Wednesday, September 29, 2010

About Tropical "Snow Days"

Kids learn a lot from their parents. Sometimes the things they learn aren't so good, but other times the things kids say about their parents can really surprise you.

Kid: Ms. Yang, I hope we don't have school tomorrow.

Me: (sigh) I hope we do have school tomorrow. ... [Kid], You know that if they have to cancel school tomorrow, that means that a lot of poor people are out of their homes, right? You know they'd only do it because they have to use the public schools as evacuation centers?*

Kid: Yeah... I know. My maid's family is staying with us right now.

Me: The whole family? How big is their family?

Kid: There are 11 of them.

Me: Everybody is staying with you??

Kid: Yeah. My mom told her to bring her family over, so they're all staying with us for now.

Me: Wow. That's really nice of your family to do that, [Kid]. But wow, that's a lot of people.

Kid: (smiles) Yeah, but it's OK... Our house is pretty big and we have 3 extra rooms.

Me: Oh, cool.

Kid: Anyway, I hope there's no school tomorrow!!

Me: (rolling eyes) I hope there is school.

*Background info: Tropical storms are really bad right now. It's supposed to rain for the next 3 days straight, and those heavy rains can cause collapsed roads/homes and (obviously) loss of lives, especially since it has pretty much not stopped raining since last Friday. So, the country generally is in high alert (and a lot of anxiety).


PS. Random tangent about snow days: Growing up in Southern California, we never had snow days. So, as a kid, I never experienced the excitement of getting to sleep in and then going out sledding. But, we did have a couple of "smog days" that I could recall, when the city's smog level was so high (you sciency people know how smog level rises with the heat) that it wasn't supposed to be healthy to go outside, so the city would cancel school for the day. Obviously, we all went out anyway. But, come on! It was 90+ degrees and exceedingly beautiful (albeit I guess smoggy) outside -- what kind of sucker teenager would stay at home??

It wasn't until years later, as a teacher in NYC, that I experienced my first snow day. I think I just slept all day and graded in bed while watching Law and Order.

Saturday, September 25, 2010

Virtual Piano!

This is such a cool idea. I can use it to illustrate basic uses of transformations in music. Yesss.

Neat links: Meow Meow song and 3 Blind Mice are both good examples of some of these common transformations. I think I'm going to give the kids a sheet of partially filled out music, have them perform some quickie reflections and translations to fill out the rest of the missing bars, and we'll use the virtual keyboard to play the music, in order to reinforce the connection of how those musical notes / patterns are affected by the transformation.

Excited! :) What a cool, multi-sensory application of transformations!

(I was telling Geoff about this, and he wasn't too impressed. Them musical people take these things for granted, I guess. He told me it's called "transposing," and that pianists do it all the time -- in real time! -- to adjust the musical keys to fit the singers' vocal ranges. But still, I think it's pretty darn cool that it's nothing more than a geometric transformation.)

Another neat link between music and math: My friend Jon has a deejay software that allows him to filter out certain parts of a song. When we get to sinusoidal functions in Precalc, I'm going to try to set up something to give the kids a sneak peek of how Fourier Transforms are used by musicians to blend / adjust music! Maybe we'll even have Jon guest-deejay for us. :)

Why Tessellations are Geometrically Educational

I am really excited about the tessellations project I am in the middle of doing with some of my kids. It turns out that doing tessellationgs with a protractor and ruler is way more geometrically intense than simply learning how to accurately use a protractor.

Take a look at a "simple" tessellation below of a triangle. As it turns out, triangles get tessellated by forming parallelograms. In order to construct subsequent segments that are parallel to segments already existing, you need to look for and measure alternate interior angles to be congruent. In the picture below, I marked the original triangle in black, and the second triangle in red, the third triangle in green, the fourth in blue. Angles marked with an apostrophe (A') indicate angles that turn out to mathamagically be congruent to original angles WITHOUT the kids needing to measure them. (This is obvious by either the Third Angle Theorem or ASA.)

It's pretty AWESOME how even the simplest tiling construction involves a series of geometric theorems that we can discuss later. And also awesome that these subsequent triangles are constructed without the use of a ruler (just straight edge and protractor).

The math is even juicier for tessellating quadrilaterals. As it turns out, quadrilaterals can be tiled by forming hexagons whose opposite sides are parallel and congruent. (Not unlike parallelograms.) In order to construct the new quadrilaterals, it turns out that we still measure only two pairs of alternate interior angles, but we also have to use a ruler to make sure the resulting parallel sides are also congruent. By some sort of SASAS rule (is that actually a theorem in introductory geometry?), the resulting quadrilateral will be congruent to the one you started off with. Again, in the diagram, A', B', and C' are angles that were not measured in the construction of those new quadrilaterals. They were mathamagically natural results of other parts of the construction!

Truly phenomenal. I am thoroughly impressed by how something that seems so simple is brimming with math content.

Addendum: SASAS exists!

Friday, September 24, 2010

Workshop-Styled Presentation Tips?

My department head has commissioned me to put together a presentation on available math teaching resources on the web. He said his goal is for "people to really fall in love with the idea of these web resources" by the end of the (relatively short, less than an hour) presentation.

Suggestions?? I am thinking of launching it with a fun activity -- one of Dan Meyer's graphing stories (ie. make the teachers actually try graphing it), and then giving them a few great worksheets I found from Dan Greene's blog (they won't do these... they'll just look at it, compared with the solution), then giving them the gist of WCYDWT, followed by Dan Meyer's escalator/current movement video as an example.

Then, I think for the hands-on portion, I am going to direct the teachers to Sam Shah's virtual filing cabinet and divy them up by discipline, and they would poke through the list of links to find a few interesting examples of a different way to present something that they already teach in their classes.

In the end, we will have a shareout, and maybe play with a GeoGebra applet (if time allows) and I'll give the teachers a short list of (what I consider) the most useful websites.

What do you think about this plan?? Suggestions are very welcome! (I've still got weeks until the presentation, but my department head has already spoken to me about it SEVERAL times. I think he really wants to get the most out of this.)

Wednesday, September 22, 2010

Tessellations in MSPaint

I am starting a new tessellation project with my Geometry kiddies this week (honors) and next week (regular). Already it's looking like the tessellation of triangles and quadrilaterals (using rulers and protractors) is going to take a lot of time in class. But, I think it's still worth it, because it's bringing up a lot of good issues about how to construct parallel segments using alternate interior angles. Kids are feeling frustrated because they can visually see that subsequent lines and shapes are not where they need to be, when their angle measures are off by only 1 degree. In a period of working solidly for 40 minutes, most of my honors kids only tiled their triangle correctly 3 or 4 times. (They now need to finish that part for homework.)

When they eventually get to the "fun" (ie. artsy) part of the project, I've prepared some examples of "fancy" tessellations (which I whipped up in MSPaint today! I love making vector drawings.) I'm looking forward to that part of the project, of course.

Tuesday, September 21, 2010

Architecture and pressure

By the way, Geoff's dad is an architect, and one time he explained to Geoff and me that there are certain architecture rules for planning enough floor area into a building in order to withstand a certain estimated weight (estimated max capacity + furniture). There is wiggle room, obviously, by building in additional support structures into the frame of the house, but in general you have to allot for more area in a library building, say, than inside a regular house.

--DAMN! Had I remembered that earlier this year, I would have included it into my architecture project.

Sunday, September 19, 2010

Where They Falter

I just gave / graded my first Geometry exam for the year to all classes. I thought I'd post a comparison of how my regular Geometry kids did on a few select problems. I am very interested in seeing where their thinking breaks down, so most of my questions are fairly tricky. There is always a good chunk of explanation / multiple-choice questions on each test, where you have to be absolutely sure of yourself to not fall for any of my "common algebra misconception traps."

Can you see where kids would falter? (You might be particularly interested in this if you yourself also teach Geometry, but maybe not.)

Pg. 1 turned out to be surprisingly easy for most kids (even though the questions are NOT simple). Most of the points I took off were because of incomplete explanations, not because of big misconceptions. Here is one kid who did particularly well (as you can tell from his total score, he isn't one of the "hot shots" of the class... he's pretty representative of the average kid). His explanations showed that he avoided all the traps that got my kids last year, which made me very pleased. :)

Pg. 2 is interesting, because it involved some multi-stepped sketching of diagrams based on verbal instructions (always harder for kids than you might imagine, even if they know the meaning of all the vocab words), plus some TRICKY multi-stepped algebraic problems.

Excellent-understanding example:

Spotty-understanding example:

Poor-understanding example:

In particular, in my two years of teaching Geometry, I find that kids always want to jump to calling all missing angles x, and then rushing to write an equation. (Even though I keep reminding them that you can't introduce your own x if there is already an x in the problem.) That's one of the biggest "gotchas" on this exam; I hope that moving forward, they will gradually fix this.

Overall, the kiddies did okay (ie. not badly enough to depress me, but with a lot of room to grow). :) My honors kids also had a decent spread, which is also good. (A small kick in the butt is always good for the honors kids. Scarily enough, some of them honors kids couldn't figure out how to find the area and perimeter of a 3/4 circle, because by just looking at the picture, they couldn't reason out that it was 3 times the size of a quarter-circle, which we had practiced explicitly as a class. ---SCARY FOR BEING HONORS STUDENTS!!!)

Ok. Off to my week! :) By the way, I've got one super-sharp kid in my honors class, who works at roughly 2.5 times the speed of others. I kept him spinning last week with a hardish Ken-Ken puzzle, and he was super happy. Got other good things to throw my way?

Tessellation Madness

I found two awesome resources for teaching tessellations: this is super cool for taking your kids down to the computer lab and quickly creating their own custom tessellations handouts, followed by this as an uber-fun tessellations project. (I am going to add more parts to the tessellations project, obviously. Minimally, before the kids can get to the cool cartoon tesselations, they'll have to show me that they can tessellate by hand triangles and quadrilaterals. I'll probably write more about it once I know how well it works out in my classes.)


Today's the 1461st day that Geoff and I have been officially dating. :) (That includes a leap day somewhere.)

Since it's easier to celebrate on Saturdays than on Sundays, we went out last night to one of the fancy Japanese restaurants we've only seen but never tried. It's on top of Torre Futura, which has an amazing night view of the city from all sides. The dinner turned out to be delicious! :)

Thursday, September 16, 2010

Fun with Coordinate Transformations

I had a lovely time using my remixed version of Dan's snowflake prediction activity with my Geometry kids this week. It turned out to be a lot of fun for all the kids, and super educational! (We only cut out the first and third designs on the first page, plus the first design on the second page. For the rest, I had some pre-made cuts that I unraveled step-by-step in front of the class, once they've already made the predictions.) The faciliation turned out to be a good exercise in throwing more geometric vocabulary words at them (ie. "If you open it up here, where will be your line of symmetry?").

Lovely! :) It makes me so happy that it worked.

The other thing that I did with coordinate transformations this week that worked exactly the way I had envisioned was a partnered activity, where each partner has to describe orally how to transform existing polygons into their new images. They don't get to refer to the new coordinates OR to show the desired result to their partner, but they get to describe the CHANGE using words. It's a lead-in into how we specify transformations using numbers. It worked really well, and the kids loved it. :) By the time that they tried this activity, they hadn't seen at all how to specify the amount of translation, dilation, rotation yet. Many groups ended up figuring out the most efficient ways to describe the changes -- all by themselves!

Here are the files. Try them out!

(Notice that the back of each partner's worksheet is the space provided for sketching the solution to the other partner's verbal description.)

More goodness to come. (This week, I've been very busy! Giving tests in all classes, finishing up projects in 3 classes, and I went away to someone's lake house all of yesterday, during the Central American Independence Day holiday.)

Sunday, September 12, 2010

Architecture (area and perimeter) project

Here is a student work sample from a recent area / perimeter project that I assigned. It's the second year in a row that I've done this project (last year with regular Geometry, and this year with both regular and honors Geometry), and the kids really enjoy it across all the sections. This girl's math work and her explanations are particularly fabulous, so I decided I'd scan her project in and post it as a sample of the skills I desire in every child. (I am very nit-picky with explanations on projects; they have to give formulas AND show substitutions again AND explain their reasoning for modifying certain geometric formulas, in order to receive full credit.) All groups were assigned different floor plan layouts (I had five pretty different designs), so their answers are all a bit different, but each design has two semi- or quarter-circle curves and a couple of diagonal/triangular sections. It's a good review of what's what with regards to some common geometric formulas. It also teaches them an important skill of looking at a picture and dissecting it into pieces. (ie. How do you know when to add or subtract an area from the total? --NOT obvious to incoming 9th-graders, mind you.)

The project takes about two days of class time. (One day, they do all the perimeter calculations and explanations; can finish up for homework, so that we're all on the same page at the start of Day 2. Next day, they do all the area calculations and finish the explanations for homework.) For me, the hardest part of managing the project is finding enough class time to go through every kid's rough draft (the explanation parts) to nit-pick and give them individual oral feedback before they type up their final drafts at home. I did it this time by assigning a multi-step Pythagorean word problem at the beginning of the day (on Day 3), and also giving the class some time at the end of the day (on Day 3) to work on reading origami instructions and building regular hexagons. During those individual work times, I walked around and quickly conferenced with each kid about their project (while trying to also help other kids with their Pythagorean problems). It made me feel extra ADD/frazzled to be doing two completely different things at the same time, but it worked.

Do you do something similar to this area/perimeter project in your class? What does yours look like?

Thursday, September 9, 2010

Maras Showdown with Government, etc.

This is a quick life update for my non-mathy friends:

El Salvador is crazy. I am mad that the maras decided that they could shut down the entire public bus system as a fist-waving gesture to the government. Some of you might remember that back in June, as part of their extortion schemes, the maras burned down a bus with over a dozen people trapped inside. The outrage continues, when they issued a threat (via pamphlets and other things) earlier this week to do more of the same. So, in response, the entire country's bus system was shut down 2 days ago.

As usual, this doesn't impact the well-to-do, who travel in private vehicles. It only impacts the poor Salvadoreans who now have to get up early in the morning and walk hours across various towns to get to work, or to take camiones if those can be found. It sounds like as of yesterday, parts of the transportation system have resumed operations, but the system continued to be affected today.

What a ridiculous situation! Supposedly the reason for this wave of mara threats is a retaliation / protest against a newly passed bill outlawing Salvadoreans from joining a gang. Well, it looks to me like the mara are playing Quien tiene mas grande huevos with the government, and they're not doing so badly in this contest. I was talking to Geoff about this, and I was saying that in the States, those gang threats would have been SQUASHED by the government. He reminded me that this situation would never have even occurred, because the gangs simply don't have the same power over there. What a different world it is that we live in. :(


In brighter news, Geoff and I finally got our car back! Yay. We might have to send it away again soon, to fix the shocks. Who knew that there were 4 sets of shocks in a car?? I don't like the sound of that. We need to go back to driving Flintstone cars.

But, having a car allowed us to go back to our salsa class last night. SUPER FUN!!! :) :) We did some closed-eyes dancing, which is always incredible, because you have to rely on the raw connection you have with your partner in order to follow their lead. It made me miss swing dancing like crazy.


By the way, our first progress reports are due tomorrow. Time in school is flying by!! Time to finish grading those quizzes...

Tuesday, September 7, 2010

How I do Assessments

It's Week #4, and I am finally ready to tackle writing about how I do pacing and assessments. I'm no expert at this, but I've got a system that I think works pretty well for me.

I did a lot of blog reading this summer about other people's ideas of math teaching. I'd say that a majority of them are keenly interested in writing about their venture into what's called the Standard-Based Grading, which is basically re-testing/remediating the hell out of every kid until they get every concept right -- even if that means you've already tested that same kid 10 times on the same topic. Also, it seems like SBG believers don't count homework into the student grades, furthering the strength in their belief that they assess only knowledge, not effort.

Personally, I decided that this would not be the right time for me to implement SBG. Knowing myself, I fear that it would take my focus away from my own planning for content and focus my energy on the logistics of assessments. Likewise, I am not sure my freshmen have the responsibility it takes to only have optional homework assignments. Maybe some day SBG will happen in my classes, but in the meanwhile, I decided that it would still be feasible for me to incorporate some nice features of SBG into my weekly planning.

Here are the things I already implemented throughout last school year (2009-2010):

  • I consistently give the kids time in class to work on practice quizzes and practice tests, at least one day before the real assessment. This serves two purposes: it helps kids who have trouble decoding instructions to see them once, the day before an exam; it also helps all kids to focus in on the topics and skills I think are the most important; it also helps me do last-minute troubleshooting of conceptual issues before the actual graded assessment.

  • I consistently give kids opportunities to make up points from the quizzes, by doing corrections the day after. I typically allot 10 to 15 minutes at the beginning of class to allow kids to do corrections on a couple of questions (and I cap the number of points/questions they can make up, in the interest of precious class time). I circulate during this time and kids can ask me or each other for help.

  • Every quarter, before the end of the quarter, we spend a full day in class making corrections for the tests we've taken that quarter. This way, kids can make up some more points and they also spiral back to some material they didn't get along the way.

Changes I've made this year (some are inspired by SBG):

  • Every week, the kids will have either a quiz, a test, or a project due. So, every week, I am formally assessing them in some way. Some weeks, like this current one, they have multiple things due.

  • We are still consistently doing practice quizzes even with the near-weekly quizzes. I only see the kids 4 times each week because of a hybrid block schedule that we have, so that means that we're spending about 30 minutes one day reviewing for a quiz, and 20 to 30 minutes another day taking the quiz. That sounds like a lot of time spent on assessments alone, but the truth is that I get to walk around and give them feedback during the "practice quiz" time, so I see it just as any other regular mixed-review work time, except with more focus on the exam topics (and therefore, more urgency from the kids).

  • I no longer spend time in class doing quiz corrections. Instead, the quiz correction option is always available to all kids to complete on their own time, and I make kids who fail a quiz stay with me during that week to make up the quiz points after school. This concentrates my efforts on the kids who really need the remediation on those particular topics, and ensures that the remediation happens more-or-less right away for those kids.

  • Instead of doing quiz corrections as a class, each quiz (and corresponding practice quiz) I include one problem from the materials of the previous quiz. I only re-test the one type of problem that I feel that as a class, the kids struggled with the most. In reviewing for the new quiz, I can highlight again that old skill, and they have a new opportunity to show me that they now understand this topic.

  • We're not that far into the year yet, but I plan on re-testing most commonly missed problem types on subsequent tests as well.

A brief note about why I make these choices: I feel that assessment needs to come in many forms, and be constantly happening. Part of the reason I don't make homework optional is because when I go around and check kids off for homework each day (during their Do-Now), I am actually looking at their work and giving on-the-spot feedback on major conceptual errors. I agree that we should re-test kids on certain topics so that kids feel like they are responsible for old material, but I don't think there is a point in re-testing something to the whole class if 85% of the class got it right the first time. (Example: Well more than 80% of my kids got both algebraic problems completely correct for Segment Addition Postulate vs. Bisected Segments on the last quiz. It's not an efficient use of our class time for me to put it again on the next quiz for everybody.)

I also think remediation is important, but I struggle with spending too much class time remediating all the time. Pacing-wise, my preferance is to have the quiz a few days after I've finished teaching that chunk of material, to allow kids time to ask me questions after class or during lunch, if they still need help. So, pacing-wise, this looks like I will teach something until Friday, give a short practice quiz in class on Monday, and give the actual quiz on Wednesday. Meanwhile, I'm teaching new material the rest of Monday, all of Tuesday, and the rest of Wednesday and Friday. (Remember I only see each kid 4 times a week.) This leaves me in a pretty good situation to quiz again the following week. Projects don't conflict with that schedule either, because I give kids plenty of in-class time to work on projects, and for the most part, they shouldn't need to see me outside of class for help on projects. So, my own outside-of-class time is carved out to help only those kids whom I've identified as needing remediation on the most recent quiz.

I like a lot of things about the SBG, but I don't get how SBG teachers can convince their kids that remediation needs to happen immediately, if they allow their kids the option of re-testing all the way through a quarter. Also, I don't get how they manage to keep their own sanity in place near the end of a quarter, with all the student requests for re-testing. So, until I can answer those questions for myself (in a satisfactory manner), I'm going to work on continuously revising my own assessment system so that it adopts more of the good things I see in SBG...

Sunday, September 5, 2010

Snowflake Scaffolding

If, like me, you are someone who loves the idea of using Dan Meyer's snowflake prediction activity to teach reflections, maybe you would find this post helpful. I read through his old blog post on this activity and it seems like he is saying that the activity is cool but needs some more scaffolding in order to reach all learners, so here is my supplement to his activity. I think this scaffolding will work (but we will see in a couple of weeks, when my kids actually do it). You can still have your kids cut out the first couple of the shapes to verify kinesthetically why the reflection happens the way it does, but hopefully they are then building the abstract understanding of reflection as they progress. If you compare the shapes I scaffolded below and his worksheet, you'd see that I didn't scaffold the last ones of each series of snowflakes. I want the kids to be able to do the last ones without all the scaffolding.

Let me know what you think, if you do use this supplement to Dan's activity! The file is here if you wanted to grab it for use in class.

Friday, September 3, 2010

Slopey Linear Goodness

It's the end of Full Week #3, and I am in the midst of linear functions with my Precalc class. I decided to try something a little different with graphing lines this year. I'm tired of teaching every kid to re-arrange terms into y=mx+b form only to have them then not remember how to graph a line of that form. This year, I've been telling kids to just come up with TWO points that fit an equation, and then to connect them. This allows them to find the slope immediately via looking at the graph, and then if they want to, they can either extend the line (if it's a simple case) or plug in a point to find the y-intercept, as per usual. I like this "graphical" approach for two reasons: 1. It re-inforces, at every step, the idea that a line is the locus of solutions to that equation. 2. Most kids with a decent number sense can look at a linear equation (of any form) and think up in their heads at least one (x, y) ordered pair that satisfies that equation. If a kid can't think of the ordered pairs intuitively, then at least the idea of plugging in an x and solving for its related y value makes SENSE to the kids. As opposed to the whole re-arranging terms thing, which they only half-understand, at best.

So, for example, today we saw in the textbook a problem that said, "Find the equation of the line through (2, 3) and parallel to 3x - 2y = 5." The way we approached it was we first took a look at the line 3x - 2y = 5. Some kids figured out that (3, 2) and (1, -1) were two points on that line. We graphed and connected the two points to graphically find the slope of that line = rise/run = 3/2. Then, for its parallel line that we're looking for, we graphed (2, 3) then moved towards the y-axis via the same rise/run until we graphically found its y-intercept, which happens to be zero. So the equation we're looking for is y = 3x/2 + 0.

Obviously, out of habit, some kids want to immediately re-arrange the given equations into y=mx+b form and to do the whole problem via algebraic manipulation, which is all good with me (as long as they can do it correctly). At this point in their junior year, I feel like I want to just try different strategies to fill in the holes of their understanding. Maybe some kids are already good at the traditional methods, and that's great. For the rest, I want to offer them a little bit of a different strategy that might make more sense to them.

Sprinkled into this week was Sweeney's Slope Rida sing-along (yes, I sang with my kids to the instrumental version... I also showed them Sweeney's rapping video, even though I couldn't do the rap myself... and I even got another teacher's econ class to come in, be their audience, and to sing along!!), as well as his trick for remembering the difference between zero and undefined slopes and the first of the graphing stories from Dan. (I plan on doing a graphing story per week. Keeping it light, maybe on Fridays!)

Overall, it has been a good week. :) There's a lot of stuff I'm throwing at my kids this week, so I won't really know until next week how well they are doing, exactly. But, I think it'll be OK!

Wednesday, September 1, 2010

Always, Sometimes, Never True?

This is an old worksheet that I made last year as an intro to points, lines, and planes. I read on Math Teacher Mambo about an awesome True / False exercise that is similar in format and based on the same topic, so I thought I'd post this as a somewhat longer activity variation. If I remember correctly, last year it had generated some good discussion among the kids (I had let them work in groups), even though the activity was fairly time-consuming and (ultimately) a bit dry for those highly ADHD kids.

Anyway, this year I decided to skip this activity, only because it ended up dragging out the points-lines-planes topic into 1.5 or 2 days in my class, which I thought was too much for that simple of a topic. I still like the exploratory nature of the Always, Sometimes, Never True questions. At some point, it'd be nice to incorporate this type of activity into other topics that I teach...


By the way, I more or less finished the Geoboard area activity with my honors kids, and they said that they loved it! :) :) (Most kids needed a little bit of help coming up with the quadrilaterals of decimal areas. That was really cool; I think it really helped to solidify their understanding of rectangular and triangular areas.) I also did the pentagon origami with my regular kids. --This week is so much fun!! My regular kids really didn't like the pentagon activity (they were frustrated even with my helping them out and their helping each other out), but they were pretty excited when I told them that we're going to learn to make hexagons next, in order to build a regular tetrahedron that requires no glue! (I showed them the one I had made earlier as a trial run, and they thought it was the COOLEST.)