The Dean's Column:

The Flight of the Bumblebee

By Dr. Aldred S. Posamentier

Problem solving is not only done to solve the problem at hand;
it is also provided to present various types of problems and, perhaps more
importantly, various procedures for solution. It is from the types of solutions
that students really learn problem solving, since one of the most useful
techniques in approaching a problem to be solved is to ask yourself: “Have I
ever encountered such a problem before?” With this in mind, a problem with a
very useful “lesson” is presented here. Do not let your students be deterred by
the relatively lengthy reading required to get through the problem. They will
be delighted with its unexpected simplicity of the solution.

Two trains, serving the Chicago to New York route, a distance of
800 miles, start towards each other, at the same time (along the same tracks).
One train is traveling uniformly at 60 miles per hour, and the other at 40
miles per hour. At the same time, a bumblebee begins to fly from the front of
one of the trains, at a speed of 80 miles per hour towards the oncoming train.
After touching the front of this second train, the bumblebee reverses direction
and flies towards the first train (still at the same speed of 80 miles per
hour). The bumblebee continues this back and forth flying until the two trains
collide, crushing the bumblebee. How many miles did the bumblebee fly before
its demise?

Students will be naturally drawn to find the individual
distances that the bumblebee traveled. An immediate reaction by many students
is to set up an equation based on the relationship: “rate times time equals
distance.” However, this back and forth path is rather difficult to determine,
requiring considerable calculation. Just the notion of having to do this will
cause frustration among the students. Do not allow this frustration to set in.
Even if they were able to determine each part of the bumblebee’s flight, it is
still very difficult to solve the problem in this way.

A much more elegant approach
would be to solve a simpler analogous problem (one might also say we are
looking at the problem from a different point of view). We seek to find the
distance the bumblebee traveled. If we knew the time the bumblebee traveled, we
could determine the bumblebee’s distance because we already know the speed of
the bumblebee. Again, have your students realize that having two parts of the
equation “ rate x time = distance ” will provide the third part. So having the
time and the speed will yield the distance traveled, albeit in various
directions.

The time the bumblebee traveled can be easily calculated, since
it traveled the entire time the two trains were traveling towards each other
(until they collided). To determine the time, *t*,
the trains traveled, we set up an equation as follows: The distance of the
first train is 60*t* and the distance of the second train is 40*t*.
The total distance the two trains traveled is 800 miles. Therefore, 60*t* + 40*t* = 800, so *t* = 8 hours, which
is also the time the bumblebee traveled. We can now find the distance the
bumblebee traveled, using the relationship, rate x time = distance, which gives
us (8)(80) = 640 miles.

It is important to stress for students how to avoid falling into
the trap of always trying to do what the problem calls for directly. Sometimes
a more circuitous method is much more efficient. Lots can be learned from this
solution. It must be emphasized to your class. You see, dramatic solutions are
often more useful than traditional solutions, since it gives students an
opportunity “to think out of the box.”#

*Dr. Alfred
S. Posamentier is Dean of the School of Education at City College of NY, author
of over 40 books on math including *Math Wonders to Inspire Teachers
and Students* (ASCD, 2003)
and *Math Charmers: Tantilizing Tidbits for the Mind *(Prometheus, 2003), and member
of the NYS Standards Committee on Math.*