Saturday, July 31, 2010

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. :)


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.


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.)

Friday, July 30, 2010

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.


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!

Thursday, July 29, 2010

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...


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.)


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!)

Wednesday, July 28, 2010

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. :)

Tuesday, July 27, 2010

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!


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.)

Monday, July 26, 2010

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.)


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!!


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??

Friday, July 16, 2010

Student-Art Combinatorics

Pretty momentous: Yesterday, after Geoff's morning run, he came back in and said nonchalantly while laying out his towel for doing situps, "Happy 1 year in El Salvador!" --Wow! He's totally right!

On a funny expat note, just the other night when I was making tea for the both of us, I had asked Geoff what type of tea he would like. For a brief moment, his face went blank and he mumbled after a bit of hesitation, "Manzanilla." I started to giggle because I realized then that he had remembered the Spanish word but not the English word for chamomile!


I don't teach combinatorics, but a fun lesson idea occurred to me while I was skimming through a discussion online, and I thought I'd sketch it up for the possible benefit of others.

The idea involves setting aside a day when the kids would each create/bring in a full-body, colored cartoon drawing of a person, and we would cut them into three parts (head, torso + arms, legs), and make three piles in the front of class, one pile for each "body part".

The kids would then each randomly pick a head, a torso, and a pair of legs out of the respective piles, and paste them together on a page, no matter how funny it looks, and write about the combinations / probabilities involved in this problem:

    Assuming that no student has yet gone up to pick from the piles...

  • How many combinations of body parts are possible?

  • What are the chances that you would pick out an "entire" (contiguous) paper person, with all three parts that belong together?

  • What are the chances that your paper person would have at least one mismatching body part?

  • What are the chances that your paper person would have exactly one mismatching body part?

  • What are the chances that you would get all three parts of your own original paper person back?

  • Let's say that Jenny is the first student to go up to the front of class to pick out three parts at random, and James is the second. Explain why and how the results of Jenny's paper person might impact the probability of James to then pick three parts that belong together.

  • Compare the chances of piecing together an "entire" (contiguous) paper person for the first student of the class (Jenny) vs. the last student of the class (you). If you go last, will it be "unfair"?*

*This question is my favorite, because it sounds deceptively simple but it begs an intuitive wrestling between what is commonly perceived as unfairness ("I am the last person to go up; by the time it gets to me, I've got no influence over whether I can piece together a contiguous person!") and a different story told using numbers (All participants, regardless of eventual order, had an equal chance of winning prior to the picking).

It could be a fun and light activity, while still keeping the rigors of a combinatoric lesson -- and it may also contribute towards decorating your classroom! ;) Obviously, the kids would write an explanation for each question in addition to showing the math work...

--But, sadly, I don't currently teach combinatorics, so if you have stumbled upon this entry and happen to enjoy this idea and decide to implement it in your (future?) classroom, I would love to hear how it goes! Let me live vicariously through you.


Off to the highlands of Peru! WOOHOO. :) See you back here in 10 days or so.

Thursday, July 15, 2010

New Toy!

I got excited(!) late last night sketching up a trig problem that deals with rotating circles. I thought that I could maybe whip up a demo in GeoGebra today, to help me learn the geometry software better. --So, tada! :) What are the maximum / minimum lengths of the red spring as the circles rotate at the given rates?
(Or, for those of you too lazy to load the page, here's the pertinent info about this problem.)

Circle A has a radius of 8, and circle C has a radius of 7. The two circles are lined up vertically, and the shortest distance between them, EF, is 5 units. At time t = 0, the points B and D are placed maximally far apart on the circles. As time progresses, Point B rotates about A counterclockwise with the speed of 5 seconds per revolution, and Point D rotates about C with a speed of 6 seconds per revolution. As you can see, a red spring is tied between B and D. Sometimes it stretches, and other times it compresses...

Here is a screenshot (but the visualization demo I made is much more interesting, because it actually rotates...). To create the java applet (or a similar one) yourself in GeoGebra, you would have to do some simple trig involving sine, cosine, period, and scaling/translation, in order to specify the location of the points as a function of t. (Might not be a bad exercise for your kids to animate something like this themselves!) I originally created everything on a coordinate plane, and then in the end I hid all of the coordinate info, so that kids working on this problem would have to do a little more work piecing it together themselves.

Anyway, I started to brainstorm what questions I could ask, and in the end, I was pretty happy with Question C (see the bottom of my linked-to HTML page above), because it can be answered so many ways, as simply as thinking about the fact that one circle has a period of 5 seconds, and the other has a period of 6 seconds, so their LCM of 30 seconds must be the common period. You can also verify this visually by toggling the t value in the GeoGebra applet. :) And, graphically, it looks beautiful! (I had to try a few different number combos, I admit, before I settled upon these final radii and periods.)

This is what it looks like if you graph the length of the red spring over time. With a wee bit of graphing-calc proficiency, you can use a graph like this to verify the period = 30 secs and to find the local max and min, which represent the longest and shortest lengths of the spring, respectively. (Click to open enlarged graph in a different window.)

I think you can also lead a rich discussion about what's going on with the two circles using this graph. (What's happening with the circles at points A, B, and C? Why are there parts of the graph where the oscillations are sharp, versus other parts where the oscillations are comparatively minimal? WHY does the oscillation graph show symmetry at t = 15, t = 45, t = 75... even though the two circles are both rotating counterclockwise?) --The best part? The kids can then verify/refute their hypotheses visually by toggling on the t values and comparing what they see in the visualization applet with the data from the graph. Try it! You may be surprised by what you see happening with the circles at t = 15, t = 45, t = 75... (I know I sure was!) :)

...I heart math!

Some variations of this problem I may try to cook up later-later (hopefully in GSP next time): What happens if the circles overlap? What if you have three circles, each controlling one vertex of a triangle? (Or as a computer activity for students: How can you animate a circle that has a specified location, a given radius, and a specific rate of motion?)

Wednesday, July 14, 2010

Fun with Circles

I was doing my usual blog-skimming this morning when I became very intrigued by this problem. Now, my calculus is admittedly rusty, but I think I've come up with a pretty simple geometric approach to the problem! I'm illustrating it here for the particular cases of n = 9 and n = 10. (Again, if you click on the pictures below, you can see enlarged versions in a separate window.)

Of course, I leave it to you to find out what the products would be. (Not difficult, obviously, considering all the lengths are already labeled for you.) What's getting me stuck is how to simplify this process into one neat formula, given any n, in order to prove the conjecture. (If you click through Sam Shah's post, you'll see at the bottom that I tried to formulate it at least in words, but my calculus fails me and I'm not sure how to write down one elegant formula for the whole thing!)

Anyway. I thought I'd share it with you, especially because GeoGebra helped me make such nice unit-circle diagrams, complete with color-coding! :)


Incidentally, I've been (true to my word) researching Geometry stuff for next year. In particular, Nancy Powell has some really lovely Geometry projects, that seem extremely rigorous and FUN! This was the first link that I came across when googling Geometry projects, so it's probably old news... but in case you haven't seen it, you MUST check it out!!

And, here is a cute SAT problem from David Marain -- this one is good for your middle-schoolers (and/or good for discussions about prime factorization)! :)

Tuesday, July 13, 2010

Roundabout Math, Day 2

Note: For organization's sake, I decided that I would go with the original plan of finishing up this post today, even though I am deeply boggled by something else altogether. (You can skip to the end if you don't care for the math talk.)


If you're just tuning in, perhaps you would want to start with yesterday's entry, but basically we are trying to break down the math about roundabouts ("traffic circles"). We left off after having intuited that the maximum velocity of a car should depend on the curvature of the road (ie. radius of the roundabout), and I brought in an external piece of information for calculating that max velocity. Now, onwards!

7. Besides the curvature of the road, what else limits the speed of your car as it actually drives through a roundabout? (Crowdedness, obviously.)

At this point, you could start a guided discussion with the kids about how we might mathematically represent the "crowdedness" of an intersection. My personal feeling about this is that, for high-schoolers, we should use a simple metric, such as what percent of the road is covered with cars? The kids can then go out to the parking lot and measure an average car's length and width, and use it to estimate how many cars can maximally fit into a given roundabout. If there are roundabouts in your neighborhood, now would be a good time to introduce the artifact (aerial photo?) and its dimensions.

8. So, the kids can create a graph that looks like one (or both) of the following, depending on what metric they use for modeling the crowdedness. (If you click on the graphs, they'll open up in another window and make it easier to read.)

In either case, the kids will have to do some geometric calculation in order to figure out the translation between the number of cars and the percent of road crowdedness. I am thinking of something in the neighborhood of:

(Notice that the "drivable area of a roundabout" is no more than finding the area of the ring, with radii r1 and r2 as given in the previous photo.)

9. If we assume -- kind of a big assumption here, but, what the hay -- that the crowdedness linearly impacts the actual speed at which a car can travel, then in theory, by knowing how many cars are in the intersection at any given time, we can construct the average speed of a car and calculate how long it would take for a car to get through the intersection. So, at this point, you can introduce some data that looks like this:

Note that this is a rough sketch of the type of data you would want to provide. You probably want to modify the table to show that at some point during the day, the traffic circle reaches its max capacity.

10. And then, you should be able to ask your kids some relevant questions!

  • Assuming that you come to this roundabout three times a day: at 7am, 12pm, and again at 6pm. There is always a 3-second delay as you approach the roundabout and come to a stop, hoping to merge into the traffic. Assuming that you're extremely lucky and are the first person in line waiting at the intersection to enter the roundabout, how long would it be until you go through the roundabout to the other side?

  • Assuming now that you come to this same roundabout at 5pm the next day, but find yourself to be the 10th person in line waiting to enter the roundabout. How long would it be until you go through to the other side?

  • Assuming that you are considering building a "regular" intersection that is as big as your roundabout, and each full traffic light rotation is 2 minutes. The posted speed limit is 50 kilometers/hour. Estimate the max, average, and least amount of time it would take for your car to go through the intersection, if you are the first in line waiting at the light.

  • Explain whether this wait time changes dramatically if you are the 10th car in line at the light.

  • Why do people sometimes build roundabouts that also USE traffic lights? What advantage(s) might there be to this hybrid approach?

...In the end, I am still not at all happy with my organization of this as a potential topic for teaching. Even though I think the problem is inherently interesting, there are way too many variables even when it is broken down like this. So, I am going to leave it at that. If you can think of a way to pare down some of the variables further while still keeping the juicier, intuitive parts intact, please - feel free to give me your thoughts! Otherwise, I hope this had been a worthwhile read for you, to follow my venture to nowhere.


In the mean time, I will go back to grieving/seething over this bit of local news. I have read parts of the original Spanish article, and it is even more heart-breaking than what Tim has already summarized. The family has 7 children, and because the dad only earns $4 a week driving sand trucks, he cannot afford to feed all 7 of the kids. He actually insisted on making tortillas out of the seed corn, even though his wife had reminded him that there was poison (insecticide) on them. He had told her to just wash the corn well, to get rid of the insecticide. In the end, only two of his seven children were not intoxicated, possibly because even with the tortillas made from the seed corn, there was still not enough food to go around at home. The Spanish article also has a picture of his two kids that died, a 10-year-old and a 12-year-old. Truly, truly devastating.

Needless to say, it horrifies me to read about this. The Salvadorean government does little to help the extreme poverty that they know to exist right on their doorsteps. ugh.

Monday, July 12, 2010

Roundabout Math, Day 1

I had an idea for a possibly interesting math-modeling situation, so I went ahead and googled it to see what there is already out there on this topic. The results were both encouraging and disappointing, because 1. it confirmed my impression that the material is inherently interesting, because there have been loads of math written about it, but 2. the math is perhaps way too difficult to be explored at a high-school level.

So, I have decided to spend a couple of entries this week just looking at the possibilities of presenting Roundabout Math to high-schoolers. It'll be done in installments, as I need some time to think this through. Feel free to jump in at any point on my hypothetical lesson.

Here is the premise: Roundabouts, or "traffic circles" as they are known in some circles, are extremely popular in El Salvador. They drive Geoff nuts, really, because they cause traffic congestion everyday right outside of our apartment. And, because the entire city utilizes roundabouts, it can sometimes take you 30 minutes to go a block or two, during PM traffic hours!

So, let's just take a look at some simplified setup for this problem:

1. Here is a standard roundabout. I'd say they average out to have about two lanes and about four streets coming in/out of them. Notice that they DO NOT USE TRAFFIC LIGHTS!

2. Here is what an average driver might want to do: go straight through the roundabout. You can only go in counter-clockwise direction around it, and you can only make right-hand turns to exit.

3. Of course, this gets more complicated when there is traffic already in the roundabout. The standard code of conduct is to wait until you think it is safe to enter the roundabout (ie. yielding to all oncoming traffic). Once you are in the roundabout, you have the right of way (at least in theory) over the new cars trying to enter from the side.

4. Let's take a detour and look at how curves affect the speed of a vehicle. Let's say that the picture below represents the top-down view of a NASCAR racetrack. Where would the driver need to slow down the most (and why)?

5. Now, let's assume that we super-impose circles on top of the race track. How does the radius of a circle (or arc) affect the speed of the vehicle as it travels through that section of the road? (Is it a positive or a negative correlation between the radius and the speed?)

6. Well, fortunately we have some information to work with. As it turns out, the maximum velocity (in m/s) that you can reasonably achieve when traveling through an empty roundabout depends on its radius (in meters) as such:

This regression equation is provided by a National Cooperative High Research Program Report, dated (unfortunately) about 30 years ago. It should match your intuition that the larger the radius is of a circle (or circular path), the higher your maximum speed will be when you travel around that circular path.

Stay tuned. Again, this is all uber-simplified stuff, because I am trying to get a math model going that would make intuitive sense to high-schoolers and get a reasonable analysis going. If you're hoping for a more detailed/accurate model, consider reading this instead. (It's a nicely written piece about the math of roundabouts, but it does involve some differential equations, which are eons ahead of what my kids can do.) I feel a little guilty doing all of this over-simplification, but considering that much of high-school mathematics is an over-simplification, I don't feel tooooo bad.

Mumble Jumble Peru Mumble Jumble

Geoff and I are counting down the days until we go to Peru! We are supposed to be hiking and camping for 5 days / 4 nights and ending up at Machu Picchu. I am excited!!! :) I am a little nervous about the whole altitude thing, since we won't have a lot of time to spend in Cuzco before embarking on the hike. (The timing just doesn't work out that well, even though we know that it is recommended to arrive early in Cuzco to let your body adjust to the altitude before climbing to higher grounds.) But, we'll see how it goes!

This past weekend, we went down to Costa del Sol, which is a ritzy beach area where the rich and the powerful Salvadoreans bring their families on the weekends (to stay at their luxurious villas). It was Geoff's and my first time down there, so we drove around for a while before we found Hotel Bahia del Sol, which we had read mixed reviews of. It turns out that they only offer all-inclusive deals ($69 per person per night, which is pretty pricey by Salvadorean standards), but since we had never done something like this before, Geoff and I decided to spring for it!

As you might imagine, we had a really lovely time. :) We actually didn't spend much time in the ocean this time, but we did lie under the coconut trees on the warm sand for a while. The rain managed to hold off for the most part -- it drizzled a good amount during the evening that we stayed there, but not enough to spoil our spirits. The rest of the time, we dipped in their lovely pool (it was shaped like a circular river, and even ran underneath a bridge at parts) and relaxed. Geoff also drank like a fish, because he felt the need to get both of our money's worth. :) So, it was amazing! Definitely a do-over at some point, maybe with more people next time.


After our trip to Peru, things will definitely pick up their pace! The new hires will arrive, and the year will start to glide into gear... If I can get myself motivated, I'll try to work my way through some Geometry researching / planning this week, while I still have time to myself.

Also, I am back on the yoga bandwagon.* My goal for this week is to be able to consistently hold my headstand for 1 minute, so that I can start working towards 2 minutes! :)

*My yoga teacher is officially on maternity leave! She hasn't actually delivered her baby yet, but she's putting off yoga instruction for now. I am sad and relieved at the same time. Up until a few weeks ago, she was still doing some demonstrations and spotting people on their scorpions, etc. Some of the girls in the class were more scared of accidentally kicking her than they were of falling!


I remembered this morning a story my mom had told me about how poor the people were in Taiwan when she was growing up. Only when they had guests over, would her mom send her down to the store to buy one egg. They would stir-fry the (one) egg for the guest, and all the kids would stand by enviously as they watched the guest eat. They also bought cooking oil by the scoop. It wasn't until she was much older -- maybe high school? -- that eggs were sold by the dozen, and oil by liter-containers.

I've been thinking about writing down some of these stories, because I am very forgetful and, at some point, they will be lost from our family. You might read about some of them here, as I feel randomly inspired to transcribe them.

Sunday, July 11, 2010

Reflections Drawn from SAT Problems

This is the process that I go through whenever I complete one of David Marain's SAT problems of the day:

1. I do the problem. In particular, let me use this one as an example, because it helps to illustrate my point:

2. I consider realistically how many of my students would be able to do this problem with reasonable ease/accuracy, and how many kids in general taking the SATs could do this problem. (My uncorroborated rough estimate for this problem is about 45% of all kids, maybe about 30% of my students, and maybe 25% of the kids in general at my school.)

3. I link the problem to a very simple concept that we all already teach in our math classroom -- a concept that is perhaps very pervasive in the high-school curricula. In this case, it is plugging in a solution (or part of a solution) into an equation, because we know that all solutions have to "fit" their equation.

4. I consider the conceptual scaffolding that it would take for the majority of our kids to "get" this problem.

  • Figuring out that solution points have to fit the equation.

  • Figuring out that you can find the missing coordinate, as long as you are given the equation and the other coordinate. ie. You can find the y-value of (1, y) in this problem by simply plugging in x = 1.

  • Having some really basic algebraic skills for simplifying.

  • Figuring out that most SAT equations are merely monstrous twists of simple ideas.

  • Review this every 6 or so months, for the same concept.

Sounds simple, no? Apparently, something along the line goes awry, because (I believe) more kids than not will likely get this problem wrong on the test. And, why is that the case?? The three first bullet points are basically given. We already do that inside our classroom, don't we?

...Obviously, I do have some less-than-amicable personal feelings about some of these SAT problems. I don't think the way that they are posed supports the juicier things we, as math educators, try to do inside the classroom: helping kids appreciate the meaning behind numbers, creating problems that are inherently interesting, building a connection between intuition and the numeric solution, etc. But, that does not change the fact that I, as a teacher, need to enable my kids to be able to deal with these abstract and intimidating forms in which a simple concept may appear. The SAT problems should not dominate our classroom, but we need to keep in mind that they are in store for our students.*

So, that is one of my goals for next year. I am going to work on my own exposure to SAT problems, and to increasingly sneak those problems in there once the kids have mastered their fear of the concept.

*And I say this for middle-school teachers as well as high-school teachers. I speak from my personal experience only (so maybe this does not reflect other teachers), but we are often so bogged down trying to shoot for fulfilling state standards and covering what seems to be an extensive curriculum, that we skip out on the extra little abstract analysis that could really help our kids down the road on the SATs (and maybe other things)! Is there any reason why this particular problem could not be done with middle-schoolers who have already seen quadratic basics?

Thursday, July 8, 2010

My Second-Favorite Decade

This one is a special shoutout to my awesome friend Amy, who is incredibly busy finishing up her electrical engineering doctorate from the prestigious CMU, :) but who enjoys some silly dance moves now and then:

--Do I hear a 90s dance break? :) :)

By the way, here is a neat list of all the math-related strips from xkcd. Regrettably, most of the others don't make you want to dance like you've got no shame.

...Sometimes I think we miss our childhoods because things back then actually were better. Lady Gaga's got nothing on these guys!

Wednesday, July 7, 2010

Who's the Sucker?

After looking at some of the standard trig problems from the Precalc textbook, I adopted one textbook problem about statues to seating in IMAX theaters. Here's the picture I pulled together in MSPaint, that would serve as motivation for the lesson:

ALL of our kids go to the movies regularly. The question I will ask when they look at this picture is, "Who's the sucker? Where (in this picture) would you NOT want to sit?"

Then, hopefully they'll figure out that the ideal seat(s) would maximize the viewing angle. So, that brings us to a standard maximization trig problem. An IMAX screen is 53 feet by 73 feet. We all know that we would want to sit in the middle of the row if we can, so that isn't so interesting. What is interesting is WHICH ROW IS THE BEST?? If we now assume that, when you sit, your eyes are about 4 feet away from the ground and that the bottom of the IMAX screen is about 7 feet off the ground, you can draw a side-view diagram to help you set up an equation, in order to maximize the viewing angle, V, by adjusting horizontal distance from the screen, d.

The MOST interesting part of this, for me, is that when you set up this hypothetical problem and you then graph V as a function of d, you get a graph that looks like this, which has a domain of d feet away from screen and a range, V, of viewing angle:

Still not interesting for ya? Look carefully at this graph. The graph assumes that the viewing angle decreases as you move further away from the screen -- true in old-school movie theaters, but why isn't it true in modern theaters? What do they do differently nowadays that fixes this problem?

(The difference is obvious if you've ever been to the Egyptian Theater in Seattle's Cap Hill, or any other old-school, ma-and-pop theater. They don't have a slanted floor!! HOW CAN THEY NOT HAVE A SLANTED FLOOR?!?!)

Anyway, that was a lot of rambling, but I was excited after I had worked through all the math to discover the connection between the graph and the real-world movie experience! :)

Cameras, Shutter Speeds, and Suspicious Half-Chickens

In the interest of taking motion-blurred pictures of rotating objects (a la this, this, this, this, or this), I did some mini-research about my digicam and about the iPhone. I'm not one to be particularly bogged down on electronics or to be nitpicky about what tools I own, etc. But, the results of my research have spurred me to want to get another camera (or one of the newer iPhones)!!

It turns out that the Canon Digital Elph -- which Geoff and I have and love -- only has two modes: one being instantaneous exposure (with flash) and the other being long exposure (1 second or longer). For the purposes of math-teaching, if I were to take motion-blurred pictures myself, the exposure should be somewhat variable in the 0 to 1 second range. And, more depressingly, it looks like because Geoff has an older iPhone (iPhone 3G, with OS version 3.0), its built-in camera does not have all the spiffy features that I would need, either. --Doh!! I'll have to keep looking around for another solution, I guess. In the meanwhile, the Flickr photos will have to do (provided that their owners are kind enough to share their shutter speeds with me).


By the way, it's pretty amazing that Geoff and I have not gotten really, really sick from food poisoning in our 12 months of living together. Case in point, two nights ago I decided to try cooking a new chicken dish. Now, I'd say that I'm a pretty decent Chinese-food chef, but whenever I try a new dish, it's still really nerve-wracking because my mom's recipes are hand-wavy at best. In this case, it was something like, "You boil half a chicken with some salt, and then when that's done, you scoop out the oil at the top of the broth, and you add the oil to some chopped ginger, scallions, and you add some salt to the dip. You take a big butcher knife to chop up the chicken into slices, and -- tada! You'll have what's called the Scallion-Oil Chicken." Sounds pretty easy, but it turned out that for some reason, part of the chicken we boiled was pink even after a while of cooking (and we were pretty sure, by the way, that this chicken was well-defrosted before cooking). We sliced it up pretty well and threw it back in the broth for some more time, and then took it out. After a couple of bites, Geoff and I decided to be safe and to microwave the chicken before eating.

I'll never know if that chicken was fully cooked or not. It tasted oddly tender and looked oddly pink even after the microwaving. But, I'll say that this is the first time I've topped off any cooking feat with microwaving! It's royally sad. And, amazingly, Geoff and I didn't get sick from that meal; we just might have stomachs of steel!

Tuesday, July 6, 2010

Stormtroopers Fun

Cute, and possibly in the "What Can You Do With This?" teaching department:

For those of you non-teachers, WCYDWT (What Can You Do With This?) is an innovative way of formulating lessons that I am going to try to incorporate into my classes this year. It involves taking a piece of interesting, rich media (ie. video or picture), and asking the kids a question that naturally arises as they observe this. Then, the kids brainstorm the pieces of data that they will need to collect before you reveal the available info to them piecewise. The calculation comes in the final part of the problem, not immediately, thereby maximizing the amount of meta-processing they need to do. You should anticipate me updating regularly throughout the year, to reflect my efforts in this area.

(Pictures taken from What Stormtroopers Do on Their Day Off.)


By the way, this is a cute application for the sum of a Geometric sequence, that I found over at the NCTM site.

Each spring, a trout pond is restocked with fish. That is, the population decreases each year due to natural causes, but at the end of each year, more fish are added. Here’s what you need to know.

  • There are currently 3000 trout in the pond.

  • Due to fishing, natural death, and other causes, the population decreases by 20% each year, regardless of restocking.

  • At the end of each year, 1000 trout are added to the pond.

Find a mathematical model for the trout population over time.

Monday, July 5, 2010

Lovely weekend

Geoff and I had a lovely July 4th weekend sans BBQ and fireworks. :)

Despite it being the rainy season, the weather turned out to be beautiful all of Friday, Saturday, and Sunday. On Friday night, we went out for a dinner and a movie, and watched Kick Ass, a cheesy but very entertaining film about wannabe superheroes. On Saturday, we checked out La Puerta del Diablo, or "Devil's Door", a famous site for executions during the Salvadorean Civil War. It was interesting -- we met a coconut vendor who speaks flawless English, and he initiated a very frank conversation with us about El Salvador's present and future. Much of what he said corroborates my existing belief that there is little hope for foreseeable change in this country. He confirmed for us that the police fears the maras and added that the government's lack of funding in education means that the poor is without hopes even in the long run. So many people sell things on the street, he said, because that's the only job they are capable of doing. It was a profoundly depressing conversation...

On Saturday night, after a failed attempt to locate a house party (the host had sent out very poor directions), Geoff and I went to Alambique for a drink. We ended up drinking a bottle of wine between the two of us, dancing a bit, and then deciding to come home. We hung out and talked in the hammocks, under the soft lighting of our patio, until it got to be really cold and really late -- about 4am! Finally, we went to sleep, and blissfully slept in on Sunday.

Sunday was a beautifully lazy day. We got up late. Geoff played his guitar while I hung out in the hammock next to him. Then, we took a walk to a nearby park, ate a delicious lunch at Kreef (--YESSS for Prosciutto and salami baguette sandwich!), and then took a long nap and watched some TV before going grocery-shopping. We dug up a recipe that we liked for a dish called the "Drunken Tuscan Pasta," and used a whole bottle of wine to cook our meal. --It was delicious! By the time we finally cleaned up after dinner and were ready for bed, somehow it was already 11pm.

I love aimless weekends spent with Geoff.

Friday, July 2, 2010

Sharing Teaching Materials Online

Since summer is slow, I have been putting myself to work looking around the web at other math teachers' blogs, to see if there are ideas that I can adopt for my own class. I've spent three full days doing this, and although I am nearing the point of temporary saturation, it has definitely helped to put a bunch of ideas in my head.

One thing I think I am going to try to do a better job with is in sharing my teaching material on this blog. Already, all of my lessons are digitized (you wouldn't expect anything less, would you?) except for the few extremely brilliant things I found in resource books, such as the 3-d spatial puzzles that I had spun into a multi-day computer project.

I have to be careful though, because ultimately this is not a teaching blog. I mention a good amount of teaching in here, because -- well, I am a teacher and I love my job. But for me, my experience as a teacher is intrinsically tied (for now, anyway) with my experience living abroad. So, for now, I have no plans of forking off a separate blog for just teaching materials. But, what it means is that, when I do discuss teaching, I'll try to be more thorough and to provide digital examples of my work. And I'll file all of those entries under the tag math stuff, for easier retrieval. (In fact, if you go back and look at the older entries, you will notice that I have already updated them to contain some screenshots of my older work! I am committed to this...) It is indeed possible that in the future, I would want to create a comprehensive digital portfolio of the blogs I read, the bookmarks I keep, the blog entries that I write, along with my favorite digitized lessons. But, I don't see that happening before we settle back in the States, because I'm already having a hard time keeping up with the maintenance of one blog, let alone two!

Anyway, a running list of ideas for teaching about circles (some of these might be too difficult for 9th-graders, but I thought I'd jot them down anyway):

  • Take a motion-blurred picture of a fan, and using the shutter speed (and the picture) to figure out how many times the fan blades revolve in a minute while on different settings. Inspired by Dan's post about tennis ball dropping freeze-frame lesson.

  • Discuss how we can measure the amount of material it takes to build a basketball and baseball, and using the discussion to highlight the difference between surface area (rubber of basketball) and volume (filling inside baseball) of a sphere. Courtesy of Ms. Cookie!

  • If you stand at different latitudes on the surface of the Earth, how fast are you traveling at each location, as the Earth rotates about itself? (Alternately, if the ozone layer hole stays relatively constant in its location relative to Australia, then how fast must the ozone layer be traveling near that part of the world?)

  • Taking a picture of a circular lawn and developing a lesson around how much fertilizer you would need, how long it takes for you to walk around the lawn, how long for you to mow the lawn, etc. Dan's post for how to do this correctly.

  • If you slide a beverage glass on the floor, it always comes back to where it started after making a circle. Why? Which glasses make a bigger circle (and how much bigger)? Another of Dan's popular posts.*

  • Circular probability - dartboard, Wheel of Fortune, etc. (This is very project-worthy.)

  • Wheels, distance traveled/pedaled, and gear ratio. How do bikes work?

  • Clock hands - how far (in degrees or distance) do they travel in a given amount of hours + minutes?

  • Intensity of radiation as a function of distance from source.

  • Circular mirrors (ie. Christmas ornaments) and their tangent lines. Where do you have to stand in order to see your friend reflected in the Christmas ornament?

Am I leaving out any good ones that are immediately obvious? The exciting thing is that I can teach basically everything related to circles using these examples. The obvious drawback (as always) is that it would take a long time to get through all of these activities, for just one unit! Sometimes I wish we had unlimited amounts of time.

*By the way, you might have noticed that I really like this guy Dan. He's awesome! He has re-inspired me to look at how I am teaching every topic more critically. Don't over-scaffold and think you're helping the kids; guide their thinking so that they can figure out what questions to ask and what type of data to collect! Love. It.