The activities were given to 25 engineering students from 18 to 19 years of age who were
taking a first calculus course. The activities were implemented in bi-weekly sessions of two hours
each during the regular semester.
The material and equipment available to students included:
1) For the individual exploration work, the students used a scientific calculator or computer
(according to the number available), paper and pencil.
2) In the group work they used scientific calculators, EXCEL (according to availability) and
paper and pencil for the written reports.
3) For the discussion phase the group used the blackboard, paper and pencil.
4) The sessions were tape recorded.
5) The work with the calculator or computer was saved on files for later analysis.
During the work sessions the students had the opportunity to use a scientific calculator, Excel
and/or paper and pencil. Students worked initially on individuals bases. In this stage they were
asked to hand in a written report of their ideas and experiences with the task, explaining and
justifying their results. A second stage consisted of work in groups of three in which they
exchanged ideas in order to come to an agreement for the solution of the problem. They then
gave a written report.
The group work included, in its final stage, a discussion of the related mathematical ideas.
The discussion was moderated by one of the members of the group and directed by the instructor.
The problem of the Towers, (Taken from Fey, 1995, p. 73)
The problem provides a formula to calculate the largest distance that can be seen from the top
of a building: if h is the height in meters from the point of observation, that is, the distance of
visibility in kilometers as a function of the height is obtained from the rule: s(
h
) = 3.532 h .
This, students were asked to:
a) Calculate the distance of visibility that can be achieved from the top of different
buildings. (See Table 1)
Structure Locality
Height,
(m) Distance of Visibility, (km)
CN Tower
Toronto Canadá
555
Sears Tower
Chicago, USA
443
World Trade Center, N
New York, USA
419
World Trade Center, S
New York, USA
419
Empire State Building
New York, USA
381
Amoco Building
Chicago, USA
346
John Hancock Building
Chicago, USA
344
Centrepoint Tower
Sydney, Australia
325
Texas Commerce Tower
Houston, USA
305
Allied Bank Building
Houston, USA
300
Eiffel Tower
Paris, France
300
Table 1. Buildings and their corresponding height.
b) Make a table with the relationship of the heights to the values of s(h).
c) Figure the interpolation for the determination of the distance of visibility taken from a
graph of the function s(h).
d) Determine the height of the building in order to be able to see x kilometers.
e) Analyze the change in distance of visibility when the height is increased.
f) Describe the tendency of the changes of the incise d).
Convention