very hard in class everyday, and I am pacing the material slowly enough for it to be absorbed piecewise. But, it seems like I am having to teach them how to run when some of them can still barely walk. (It's already halfway through the year, and every time they see a new word problem, half of the class's attitude still is, "I don't know how to do this!" without having drawn diagrams, made tables, or really even attempted to set up any equations. In terms of persistence, they're miles behind my regular 9th-graders.) With the end of the year so quickly approaching, I just wonder if I am doing/will be able to do enough before June to prepare them for their future classes.

sigh. The scariest thing is that if we don't struggle with these "small" details now, they could very well get to Calculus, do a bunch of differentiation and maximization/minimization, get some x value, and have no clue what that means. SCARY!!!!!!! So, my taking some extra time with them is (sadly) absolutely necessary. But, it just seems... exorbitant.


Anyway, recently we've started looking at function transformations in Precalc. I created an activity meant to introduce kids to the basics of transformations via GeoGebra, and I really wasn't sure how it was going to pan out. It was long -- we had a 50-minute class on Friday, and most kids only got a little over 2/3 of it done. We're going to finish it on Monday, and then ease our way into practicing graphing by hand with "irregular" base functions drawn in the textbook.

Here it is. Check it out! I was actually extremely pleased with how it went in class. Kids (working in pairs) were making actual predictions every step, then verifying their predictions via GeoGebra, and asking me questions if/when they couldn't figure out why their prediction was incorrect. Lovely mode d'etre! And, as it turns out, GeoGebra is a super nice tool to use for this activity, because it allows you to define some base function (say, f(x)=x^2), and then to define other functions in terms of f. ie. a(x) = f(x + 3). Once both functions show up on your screen, on the left side you can see the vertex form of the dependent function. I think this ability to define functions in terms of other functions really helps to generalize their transformational understanding, right from the start, to any form of base function.

Thoughts? (In this introductory activity they only explore effects of transforming quadratic, absolute-value, and square root functions. But, my hopes are that they can then transfer this understanding to the other function types.)

Friday, February 11, 2011

Some Nice Activities Books!

I've mentioned before that I've made it a personal goal to go through our school's stash of books and stuff. I'm making good on my claim, and have already made repeated visits to the library and the Math&Science Center at our school. These are a few that I like, from my browsing of my school's collection so far. They're buy-worthy, I think.

* Informal Geometry Explorations from Dale Seymour Publications is a treasure trove of geometry puzzles. Through hands-on activities they encourage students to develop inductive reasoning and great geometric vocabulary.

What I liked about it: I would totally use this in my class to jump start (or to verify understanding of) a topic!! Some of them are just fun thinking-outside-of-the-box activities that build spatial awareness, but many of them require working knowledge of geometric vocabulary as well. The activities vary in the amount of intuition required, so you'd find different things that are appropriate for regular vs. honors students.

Things to watch out for: None that I can see from just looking through the activities superficially. This book is fabulous.

* Graphic Algebra: Explorations with a Graphing Calculator from Key Curriculum Press is a nice workbook with a lot of nice word problems. I think the contexts are relatively realistic (although they're not always flashy), and they scaffold it well to build up to various types of graphs and functions -- quadratic, rational, exponential functions. Besides building a solid understanding of graphs, equations, and tables, the focus of this workbook is to allow kids to navigate their graphing calculators with ease, so they've also provided ready-to-xerox handouts that work specifically on those tech skills, asking kids to analyze the function (Trace and Zoom, that kind of thing) from a graphical perspective.

What I liked about it: If you work with your kids (maybe 9th-grade level, since there is some function notation in there) sequentially through all of the problems, I feel that it would be an excellent preparation for the more complex analysis that they would have to handle in high school. It provides a good amount of depth for the topics it does cover, and the word problems provide a smooth transition into/motivation for graphical analysis. You can also adopt it for use with higher-level kids, but in the interest of teaching time you probably would have to condense the explorations for those older kids...

Things to watch out for: There is almost too much scaffolding, down to setting up tables for the kids. If I were to use this book (which I totally would do), I'd keep the word problems and take out most of the excessive scaffolding, so that kids are left with some struggling room. The questions themselves are good. If you're already pretty familiar with the applications of these standard function types, some of the word problems are not going to be new to you. (But, it's definitely a super nice tool set for less experienced algebra teachers!! And, for me it still provided a couple of new ideas for new contexts that utilize the same-old functions.)

* Real-Life Math Problem Solving by Mark Illingworth is a workbook that is truly story-based. It's designed for younger children (maybe 6th-grade? ...Also applicable to 7th-graders as intro to various algebraic methods...), and I think it could be excellent for what it is. The stories are definitely charming (if a bit "young", some of them) -- they make no claims to be super realistic, but they're half children's stories and half word problems -- and for me, that's totally part of their charm. :) They remind me of those cute Alexandria Jones stories, whose context is as important as the actual math for keeping kids' interest.

What I liked about it: The really nice thing about these story problems is that they make no mention of a strategy to use. The kids can use whatever strategy it is that they wish/need to use, in order to solve the problem. --It also means that the kids are absolutely forced to read through the entire problem in order to figure out what's being asked, which is great practice for higher-level problem-solving. (Believe me; I tried skimming through the problems, because I myself have a terribly short attention span. Skimming doesn't work. They're not formatted like normal word problems where all of the info can be extracted at a single glance.) I also liked the author's suggestion of letting the kids keep a problem-solving "portfolio" of these problems and to work on them consistently over time, maybe once every couple of weeks.

Things to watch out for: Some of the stories are a little confusing at parts in the way that they are phrased -- especially because they give you quite a bit of not straight-forward information (which is a good thing). The ideas are usually great, moderately complicated, and pretty fun, but I think they need to be edited a little bit in order for your kids to totally get what info's being given. If you use these, definitely do the problem beforehand and check the solution in the back, because what you think the problem is saying isn't necessarily what is meant by the author.

* Spatial Visualization by AIMS Education Foundation (which I used extensively last year with my 9th-graders) is an awesome activity book for 3-D visualization. You can use it with physical manipulatives or virtual manipulatives a la this applet.

What I liked about it: Super awesome intro to 3-d unit; kids really "get" after this that surfaces usually have opposing surfaces (which I think is extended in higher-level math to show that the net surface area vector of any closed 3-D figure is zero), and what volume/surface area mean. Plus, they really love the activities!

Things to watch out for: At least in the version of the book that our school has, there are noticeable errors with the provided answer key for surface area and volume. Need to do the problems yourself to have a proper answer key. Also, since they give you a bunch of different problems for every visualization skill, you should be sure to pick-and-choose problems as to scaffold/make the content accessible while keeping students feel challenged.

* Modeling Motion: High School CBR Math Activities has some lovely hands-on activities that seem pretty cool, if you've got some motion sensors sitting around at your school.

What I liked about it: There are various activities that seem very do-able inside the classroom. You don't need much else than the motion sensors and some easy-to-get supplies. I've used CBR once in grad school (to try and create motion that mimics the given graph); it was pretty cool and generated good discussions.

Things to watch out for: I don't have enough experience with CBRs to really judge whether the instructions are sufficiently detailed. And I'm pretty sure my current school doesn't have CBR's. :(

Stay tuned. I'm going to keep looking around... Maybe at math videos next time!

Thursday, February 10, 2011

Two Questions

I would like all of my 9th-graders (regular and honors) to be able to answer questions like these two by the end of the trig unit, with relative confidence:
* What is the difference between x=tan(5) and x=tan-1(5)? Draw a diagram to show me precisely what each statement means. Label appropriate side lengths and angles in each diagram.

* Evaluate tan(cos-1(5/11)) without a calculator. Show all work, and leave your answer in simplest fractional form.

If they could, I feel like I could be sure that they have a pretty good grasp of introductory trigonometry. The rest would just be icing on top of the cake! :) I did some quiz review with my honors kids today with that first problem, and they all thought it was really straight-forward. I was so happy!!

PS. Reporting from the front lines, the regular kiddies think regular trigonometry isn't so hard (and more or less did an entire assignment with algebraic and word problems in groups, with little help from me). They're ready to go outside tomorrow with their inclinometers!! YAY.

PPS. Some of my 9th-grade honors kids went wild for those "congruent halves" Geometry puzzles I had found yesterday. They were sneakily doing it today underneath their regular math work! How super cute!!

Wednesday, February 9, 2011

Puzzly Excerpts

Here are some fun puzzle excerpts from Informal Geometry Explorations by Kenney, et. al. (I will write up a more formal review of it later, but I just couldn't wait to share these! They're so much fun!!):

The directions for Puzzle #1 are to fill in the missing letter in the correct direction/position, so that when you put the cube together using this net you'd see the word MATH. I love these (the book has a lot more) because they are super good for spatial visualization training!

These next few puzzles are fun and can get a bit tricky for kids; you need to find a way to draw a path in order to divide this shape into two congruent halves. Great practice for visualization and definition of congruence!

Puzzle #2

Puzzle #3:

Puzzle #4:

(For some reason, this last one took me a few minutes. I had to actually sit down with a pencil and try different things in order to "see" it. Was it hard for you too??)

Tuesday, February 8, 2011

Reading in Math Class

Something that I think is important (but that I struggle with) is providing opportunities for reading in my math class. I've mentioned this before, but I want to formalize a way for implementing this more regularly.

From time to time kids would ask me interesting questions that I cannot answer on the spot, or that are not immediately relevant to the lesson aim. For example, one student asked me during the Measurement Unit how scientists could measure the volume of the oceans. I told her that we'd table it and maybe come back to it in a different class. I went home and did some research, and to my surprise found some interesting bits about how satellite photos are used to approximate the shape of the ocean floor, because -- believe it or not! -- our oceans bulge out in areas where there are ridges underneath. Instead of just quickly sharing the interesting tidbits that I found on the internet, I had decided to pull together a reading that I thought would be interesting to my students. (I was pretty sure they're not aware of sonar-mapping and how it works, so I threw that in there as well.)

Last week, when a couple of history teachers mentioned to me the significance of the sextant / inclinometer, I made a mental note to go home and read up about it. Most of the links on the internet were a little too dense for my 9th-graders, so as usual, I had to write up my own kid-friendly version of the reading summary. But, I think it's very worth it (and I believe these readings contribute to multiple modes of learning AND they add real-world relevance to math). Check it out!

I'm not sure if you have language-learning kids like mine (...who doesn't have language-learners these days??), but I definitely go over the reading in details after they read it, to make sure that kids are practicing extracting important information. I ask them key questions and if they can't get it as a class, I wait patiently for them to go back to skim through the article in order to pull out that information. (And I also draw diagrams on the board as we summarize the readings, to help the visual learners put together the meanings of the articles.)

PS. I still have a cool reading about the importance of ratios from my middle-school teaching days. Here is an example for how I scaffold middle-school readers a bit more than my high-schoolers, even before they get to the whole-group discussion part. They would have had to fill out the column with notes as they read along...

PPS. Something that I still struggle mightily with is how to get my upperclassmen to read the math textbook, in all its denseness (density?)! I force them to do it in class once in a blue moon -- like, maybe 3 days so far this school year. But, A.) our textbook is kind of crap and gives them algorithms in lieu of justifications and skips steps and doesn't provide a lot of explanation between steps, B.) they get mentally lazy knowing that I'd just go over the material anyway, in a more intuitive manner -- WHICH I TOTALLY GET, SINCE THAT'S HOW LAZY I WAS IN HIGH SCHOOL!! I guess until I am willing to re-write the entire textbook in kid-friendly explanation, I'm not going to likely experience real success with my efforts on this front...

Monday, February 7, 2011

The Value of Traditional Resources

One of my goals before I leave the school in June is to review as many activities booklets and math videos as I can. Speaking for myself, I am often so overwhelmed by the many resources that are available on the web, that I forget that there can be really good things in those dusty activities booklets as well. On at least a couple of occasions, I have seen something on the web, thought to myself, "Wow! That's so cool!" only to find out later that it was very similar to something I would have come across in a very non-flashy paperback booklet somewhere on an abandoned shelf.

Our school is a perfect place for me to gain exposure to those abandoned resources. Here, teachers come and go every few years, so even some really great resources that someone may have once cherished can sit untouched for many years afterwards. Last year, I spun up an entire 3-d computer project around worksheets that I had found inside a traditional booklet, and both my Department Head and the kids told me that they loved the project! (The original worksheets were designed for use with physical manipulatives, but since our school didn't have those physical manipulatives, I found a great website that had an amazing applet that did everything the physical manipulatives would do -- including letting the kids drag their model around to view it from different perspectives! That actually worked super well, and the project was really easy/fun to run. I'll blog about it later this year when we get to it again...)

Anyway, today I was flipping through one booklet in my surprisingly copious free time, and found a cool hands-on activity for teaching combinations. You may have seen it (after all, it's quite likely that someone has put this on the web at some point, although some quick googling hasn't turned up anything for me):

Place 5 pennies in the grid below, such that they each take up one space and none of them is in the same row or column. Record your result, then try to find other solutions. How many different solutions are there?

Seems like a fun intro to combinations, no? And to spice it up, you can ask the kids to figure out how many combinations would involve 4 pennies (inside the same 5x5 grid), or 3 pennies, or (generally) n pennies where n < 5! Or what about n pennies inside an m-by-m grid, where n is less than or equal to m?

See? I think there are good things in those booklets. :) With a little creativity, you can turn dusty old lessons into shiny new ones!!

What do you use? Anything I can/should add to my wishlist? There was an AMAZING middle-school activities book (many pages, hardcover) that I once owned but misplaced (think I loaned it to a teacher friend at school and just forgot to bring it home). SO SAD!!! I'm still kicking myself over this years later...

Sunday, February 6, 2011

Fractal -- Hunting the Hidden Dimension

If you are a Geometry teacher, I highly recommend getting a copy of PBS's production of Fractal -- Hunting the Hidden Dimension. It ties fractals into just about everything -- medicine, computer graphics in movies, fashion, antennae design for cell phones, maps, biology -- and is really well-done (with personal interviews and stories about the origins of the Mandelbrot set). I don't usually like to show an entire feature, but I was actually hard-pressed while test-viewing this to think of which parts I would choose to skip in class. For how much information was in the film, it was well-paced and had good visuals to keep kids' interest and to keep everything very understandable.

How lovely! Now I can't wait to teach fractals. :)

Function Transformations Nitty Gritties

I've been racking my brain about how I am going to break down to my Precalc kids the procedures for how to analyze the various transformations involved in a function that looks complicated like g(x) = -3(-0.5x - 4)^3 + 7... Transformations as in, when you compare g(x) against its parent function, f(x) = x^3, what types of shifts and scaling and reflections had resulted in this new function g. This topic is always really difficult for kids, because I am sure that when a regular kid looks at the many numbers inside a complicated transformed equation, all of the number just melt together and seem to be indistinguishable.

Well, I think I've got the big-picture concept finally nailed down to one picture. I hope this is children-safe. It's sort of like an input/output diagram of a function box, but annotated.

So, basically, the way I see it, some transformations happen outside of the parentheses (such as multiplying by -3 and adding 7, in my example function g...see below for color-coding) because -- if you consider PEMDAS -- they occur AFTER the "main" function of cubing has already occurred. At that point, it's too late to be affecting x-values, so you're naturally affecting only y-values and causing vertical changes. And since kids already know that higher coefficient = steeper graph, it shouldn't be difficult for them to figure out that this means -3 does the vertical flip / vertical stretch by scale of 3, and 7 does the vertical shift.

On the contrary, (if you again consider PEMDAS,) transformations that occur inside the parentheses had occurred BEFORE the "main" function of cubing occurred. So that means they were operating only on the x-values and therefore only resulted in horizontal changes. Again, most kids can figure out that a fractional coefficient of 0.5 makes the graph flatter, so it shouldn't be hard to make the leap that 0.5 = 1/2 stretches it horizontally by a factor of 2. (Since horizontal stretching makes a graph flatter, whereas a horizontal compression would have made the graph look skinnier/steeper, and therefore would have had a coefficient greater than 1 on the inside.) And -- keeping with the theme of "inside" operations affecting only the x-values -- the negative sign on the -0.5 necessarily makes the graph flip horizontally across the y-axis.

The only complication that remains to be explained is why the horizontal shift is 8 to the left, rather than something immediately visible inside the equation, such as 4 units. The way I've always thought about it, special things happen at f(0) for most parent functions. In order for you to capture that same special point on the transformed graph, you have to find the special value x that would result in you still evaluating zero in the "main" function (in this case, by the time you reach the actual cubing operation, the stuff inside the cube would need to be zero in order for you to observe the same special behavior). So, by setting -0.5x - 4 = 0 we get x = -8, or our special point (and every other point from the old graph) has apparently shifted 8 units to the left!

...Obviously, this reliance on order-of-operations can be translated (harhar) to other parent functions as well. Here are some examples, with color coding to show which operations come before and after the "main" function operation.

(Of course, my juniors will probably have trouble with simply articulating PEMDAS in abstract algebraic form. Whenever that happens, I make them actually plug a value into x, in order for them to write down all the steps of what operation happened, in which order, before they arrived at the result.)

What do you think? Is this explanation child-proof, or still way too complicated? I don't want to be giving the kids a bunch of blind rules to memorize that they don't understand, especially because there are already so many parent functions that they are keeping track of. I also made a long GeoGebra exploration activity that will cover most of this material without any lecturing on my part, but I'll blog about it only after I figure out how well my kids can absorb the stuff through the scaffolded (but long!) activity. Wish me luck!!

Friday, February 4, 2011


I wrote the following during the fall of my first year of teaching at a middle school in the South Bronx. I periodically look back on it and think about how all things come to pass, and how we can turn around in the worst of situations. By the end of my first year, kids were doing well in my class. I liked them, and many of them liked me. Most of them did well on the NYS Grade 8 state exams and the Regents exam that they had to take that June. I didn't feel like a failure. But, getting there had been really tough, and there had been weeks at a time when I got up every morning and considered quitting my job.

Thursday was a god-awful day in my last period. At some point, I sat down because I got tired of standing and waiting for them to be quiet, and it took them another full 10 minutes to stop talking. One kid raised his hand after I stood back up, to ask, "Those of us who don't want to learn, can we go sit in the back of the room?" and another kid said several times out loud, "I'm gonna switch to Ms. B's class."

It was soooo rude. After school, I had a long chat with my principal, and he gave me some advice on how to gain the kids' respect by playing hardball. So, I called some parents that very night and gave one kid a two-day gym detention (meaning he has to sit out of gym) right away. (My principal's exact words were: "Crush his spirit. Take away his gym, his lunch, and call his mother. Show him that you have the authority to make his life hell, and if you hear another word from him, pull up a desk and a chair outside of the room and make him sit out there for the whole period. I will support you if you think that one kid is causing your class to be out of control.")

Yesterday (Friday) was a better day, and I quietly stared down some of the kids until I felt them look away. I think the problem is not that I am weak -- because really I am not, and (more importantly) I know that I am not at all weaker than these kids. The problem is that I needed to convey to these kids in a way that is convincing to them that I am not weak. One of the kids whose mom I had called decided to pull an adult d***-move on me afterwards by saying to another kid (while standing right in front of me), "Remember when she cried?" The other kid stirred uncomfortably and mumbled, "Yo, she called my dad last night." And the first kid looked at me and said it again with an adult sort of malice, "No, but you don't remember when she cried in front of the whole class? That was funny."

I was pretty appalled by his utter assholeness. All rudeness aside, I honestly didn't expect these 14-year-olds to be capable of such real malice in such an adult way. But, still, I was unphased. The kid ([James]) has got to be out of his mind if he thinks he could shake me up with that statement. I admit, if I were actually a weak person, it would have been enough to shake me up. But, instead, I just looked steadily at him with a slight smirk, and lifted my eyebrows with the apparent disdain, "Is that all you've got?" He kept looking at me to assess whether I showed any sign of weakness in response to his statement, and I wouldn't let him have any of it. As he walked past me to go into my classroom, I whispered calmly in his ears, "Just so you know, since you were rude to me, you won't have gym all of next week." After that, whenever I looked at [James] steadily during class, he would get this look on his face like he's genuinely intimidated (that's also a very adult look, ironically enough), and he would quickly look away and do what he's supposed to. Same thing with his friend [Mark].

In class, I made my kids line up and sit down silently. When they talked, I made them get up and line up again outside to do the same thing over. I made them practice passing up their quizzes row by row, on the slow count of three. When they messed up, we passed the quizzes back and repeated the same procedure. I intentionally counted slowly because I knew it was driving them nuts. We did it maybe 10 times before they got it right. It was the most demeaning way of treating them, but the rest of the period was absolutely silent, and when a kid was talking, all I needed to do was to look at them, and they would stop. It was exactly like [my principal] had said: when you slow it down for them, they know that it is because they are behaving and being treated like children, and they actually fear your authority and gain respect for you. I will need to do that a few more times in the coming week, to solidify the respect, but after that, [my principal]'s challenge to me is to make the worst kids in my class love the content so much that just the threat of making them sit out of my class would be punishment enough.

Anyway, after school, I was talking with [my principal], and he told me two things: 1. When I need to get a quiet, instead of trying to talk over the kids, I should just whisper. He says that there is magic in being able to control kids with your own quietness. 2. Don't let the kids play me like a puppet. Now they know that I'm capable of running class military style, which is exactly what I needed. But, if I keep doing the same thing too many times, they'll know that they can manipulate me into not teaching, by acting out. 3. He was impressed by how I stared [Mark] down yesterday, and he told me that he has thought about this for 15 years to figure out why some teachers can have authority over hard-to-manage kids and other teachers cannot. He believes it's a combination of the kid knowing that you are smart, and their knowing that if they do something bad, you will remember and do something to them afterwards. They will not respect you the same way if they think you are stupid, disorganized, or a pushover, because they will think that you won't remember what they did and will not follow up with a clear consequence.

Everyday is a new day. It's always one step forward, two steps back. I don't expect next week to be easy, but the good news is that I'm a tough person so this really hasn't gotten to my spirits much. I have no doubt that I will work through this; it's just a matter of careful tactics, patience, and good lesson plans.

Everyday that fall, carrots flew across my classroom at other kids when I wasn't looking. At the end of class one day, kids suggested for me to adopt a bunny. Mid- that year, a kid made a racist poster against me that said, "Anti-Chinese! And yes, Ms. Yang, that includes you too!" that had broken my heart. ...And yet, now I am still here and loving every new day at work. :) So truly, all things come to pass. You have to just have faith that things will get better if you work at them, one day at a time. (I know, keeping that faith is really, really hard. But that's the only thing that got me through Year 1. So, if you're reading this out there and you feel like you're strugglin' everyday... my heart goes out to you in a very real way. But it WILL get better.)

Not Enough Information?

My 9th-graders are great. Today was Day 2 of trig in H. Geometry (aka. the second day ever of SOH-CAH-TOA in their lives), and I nonchalantly left this as a Do Now on the board:

1. Do you have enough information to find the height of the flag pole?

2. Calculate the height of the pole. (Ask for more information if necessary.) Round to the nearest hundredths.

3. If this girl walks towards the flag pole until she has an angle of elevation of 50 degrees in order to see the top of the flag pole, how far away is she from the flag pole then?
By the time I finished checking off their homework from last night (3 minutes into class), they were already antsy for more information. I was glad! They asked me specifically for the height of the girl, so I gave them the height up to the girl's eyes as 145cm (note the intentional mix of different units... Kids are comfortable with conversions now after the Measurement Unit!), and they cranked away at the rest of the problem. After we went over the problem and everyone felt comfortable with all parts, I exclaimed that we were going to be building inclinometers today and that we would be going outside to do exactly this.* And kids were so excited!!

The inclinometers (aka. "sextants" in history classes) were a big hit. Kids liked them a bunch. Predictably, it took a while to build them in class, so we only had time to go outside to really measure one object in the last 15 minutes of class (...Our classes were also short today -- only 50 minutes on Fridays!), but I promised that we'll return to them next week to make more use of their inclinometers. (And we'll obviously also use them when we launch the cannons!)

By the way, my history teacher friends told me that these "sextants" (Is it just me, or does "sextant" sound like a naughty word?) have quite a bit of historical significance. The astronomers used them to help navigate the ships as efficiently as possible on their trade routes. Shorter trade routes = more profit for the company, so they were a pretty big deal back in the middle ages. Cool, eh? ...Obviously, I had to share that with my kids. Math + history = pretty darn cool. :)

*Here is an example of how a relevant Do Now avoided any need for me to explain procedures down the road. When we got outside, very few kids needed very minor reminders of what to measure! I also threw in there a phrase that they had never seen, "angle of elevation," and expected them to figure it out by context.

Thursday, February 3, 2011

Trick for Teaching Basic Trig

During my middle-school teaching days I noticed that often kids would arrive in my 8th-grade class with a half knowledge of sine, cosine, tangent. There were two major problems they often had in solving for unknown sides in a right triangle using trig:

1. They couldn't visually distinguish opposite side vs. adjacent side. Many middle-schoolers I taught had a poor consistency (if any) with recognizing what "opposite" and "adjacent" meant in a diagram; it was just too abstract for them, even though I tried to explain how to look for the sides "across" the triangle, etc.

2. They couldn't figure out whether to use sine, cosine, or tangent in a given situation.

The first problem I solved successfully a few years ago, when I came up with an idea to start teaching kids to reach their hand out and to actually put their hand over the acute angle that is given in the problem. The "opposite" side (from the perspective of the given angle) is the ONLY side that their hand is not touching, since "opposite" implies "far away"; on the contrary, the "adjacent" side is the side (besides the hypotenuse) that their hand IS touching.

Trust me, if you're seeing the same problem in your classes, TRY THIS. It works like a charm. I taught sine, cosine, tangent from scratch today to my 9th-graders, and not a single person had trouble recognizing opposite / adjacent sides. (Granted, they were honors kids, but again, I've tried this with my regular 8th-graders in the Bronx. It had worked like a charm then, too!)

Issue #2 just takes a little bit of practice, but this year I found that I made a nice transition into this by having allowed a couple of days of pure similar-triangles proportions practice. Right from the start, kids really grasped the concept that in order to solve for x, you need a proportion that involved x as an only unknown, so it was super easy for us to transition to speaking about putting together x and the other known side inside the same trig ratio / proportion.

So, surprisingly, I taught all of basic trigonometry to my honors classes in one 75-minute period. (Inverse trig not included.) Seriously, those guys had never ever heard of SOH-CAH-TOA before today. Neat, eh? We'll see next week whether I can translate this success to my regular classes!!

Wednesday, February 2, 2011

Request for Tech+Math Project Ideas

Ever since I saw some thing somewhere about building 3-D objects in GeoGebra (I know, it's very specific... I suck at leaving bread crumb trails), an idea has been brewing in my head about letting my kids build some cool mathematical objects in GeoGebra and then digitally recording how they did it, and then adding voice-overs to explain the technical aspects of their creation, as well as the mathematical significance of what they did.

Again, that's very vague, but all I know is that the one student-made video I did manage to download on my incredibly slow computer was way too advanced for my kids to even attempt; I am not willing to spend weeks getting them to figure out the technology. (A few days, yes. But not weeks.) So, here's my question:

Have you done something like this? Do you have ideas for what topics would work well for a project like this? Ideally, I'd like the math to be manageable for every kid, and for the focus to be on getting them to feel familiar with an important feature of a common piece of math software. (Doesn't have to be GeoGebra, although GeoGebra is nice because then they can continue the work at home.)


Trig Love

I have an inexplicable love affair with trigonometry. But, actually this year is my first year teaching a full unit of trig! (Last year I was bogged down teaching geometric proofs for EVER and didn't get around to doing a proper trig unit in Geometry. !Que lastima! But, the up side is that this year I get to design a new unit from scratch. WHICH IS ALWAYS SO EXCITING!!!) :)

We had already discussed right triangle similarity a little bit in our Measurement Unit, when I had taken the kids outside to measure heights of objects using reflections in a pocket mirror. I re-introduced the topic of general similarity this week by asking kids to build congruent and similar parallelograms on the Geoboard. I was more than pleasantly surprised by how DIFFICULT the task was for most of them (honors kids included)!! When I went around and facilitated, I had to guide their attention to slopes of lines, just to get the angles correct in their similar parallelograms. And then, once they had finished constructing (with rubber bands) parallelograms of congruent angles, we discussed as a class why these two are NOT similar:

In one class, I was feeling pretty cheeky and I drew this on the board and said, "If you think those are similar, that's like you saying that your face would look the same either here or here."

I think that illustrates non-similarity pretty well (and has a bonus giggle-factor). Afterwards, we used the rubberband parallelograms to practice finding perimeter (Pythagorean Theorem, anyone?) and area (visualization of parallelogram --> rectangle), before launching into a textbook exercise of setting up ratios between similar right triangles to solve for missing sides. Good algebraic practice (involving simplification of square roots, no less), but obviously, this is not what I have in mind for making kids feel EXCITED about trig!

So, I spent part of the afternoon researching options on building inclinometers. The plan is that we'll do one day of flat textbook practice to introduce the basics of trig, then one day "out in the field" measuring tall objects using inclinometers and trig, and then one to two days figuring out where the heck the sine/cosine/tangent values come from, using traditional protractor and ruler. See below. (I can't take credit for this; I'm almost certain I have seen this table format from another teacher at my old school.)

THEN, we'll revisit the Erastothenes video as motivation for inverse trig. (How did Erastothenes find out the angle of the sun relative to the vertical column??)

And, at the end, my kid will (finally, FINALLY) be ready for the Pringles rocket launching goodness. The following blue print is given to me by my awesome physics teacher friend Brian. I haven't built/tested it out yet, but I plan on shooting these more or less straight up to see how high they can go. (Horizontally, I've seen them cover a huge distance, and Brian says that some cannons can send a ball flying for 70 or 80 meters!!)

(And then for my honors kiddies, obviously we'll go into some more advanced stuff, like laws of sine/cosine stuff. I haven't really thought THAT far ahead yet. We'll have to cross the bridge when we get there.)

Thoughts? Suggestions?? Shoot them my way, puh-lease!