Bet you already know the problem (all of us, more or less, suffered it sooner or later).
"Two trains start from Milan and Rome" (feel free to change the place names) "and move in opposite directions. Train A, from Milan, travels at 100 km/h. Train B, from Rome, travels 150 km/h. Knowing the distance is 600 km, in how much time the trains will crash?"
On the first glance, the problem looks really horrible (especially during childhood, say at 11, when some doubt on the mental sanity of some problem authors has not yet emerged). At best, your attention is not immediately routed to the first degree equation underlying the problem, but to the two crews and the passengers. And when you find or are told the "solution", a knot will invariably block your stomach for some time.
At 13, presented with the same problem, you maybe will count and imagine the passengers one by one. Meanwhile, you will find some very creative solutions, like:
- They will never crash because the railway management will cut power at least an hour before
- The two pilots will see each other trains, and will brake in time
- They could not crash even if they wanted, as the Milan-Rome line is two-track and trains always take the left
- The two trains actually crashed, but without crews and passengers (correct, but not so plausible as all trains have a device named "dead man detector", a button the pilot must push from time to time to prove the train s/he is still in good health; if s/he forgets doing, the train automatically stops).
These, and more of the same type, are "correct" (and highly relieving) solutions. I'd personally praise them a straight A for "superior problem understanding", but guess some teachers would not be impressed the same.
But a moment. Just a moment.
The original problem was intended to make students to identify and solve a first degree equation!
No less, no more!
It is the really horrible packaging to make it a bit crazy - and shed on physicists and mathematicians a shadow of total abstractness and distance from normal people. I wonder why "this" way of presenting the problem is so diffused.
But: adjust the packaging a bit, and you may engage your pupils in a discovery journey.
Of the many possible ideas, this is my one:
"Two trains start the same time from Rome and Milan, moving in opposite directions at respectively 150 and 100 km/h. When they will meet, they will be quite close, something like 1 m window-to-window as they move along their opposite tracks.
And their combined speed will be 250 km/h.
Because of this, an overpressure will develop as the two front waves will interfere. This pressure will act on the wagon structures (which are steel, almost invulnerable) and windows: these latter may be a weak point.
The effect will be higher, if the encounter will occur inside a gallery (on the Milan-Rome line there are many).
The windows are calculated to resist overpressure in both cases. But as they sustain it, they will inflect elastically transmitting part of the overpressure inside, resulting in a decreased comfort to passengers. The effect is negligible outside a gallery, and quite severe inside.
The remedy identified by the train designer is reduce the combined speed within a gallery to below 150 km/h, which on the other side demands an acceleration to regain the lost speed past the gallery - traducing in higher cost for the rail company, and increased power consumption. Outside the gallery, speed needs not being decreased.
Assuming average distance between Rome and Milan is 600 km, and galleries are from km ... to ... and ... to ..., will the trains meet inside or outside a gallery (that is, will they have to brake, or not)?
If they meet in a gallery, may you find initial speeds within 100 and 150 for the two trains to meet outside?
And: are you sure sure of your answer? Why?"
This way the problem looks more "real-world", isn't it?
That of making a travel at 150 km/h (or more than 300 km/h on high speed Freccia Rossa line) comfortable to crew and passengers is a real engineering challenge, and problems like these are a real concern to train designers. Passenger comfort in the end decides how many people will travel by train, and have deep economic implications (along with perceived safety, and more). This may even touch personally some of the students, if their parents work in the railway business.
As any real-world problem, this one is "open ended". Apparently there is a clear-cut solution (the same you would find for the crashing trains, plus the decision whether this point occurs inside a gallery or not).
Then, in addition, there is a chance to make the problem "parametric", by allowing students to change speeds at their will (of course, the "simple" solution will imply an "inside gallery 1"). As they trim speeds, the train meeting point also changes, something easy to do with just some trial-and-error. Or even, if you have access to computer facilities, with a tiny simulation program. Sooner or later the meetig point will occur outside the galleries.
Last, there is an invitation to challenge one own's conclusion. The key I've placed is in the term "average distance". The point here is that the Rome-to-Milan track is not exactly as long as the Milan-to-Rome: curves exist, and as any one of them is found, the track on the inside of curvature will be shorter. The combined effects of many turns will yield an error, to correct which one would have detailed information on all the curves of the track. Realizing that mathematical solutions always have an error built-in, which can (and in real world should) be estimated and controlled, is paramount in applications.
Formulated the new way, the problem is "long". It contains a lot of information, in part essential, in part not so, in part apparently not. It requires iterations (as any real world mathematical problem). Is not per-se horrible (although some horrible mathematical problems exist, as artillery table calculation to just give an example). More than a homework, I see it as a problem the teacher and the students may solve "together", the teacher acting as a guide.
Now my question is: how many classical exercises are formulated in abstract / unrealistic / emotionally unacceptable ways, which with little effort can be trimmed into knowledge journeys? Many, I imagine. I'll add more in future, and maybe collect them in the sister site.
But I also would like to hear from you! That's most important.
I feel that if we want to make mathematics more attracting, we have to put it into its natural context.
The challenge is open...
(No contest, no score: let's just amuse, and imagine)