Inception and Math

I went and saw Inception yesterday; it was fantastic! There were some inconsistencies in the plot, like what happens when people get shot (Do they die?? do they go back up one level?? Does it depend on their level of sedation??), but the fight scenes were original and the plot kicked some serious ass.

Spoiler Alert -->

And, make no mistake -- I did not miss the math references in the movie. They said that every time you go into another dream-within-a-dream, the time expands by a factor of 20. So, 10 hours in real life is "about a week in a first-level dream, about a month in a second-level dream, and about 10 years in a third-level dream." They then did a similar analysis towards the end of the movie, to figure out how much time they still have left in each dream to get stuff done. --Not to mention there was a good amount of Geometry -- transformations (reflections, rotations), and even a discussion of impossible Geometric constructions (ie. infinite-loop staircases)! Beautiful.

Awesome WCYDWT material, once I can get my hands on a copy of the movie. :)

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If you are an aspiring PreCalc teacher (like me!), you might be looking around for tools or demos that will illustrate / elucidate the sometimes-confusing concepts of radians, degrees, arc lengths, etc.

Fortunately, Mr. H has a slew of good tools you might be able to use. :) He creates things with GeoGebra that I had no idea were possible! AMAZING.

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I am frustrated that Geoff and I don't have cell phones while visiting in the States. I waited for a friend for an hour yesterday at the corner of Columbus and 82nd, but still missed her in the end! argh. I felt really bad because we were waiting at two different places, and I didn't have a phone to call her with. Boooo.

But, on a good note, I did get to hang out and catch up with my friend John for a bit, before he rushed off to the airport. (This seems to be a trend with me/him.) :) It was lovely, and totally worth the two-hour commute from Jersey to grab breakfast with him at 9am in the city! (That meant I was waiting outside for the bus at 7:10am, after catching only about 5 hours of sleep!)

In other news, Geoff and I went swing-dancing on Thursday, and it was lovely! Geoff sweated through 3 t-shirts; that's a pretty good measure of how great the night was. Now my old people knees are hurting, and I'm hoping that they will recover by next Thursday (the next Fram). :)

In the meanwhile, I am going to look at property with Geoff. I anticipate this being loads of fun. (Not.)

Triangle Area - Confusing Sub-Periods!

I found a simplistic way to find the triangle area, that does not involve piecewise functions. You basically use the angle measured from the fixed vertex in your calculation of the area. The resulting equation is still somewhat hairy and spikey, but at least it's all in one piece.

Here is the demo that illustrates the moving triangle, with only one fixed point and two vertices rotating each at a different rate around the circle.

Screenshot (much less exciting than the animation linked to above):


Graph of area over time. Notice that this is much more interesting, because you can see the times at which the two points overlap on the circle. NOTE: This graph is incorrect; corrected version is below in the Addendum!


Now, here is my question after looking at this problem for a bit and feeling like it's driving me nuts: Why are there 9 periods between t = 0 and t = 20 (by examination of the graph)? Is there a mathematical way of figuring this out? (Is it simply that one point completes 4 cycles and the other completes 5 cycles in 20 seconds, so the sum is how many times both points are collinear with E?? If so, how can you justify that geometrically?)

ARGH. I must be thinking about this the roundabout way, and I hate that! If you can see through this fog, please enlighten me. Otherwise it'll keep driving me nuts.


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Addendum 7/31/10: Thanks for the catch, Matt! Here's the fixed graph.


This time, I saved the file with all the helper functions for you to look at. I tried to name them descriptively, but ED(x) and EC(x) represent the lengths of two sides as a function of time, and thetaE(x) represents the angle at E as a function of time. AREA(x) is the final area calculated as a function of time, using Law of Sine. Cheers!

Triangle Area - Quasi-Simple

Continuing with the previous triangle area thread, here is a problem that is slightly more complex / interesting: How do you model the changing area of the outer (blue) triangle over time, as point B rotates around the circle?

Here is the screenshot, but again I would advise you to check out the actual GeoGebra animation (linked to above) to get a visualization of what we're dealing with.


Fortunately, as you can well imagine, the two areas are intimately related. By examination, the two triangles (red and blue) share, in fact, a common altitude and their bases are related by a factor of 1.5. Thus, we can piggy-back on top of results for the simpler case (see entry from two days back) and, in essence, the two graphs for the two triangles will look like mere scaled versions of one another:


Beautifully simple, no? (As this can be extended to show that any such blue/outer triangle will have an area that is a scaled version of the red/inner triangle.) Now, off to thinking about how this changes if two of the three vertices lie on the same circle, but rotate at different rates...

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By the way, here is a cutesy SAT problem, good for your middle-schoolers or anyone learning about rise / run (courtesy of David Marain):

Points A(4,5), B(7,9) and C(t,u) are on a line so that B is between A and C and BC = 5(AB). What is the value of u?

For high-schoolers, you can extend this problem a bit to ask where C would be located if BC = n(AB).

When You Go Budget...

I was reminded yesterday of something silly that had happened in Peru, and I thought I'd jot it down for future reference. It goes to illustrate how budget Geoff and I are, and the general ridonculous nature of our travels.

(As is the norm when you are or date a redhead...) G and I decided to buy some wine for the four-hour train ride between Aguas Calientes and Cuzco. Because we didn't have a bottle-opener, boxed wine seemed to be a natural choice. The tienda we went to only had a (dusty?!) box of WHITE wine (named Gato, or Cat), so whatthehell, we decided to get it even though room-temperature white wine in a box seems even more suspicious than red wine in a box.

About 20 minutes into the train ride, we realized that we didn't have cups. So, no worries, we emptied our plastic hiking water bottles and poured the white wine into them. We shared a round of drinks with our hiking mates, and while partaking in this first round, Geoff discovered the expiration date on the box: January 2009.

...Needless to say, hilarity ensued, during which we made several tipsy bad puns such as, "The cat is out of the box!" and "Hey, cats are supposed to have 9 lives," and other things I can't remember now. I'm half-amazed that we didn't die from bad chemical reactions.

That night, Geoff and I went to some local restaurant in Cuzco, where Geoff orderd a dish that looked like someone had literally thrown up all over his plate and brought the puke out as a dish. Hysterical in hindsight, but we were really worried about food-poisoning when it was all happening (especially because there was trash all over the floor of the restaurant and everything -- and I mean everything -- tasted recycled). I know it has been days since, but I am still banned from choosing a restaurant.

(Geoff was most traumatized because Peruvian food is super delicious, and this restaurant was damaging our impression of their national cuisine.)

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In other news, Geoff and I have arrived in Jersey/NYC! I am SUPER excited for Fram tonight (especially because Heather and Ryan will be DJing!). :) Oh, how I've dearly missed swing-dancing in the city!! (Not to mention the awesome ethnic foods galore... YUM.)

...I love how in NY, ya never know whom you're going to bump into, where, and how. When Geoff and I got out of Customs at JFK Airport a couple of nights ago -- at 2:30am -- I saw my old student Pamela V. waiting to pick up her uncle at the airport. How funny!

And, randomly, here are a couple of hysterical photos of Geoff's one day of mustache following our trip to Peru. (He thought that since he already had a full face of beard from 10 days of not shaving, he might as well shave it into a mustache look for a day.) ...As you can tell, I am just thrilled by the idea of his upper-lip decor. (--I will, until the day I die, never figure out why men think mustaches are cool. That night he wore this look to dinner with our Escuela Americana amigos; the gals all winced, and the dudes were all ready to start a new mustache trend! Ri-donc!)

My Take on Using Puzzles to Teach Substitution

Note: This is a break from the triangle area stuff I was talking about yesterday. I have full intention of returning to that sometime very soon.

I was reading through some math stuff yesterday and I came across a discussion of a grade-four question in Singapore math that requires a solution of only pseudo-algebra, thereby promoting an intuitive understanding of algebraic substitution.

It reminded me of one of my own favorite activities when I used to teach middle-schoolers, in introducing simultaneous equations and the idea of substitution. To preface this, I should state that my proudest accomplishment in creating these materials is that basically 80% or 90% of my (regular Bronx) 8th-graders would, within a few days, begin to realize that they can partially solve a simple system like this in their heads:

3x + 7y = 65
3x + 5y = -15

Essentially, they got so good with visualizing linear systems using shapes (triangles, circles, stars) that they can literally "see" that because the first equation has an overall right-hand-side value that is 80 greater than the second equation, and the only contributing factor is that the first equation has 2 extra y's, then that must mean that each y is "responsible" for 40, or y = 40. --To you guys, this may seem trivial, but it is not at all trivial to kids who, in many cases, have a natural fear of symbolic representation!

I presented this topic using a series of increasingly-difficult visual puzzles that you can find on the internet, and I let the kids work in groups of 3 or 4. I didn't introduce the idea of substitution, linear combination, or simultaneous systems, but I let them discover it for themselves through some scaffolded questions. Once they were able to solve the puzzles (WITHOUT guessing/checking), I then gave them algebra sheets and asked them to individually draw / "solve" the equations using shapes only. We didn't even begin to use algebra symbols until a couple of days down the road, at which point the transition from shapes to algebra was seamless for most kids. (I did this for two years in a row, and with other teachers at my school. The results are very duplicate-able.)

Anyway, here are the worksheets I have used in the past (that worked well in these first couple of days of group/individual exploration). I also modified and used them with my 10th-grade Algebra 2 Honors class during our intro to systems this year, and even those kids loved it!


Let me know what you think. I think it's similar to the Singapore method, but presented in a way that might be a bit more intuitive and generalized, supporting the learning of algebra. --And yes, sometimes I do miss teaching middle school (ie. having more time to focus on building introductory concepts), but whenever I feel nostalgic, I also hope that understanding middle-school approaches/foundation would help me be a better high-school teacher, in the long run. :)

Triangle Area - Simple

Tech Tangent: I fiddled a bit with iFrames this morning, in trying to embed a java applet directly into my blog. It worked (functionally), except it (aesthetically) messed up all the other elements that are part of the default blog layout. So, forget it for now; you're stuck with the regular web links until further notice.

Here is the GeoGebra visual of the day, accompanied by a math question. You'll notice that it says "Simple" next to the description/header, because I am ultimately interested in a more complex (but definitely related) problem. Eventually, I am interested in exploring how an area of a triangle changes as more than one of its vertices rotate around a circle.

A snapshot (but you should click on the link above to see the animation):


...By the way, in solving this problem, I had to try to figure out how to model a piecewise function in a piece of Geometry software. It looks like GeoGeobra does not have a way of doing this (at least when I searched in the Help index), and Geometer's Sketchpad makes it a helluva pain to get piecewise functions to work. Boooo. In the end, I still couldn't figure out how to get the individual pieces to merge into a single graph in GSP. This is the best that I could do, for now, in GSP. So, if you're a GSP expert and you happen to come across this post, please help!

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PS. I did a lot of math on the flight over to Peru. Can you tell? :) Geoff and I also proved the differential formulas for d(x^n)/dx, like the couple of nerds that we are. It's amazing how much math he remembers, even though it has been years since he has seen any of this stuff! (He is 29 this year, so that is at LEAST 7 years since college, and probably more since he has seen any form of Calculus! He uses a bit of math from time to time, both on his job -- graphics-related programming stuff for his Google-Maps type of mapping interfaces a while back -- and for understanding mortgage, interest rates, etc. in his investment-related leisure reading. But, still! I was surely very impressed.)

Our Peruvian Adventure

I am back home!! But, not for long. We just returned from our Peruvian trip yesterday (after some logistical snafoos), and we will be leaving again tomorrow. The day before we were leaving for Peru, Geoff found out to his HUGE disappointment that the house he had been looking to purchase in Jersey had fallen through. So, while we were in Peru, we each bought a round-trip ticket to Jersey. We leave tomorrow evening; I will be in the States for two weeks, visiting friends(!) and swing-dancing(!!), while Geoff looks around at other properties in NJ. The plan is that he will be in Jersey for four weeks total, but this way at least we are not apart for the entire time. (I will have to get back to San Salvador within two weeks, because that is when all returning teachers will start to prepare "officially" for the year. In truth, I have been on-and-off doing all sorts of prep this summer, but it will be nice to finally get a class schedule, finalized room assignment, meet the new hires, etc.)

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Peru took my breath away, both literally and figuratively. On our 5 days / 4 nights trek, the highest point we reached was about 4600 meters, or over 15000 feet above sea level. That is an astonishing feat! The temperature dropped below freezing on the first night of the trip -- in the 20s Fahrenheit range. What had happened to Geoff and me was that we had only arrived in Cuzco the day before our trek was due to start, so our bodies didn't have enough time to adjust to the altitude before beginning the climb into even higher altitudes. I threw up twice on the first day while ascending the mountain, and my chest hurt the entire time as I was walking / breathing, because my lungs were constricted. We passed by a couple of tombstones along the way, and the guide explained that one of them belonged to a gal whose asthma had flared up in the high altitudes, but who pushed on obstinately until things went from bad to worse. (I'm highly asthmatic, so that story got Geoff and me both a bit worried.) Finally, after about 9 or 10 hours of hiking, I gave up and hopped onto an emergency horse for the last hour of hike of the day.

On day 2 of the hike, the guides put me on a horse on the way up to the highest point, since the air was so thin. Geoff, too, was having a lot of trouble breathing -- and he runs marathons!! He was very pale by the time he finally got to the top of the mountain. I hopped off the horse and took a few steps on completely flat ground, and I already could not catch my breath. I had thought that descending would be easier for me to walk (even though we were still at an altitude > 4500 meters), but I clearly was wrong, because after walking for about 15 minutes, I threw up again. The guide insisted on me getting back onto the horse, and I threw up once more that day. Not my strongest showing, and I was super disappointed in myself that day. :(

That night, we got back to a lower altitude (in the 2000s meter-wise). By the time I woke up the next morning, the tightness in my chest had subsided. I was able to walk the rest of the trip, and even raced my way to the entrance of Machu Picchu to be one of the 400 people each day to receive a ticket to hike Wainapicchu (a nearby super-steep but stunningly beautiful mountain; you can see it in the picture above)! I was so proud of myself, because we had started hiking at 3:30am, in pitch darkness, and we were ascending these steep stairways that seemed to never end. Geoff was eager to be one of the 400 people, so he ran up the entire stairway and left me to climbing by myself. It was a tough mental game, but I was able to outrace a lot of people in similar (or better) physical condition than me, because I really wanted to get one of those tickets to Wainapicchu!

I can't describe how amazing Wainapicchu was, and our pictures do not do it any justice. It was like climbing up to a city in the clouds -- you have to use your hands and feet, rock wall-climbing style, in order to get up the narrow and VERY steep stairs. (I was really freaked out, naturally, because I'm terrified of heights. But, I tried not to focus on the fact that if you missed a step, you might very well tumble down the mountain and die. On the way down was much scarier, because you couldn't avoid looking at how high up you are.) When you look down from the top of Wainapicchu, even the immensity that is Machu Picchu is entirely dwarfed at the base of this mountain. It was definitely the highlight of our entire trip!

In Lima, Geoff and I also spent some time going to discotecas and bars. We also checked out a peña, which is essentially a local Peruvian cabaret, where traditional dancers come out in fancy outfits and dance during your dinner. Geoff and I found a place called "La Brisa de Titicaca" (the Breeze of Titicaca), which was cheap and you can get up in between the dance numbers to dance to traditional music! It was super fun. And, while walking around Lima during the day, we decided on a whim to go paragliding, since we had never done it before! It was awesome to fly over the cliffs of the Peruvian coastline. And, of course we also checked out some delicious Peruvian cuisine -- including cuy, or guinea pig!!

My only (HUGE) annoyance from the entire trip was that when we arrived at the Lima Airport yesterday, 3 hours before our scheduled flight back to El Salvador, the Copa Airline guy told us that the Salvadorean rule is for you to be deported from El Salvador (back to Peru), unless you can show proof that you have been vaccinated against the Yellow Fever more than 10 days BEFORE your scheduled flight from a country with active cases of the disease (ie. Peru)! In fact, the only reason that Copa eventually allowed us to check in to our flight was because the doctor on-site at the airport did Geoff and me a huge favor and wrote "Revacunado" ("Re-vaccinated") on our immunization record cards, after administering the vaccine to us! Otherwise, we would have been literally stuck in Peru for another 10 days!! Ridiculous!!

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Anyway, that's it for now. Ciao! I've got some errands to run (as is the nature of things when you keep leaving the country), but I'll leave you with this cute picture of us from our Peruvian trek. There was purple chalk on our faces and confetti in our hair, because it happened to be our hike-mate, Kate's, birthday, and that's how the Peruvians helped her celebrate!


How can you not love this country??