A Train Leaves Chicago Traveling At Who Cares?

Here is a basic assumption of my recent work, put as concisely as I can:

Even an application problem as stereotypically and quintessentially lame as the one above can become a powerful testimonial to math’s presence in the world if the problem begins and ends with an accurate representation of itself. Which is to say, in this case, if it starts with a side-by-side video of the two trains leaving their stations that continues unbroken until they meet each other.

If anyone’s been looking for an invitation to take a whack at that assumption, consider yourself invited.

See also: [WCYDWT] Bean Counting

Aside: The problem with paper is that it accurately represents a lot of things, but the world isn’t one of them. Not at the kind of fidelity your students need, anyway.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I’m going to re-make this problem with toy cars for my next semester class. Maybe have videos of the toy cars, and then actually watch crash them in class.

    A major problem with the train context is that so few of our students in non-metro areas (where I am) care to believe that trains travel in straight lines at constant speeds. I’ve never been on a train other than the herky-jerk El in Chicago myself.

  2. The interesting this about this problem is that it actually does get solved, albeit by a small number of people :) I don’t know that “who cares” is the right response but instead creating the context around when it DOES get used (ensuring that on single train lines we don’t have collisions and trains are sidelined at the most efficient time).

    Therefore, modeling it may not need to be done with real trains – I know I’ve been looking for an excuse to use Railroad Tycoon in the classroom :)

    The better problem of course is more realistic where you worry about corners, and hills and the like, but preparing students for the basic math using a simplified problem isn’t really a bad thing.

  3. I was thinking about the question: when in real life do people or vehicles approach each other at different speeds with the purpose of meeting?

    Idea 1. Long-separated lovers see each other across the airport terminal and break into a sprint to meet each other. One is faster than the other. Where do they meet? The right video might set that up, but it feels a little forced. Who cares that one is running faster than the other?

    Idea 2. NFL wide receiver and reality TV star Chad Ochocinco did a publicity stunt / charity fundraiser in which he races against a horse ( http://www.youtube.com/watch?v=VWGoSc28B1M&feature=player_embedded#at=36 ). He’s obviously not as fast as a horse so they had different distances to run. The geometry is slightly different but there’s some similar math involved. I think a race of some kind is a bit more compelling than the question of where two particular trains pass each other.

  4. Ok, I just thought of another idea for making trains compelling, without resorting to reality TV characters.

    Say the trains are approaching each other and they’re accidentally on the same track. At some point there is a switchyard where one of the trains could be switched off of that track to another track. So if Train A gets there first then they avoid the collision, but if Train B gets there first they crash.

    In real life they would probably hit the brakes or something. Also, it’s kind of a complicated premise. I definitely couldn’t communicate it in a single image.

  5. We did a version of this problem using programmable cars using Vex Robotics. We set the cars across the room and had one with a slightly higher acceleration. Students were to figure out where and when they would intersect.

    Haha that problem went from being a sterotype of pseudomath to an engaging experiment for a couple of days.

    Keep up the good work, its not just about the presentation, but the spirit of exploration.

    Before all of this talk about math reform I used to call meaningless problems in math…puzzles. I hope we can kindle that in others.

  6. Interesting conjecture. Since this problem type has been used continuously for let’s say, for at least a century, can we really say that a) it has no value because it’s really lame, and b) for modern kids it needs a high-fidelity representation before they will engage with it?

    I agree that the video could make it much more interesting, engaging and fun. But has the basic mathematical thinking that the original problem requires become redundant or too hard to attain in the modern era?

    I’m with Phil: keep up the conversation and the exploration, it’s really helping me to keep my brain active about what math we ask kids to do, and why.

  7. Jan van Hulzen

    July 12, 2011 - 5:46 am -

    A nice variation could be to show an action movie sequence and ask how fast the approaching train is driving given that the hero is stuck on the overpass for at least a minute and glances sideways at the train twice

  8. I’ve posted once before about the mathematical learning goal for which these problems fit (that time I asked about a learning goal for the lesson in which a problem might appear). I’m still curious about posters’ and Dan’s response to that, and I understand that it’s not the purpose of the blog :)

    A point about these problems: When I taught Algebra 1 several years ago with a certain, prolific red Algebra 1 book with many, many practice exercises for students I tried to have students draw a picture or act out a scenario described in a word problem. In my mind, if they couldn’t describe the situation in their own words or with their own picture or through their actions, then what good would it do to write and solve an equation to get an answer they couldn’t interpret? What I think these videos have the potential to do is to give students an entry point into problems by showing a real situation. The problem can be written and revised by students based on the video, and there’s less work for the student to guess which real-life constraints are assumed and which are ignored.

    Another point using the previous bean-counting and house painting examples: Inverse relationships are tough, even back to the fraction idea of bigger denominator means smaller pieces. It’s surprisingly difficult to convince some kids that the work will get done faster when people work together. Seeing the work actually get done faster might flip a switch that we’re not going for an average. For the trains, I would be ecstatic if my students could simply state: “Going faster gets you there quicker” or “Going faster means you can cover more distance in the same amount of time than someone going slower” (Notice that “same amount of time” gives us a useful relationship!). These are basic but important ideas that I think need to be articulated by students and the video can be played repeatedly to test out statements such as these.

    Thanks for your thought-provoking work, Dan!

  9. Gretchen Eastman

    July 12, 2011 - 7:56 am -

    One could possibly use clips from the movie “Unstoppable.” It has a lot of good what-if’s related to trains either hitting or just missing each other. It’s “Speed” but on train tracks.

  10. Paul,

    The following is somewhat related to your plan:

    It is in a bit of a draft stage, but essentially is two different size tires racing. One has a head start and one is travelling faster. The twist is that you can only directly observe rotational speeds of the wheels. It is not so easy to get the speeds, camera angles, and so forth right.

  11. Here is an equivalent problem that has a real aspect, since I have actually had to do this many times and, being a math geek even then, have tried to solve the problem while executing it.
    A column of soldiers 100 yards long is marching along the road at 5 mph. The 1st Sergeant needs to get from the tail of the column to the head of the column. If he walks at a rate of 7 mph how long will it take him to reach the head of the column? The history of this problem seems to be rather long. I have seen the same basic problem involving a column of Napoleon’s soldiers and a horseman riding around the column. This could also be the “how long does it take car A to catch car B” type of problem. The car A catching car B type problem is regularly applied in racing situations. Right now the Tour de France managers are constantly calculating this type of problem during the race to predict the time required to capture a break-away. The same problem is computed by race directors at car and motorcycle races. These situations may not be as exciting as two trains colliding but they are real world and are regularly being applied and solved. There are a lot of video clips of the Tour commentators discussing the catch-up time.

  12. @Belinda, I suppose I’d put this problem somewhere in the neighborhood of d = r * t. Its appeal (to me) is as an example of combined rates. Also, as is the case with all application problems, we’re implicitly trying to sell students on the power of math to explain the world they live in.

    Many other commenters have mistaken my efforts here at increasing engagement. My point here, instead, is to improve the testimonial, to craft a better sales pitch for the tight link between math and the world.

    I’m worried, all of the sudden, that readers of this blog are too convinced of that link already to empathize with those who are not. (ie. students who dislike math and find it pointless.) What efforts do you take to put yourself into your students’ heads? When you get in there, do you find yourself impressed by your own efforts?

    I follow up here.

  13. A few years ago there was a TV series called The Eleventh Hour on CBS. In this episode titled Subway, the main characters are trying to find out where and when 5 people may have intersected in the busy subway system. When the episode orginally aired, my pre-calculus kids had just finished a unit on parametric equations. I couldn’t hit record fast enough!! It was an excellent use of transforming a mundane topic into the “testimonial” of math interlinking with the world, albeit Hollywood style. Now, I use this activity to facilitate a great discussion on the when and the where and why it might be important. The students usually are jumping at the chance to now use parametric equations to solve the “Train A leaves…” problem.

  14. Another “real-life” scenario that mirrors the train problem involves battles to the death. You encounter these most often in cheesy action flicks, but I’m sure in-the-moment calculation skills can come in handy for the everyday citizen.

    Imagine that two people are in a warehouse gunfight, trying to kill each other from opposite sides. The camera cuts back and forth, as bullets whizz by and create sparks from the ricocheting. Soon, both warriors are out of ammo.

    But the battle isn’t over. The camera zooms in on an object that happens to lie on the warehouse floor. It is a machete. Both fighters see it at the same time. They make eye contact, each acknowledging what is about to happen. Simultaneously, they begin a mad dash to the machete.

    The catch, of course, is that the machete is not directly in between them. Maybe it lies 10 feet closer to person A. But person A can only sprint at 8mph, while person B can hit 12mph (I just pulled these speeds out of a hat).

    Who’s going to die?

    If a person can, at this very moment, perform this mental calculation, perhaps s/he an decide whether it makes sense to run after the machete or simply flee.

  15. Very good question! Clearly I did not think this all the way through. But I just have this classic image of the two people fighting over the ultimate weapon from all the action flicks I’ve watched. Perhaps you can judge a person’s speed by their appearance. How fit do they look?