## Wednesday, September 30, 2009

### Two Trains and a Tunnel! Is There Room For This In The Tunnel And In Your Curriculum?

At the same instant of time, trains A and B enter the opposite ends of a tunnel which is 1/5 mile long. Don't worry -- they are on parallel tracks and no collision occurs!

Train A is traveling at 75 mi/hr and is 1/3 mile long.
Train B is traveling at 100 mi/hr and is 1/4 mile long.

When the rear of train B just emerges from the tunnel, in exactly how many more seconds will it take the rear of train A to emerge?

Click on More to see answer (Feed subscribers should see answer immediately).

1. Appropriate for middle schoolers even before algebra? Exactly when are middle schoolers in your district introduced to the fundamental Rate_Time_Distance relationship?
2. What benefits do you think result from tackling this kind of exercise? If it's not going to be tested on your standardized tests, is it worth all the time and effort?
3. How much "trackwork" needs to be laid before students are ready for this level of problem-solving?
4. As an instructional strategy, would you have the problem acted out with models in the room or use actual students to represent the trains and the tunnel? OR just have them draw a diagram and go from there? Do a simulation on the TI-Inspire or TI-84 using graphics and parametric equations for the older students?
5. If you believe there is still a place for this type of problem-solving, should it be given only to the advanced classes and depicted as a math contest challenge?
6. I'm dating myself but I remember seeing problems like this in my old yellow Algebra 2 textbook? Uh, I believe this was B.C. -- before calculators! Can you imagine! Do you recall these kinds of problems? Do you recall the author or publisher?
7. Of course, the proverbial "two trains and tunnel" problems are frequently parodied and used as emblematic of the "old math"! They've been replaced by "real-world" applications. "Progress makes perfect!"

Answer: 9.4 seconds (challenge this if you think I erred!)

three of clubs said...

Am I wrong in calculating that train B has to travel 9/20 of a mile which it does in 9/2000 of an hour or 16.2 seconds?

Train A travels 8/15 of a mile in 19.2 seconds. So isn't the difference just 3 seconds?

Dave Marain said...

three...
I calculated A's time to be 25.6 seconds, not 19.2. Did you remember to divide A's distance by 75 mi/hr rather than 100 mi/hr?

I'm also interested in your thoughts about the appropriateness of this not very difficult problem for 6th or 7th graders(or 4th & 5th in Singapore!!)??

I tried to write an "RT = D" problem which did not require algebraic methods other than manipulating the basic formula to find time.

three of clubs said...

caught my mistake...thanks

I hadn't really focused on the question of appropriateness, being too caught up in trying to teach each student at his level and making sure there was something constructive for each member of the class to be doing.

It is, though, the most seminal question. You've certainly thought through more of the "should and ought" than I have. Some of your problem depends on formal teaching, but the part I find most interesting is informal understanding ... the stuff which we expect people to get, except, they don't.

At what age do we expect people to be able to know that you need to add the length of the tunnel to the length of the train? Or, better still, reverse their thinking, and have them going in the same direction?

When confronted with a class what should you expect and what should you assume? When I first started teaching high school math, I assumed my students knew the material from the previous term. That expectation was despite finding college juniors and seniors in advanced math classes still trying to turn (x+1)^2 into x^2+1. My expectations are running kind of low right now.

With regard to your problem, incidentally, I'm taking a year off from teaching, or I would field test it for you. My own guess though, fewer high school seniors could solve it than identify the Bill of Rights. So, appropriate? or, desirable?

mathmom said...

Is this a "contest problem"? No, IMO, this is a straightforward problem. I think our pre-algebra kids who have worked with ratios and can solve a simple equation with cross-multiplying could solve this problem, though I think the conversion from hours to seconds would mess up a lot of them. I would hope they would do something like:

100 miles          9/20 mile
---------      =      ----------
3600 seconds      x seconds

And solve for x. This doesn't require remembering that D=RT or any other version of the formula, and I always prefer methods that don't require kids to remember a formula

Dave Marain said...

Three of clubs--
Thank you for your insightful and supportive comments. Considering that I wrote this problem for average middle schoolers and that algebra and formulas are not required, it's sad to think that some high schoolers would give up on this.

Mathmom--
I absolutely agree that this this should not be considered a difficult problem, never mind a contest question! Your proportion model is exceptionally important for middle schoolers. Do you believe that most middle schoolers would be able to solve it this way or solve it at all?

mathmom said...

My older middle schoolers (~7th - 8th grade level, either in a second year of pre-algebra, or starting Algebra 1) are pretty good about recognizing ratio problems. They also know the D=RT relationship. But I'm guessing that there are enough "tricky bits" (figuring out how far each train has to travel, which is stated in an indirect way, translating between hours and seconds) that I'd only give them a 50% chance of solving this one correctly on the first try, without any hand-holding. Maybe I'll try it out with them sometime -- if I get to it, I'll let you know.

I wouldn't give this to my younger middle schoolers (5th-6th grade level, just starting pre-algebra) yet. Sometimes I do a lesson on distance/rate/time problems and if I do, I'll throw this one in to see what happens.

Dave Marain said...

Mathmom--
Let me know if you try it with your sons or other middle schoolers.

I believe that some high schoolers will struggle with the "tricky bits" or, as I would like to say, "the devil is in the details." There is a lot to process here. The student taking physics will definitely be more comfortable with the "rate" approach since they do many motion problems and the units issue should be trivial for them.

Going back to your proportion method, I believe that the units used, "miles over seconds" or "mi/sec" is a powerful way to introduce rate. I believe math teachers need to stress the importance of units as much as science teachers do. Formulas such as R = D/T emerge naturally from such discussions. Once the student buys into (mi/hr) x hr = mi, for example, formulas like D = RT express themselves!

The mathematical basis for dimensional analysis however is noot always fully explained in science classes. If RATE is constant, then the middle schooler needs to see that D = RT express the idea that Distance is directly proportional to Time while Rate is the constant of proportionality. The 1st year algebra student needs to make the connection that D = RT is just a special case of y = mx, a direct variation function!

Let me extend this to "varies directly as the square of". Science and math students later on will learn that y = ax^2 expresses this relationship and they will graph it and solve applied applied problems. However, how many students look at a formula like A = πr^2 and realize that this is a special case of direct square variation where π is the constant of proportionality! Sometimes we have to model "making connections"...

Anonymous said...

That's a fascinating problem, because I think the level of difficulty and appropriateness depends a lot on how it's presented and what tools students are prepared/encouraged to use. For myself, I'd be tempted to tackle this one using a space-time diagram, in which the tunnel would be a vertical stripe and the trains diagonal stripes. with different widths and slopes. I'd think that would be a good way to make clear visually what operations need to be done to find the answer.

Dave Marain said...

orawnzva,
Sorry it took so long to get back to you.

Thanks for the comment. I like your creative approach, although I need to fool around with it more to decide how it might play out in the classroom. Is this something you've used personally in a classroom or just in your own work?