Presentation of Results
The following describes the work exhibited by a small group of students (Alexis, Iris and
Gina) in the towers problem. The interaction of the group with the problem, the role of the
computer and the instructor is documented. Students worked in this task during three sessions of
two hours each.
Understanding the problem
In the solution of the problem the students had to identify that the heights of the buildings in
Table 1 in meters and obtain the distance of visibility with the formula s(
h
) = 3.532 h in
kilometers. The constant 3.532 included the conversion factor (meters to kilometers).
During the first session the questions that directed students’ interaction with the end of
generating a common context for the problem were:
How is the answer verified when the formula is in meters or in kilometers?
Must a conversion of units be realized? That is do height and visibility needs to have same
dimensions? Do small variations in the height produce significant changes in the distance of
visibility?
After a discussion, the students came to an agreement on what they had to consider as the
distance of visibility and that given the scale which was employed, a variation of centimeters in
the height did not substantially modify this distance of visibility.
During the interaction of the students when discussing their individual solutions, they were
confused with the units applied for the height and the distance of visibility. The statement of the
problem indicated the height in meters and the distance of visibility in kilometers. The discussion
turned about the need for converting the height of the buildings given from meters to kilometers,
as is seen in the following:
Alexis: I, for one, think that the question tells you that on a clear day at the height that you
are on the tower you can see up to 50 km. Therefore, I see it as a rectangular triangle.
In other words, you have a height but maybe your distance is much farther, depending
on the height from which you see the distance and that which you can come to see.
Alexis: I think that you have to convert the height into meters, if you use the formula s(
h
) =
3.532 h When you apply the formula for the first height of 555 meters this will give
you a value of 83.20 meters (emphasizes the word meters). And when you convert
them you use the rule of three.
Gina: But it would be illogical that we were in a building so tall and if we say that we can
only see 83.20 meters, or in other words, it is a very short distance.
These dialogues show that Alexis thinks the distance of visibility and the height form a
rectangular triangle (this was mentioned by Alexis), considering how it can be seen in their later
dialogues that the variation in the distance of visibility is linear. In thinking of a rectangular
triangle Alexis does not take into account the curvature of the Earth and the given mathematical
model s(
h
) = 3.532 h .
However, after Gina’s participation in the above dialogue, Iris and Alexis revised their ideas
and found, without formal verification, that the results obtained using a change in units did not
make sense in the context of the problem.
Alexis: Then this formula gives the kilometers you, good.
It is through the dialogues of the students that we find that Alexis initially mentioned that all
the factors influence the distance of visibility s(h), including the height of the person. Iris
intervened with the comment that it maybe because of this they were working the distance s(h) in
kilometers, because this way the variation in the height of the person was not significant.
Convention