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.