Exploratory Classroom Problems in Calculus
Org: Peter Taylor (Queen's)
- SERGE D'ALESSIO AND IAN VANDERBURGH, Faculty of Mathematics, University of Waterloo
Chasing Imaginary Triangles
This presentation concerns right-angled triangles. In particular, we
are interested in computing the length of its hypotenuse (h) given
the triangle's perimeter (P) and area (A). One method of
determining h involves the formula:
However, as we will demonstrate, this formula can lead to an incorrect
result. To resolve this a condition between A and P was derived
that dictates when the above formula can be used. Other interesting
issues surrounding right-angled triangles will also be explored along
- RICHARD HOSHINO, Dalhousie University, Department of Mathematics, Halifax, NS
A Heart-Stopping Solution
During a first-year calculus class I taught several years ago, my
midterm took place on Valentine's Day. To fit the occassion, a
problem on the exam asked the students to determine the polar equation
of a heart. Overall, the students found the question to be very
challenging, as we had only discussed how to sketch a curve given the
polar equation; and this question required them to go backwards. Some
of my students came up with the correct answer I was looking for,
namely r = 4 - 4sinq, where q runs from 0 to 2P.
However, one student came up with a completely different equation,
using various transformations and translations of the absolute value
function. In this brief session, we will determine "Steve's
equation", and unpack the sophisticated ideas involved in his
solution. We hope to present this heart-stopper as an example of a
beautiful and pedagogically-rich problem in the first-year calculus
- VED MADAN, St. Mary's University College, 14500 Bannister Rd. SE,
Calgary, Alberta T2X 1Z4
Information Technology-Impact on Calculus Problem Solving
Changes in mathematics instructions over the past three decades have
been necessitated by increasing student numbers at post-secondary
institutions. Advances in technology have benefited mathematics
instruction, particularly in the area of calculus due to its analytic,
graphic and numeric approaches to solving real life problems.
Until the 1970s, Calculus was taught with traditional text books. In
the 1980s, non-linear processes in Reform Calculus involved the use
of graphing calculators and computer spreadsheets. Websites designed
to promote the sharing of information and ideas became popular in the
1990s. In the 21st century, increasingly sophisticated technological
tools such as teleconferencing and chat rooms are enhancing calculus
problem solving skills with one-to-many and many-to-many web-based
collaborative instructional strategies for student-centered learning.
The pitfalls of advances in technology also need to be recognized.
For example, students today may be lacking basic skills that would
allow them to perform calculations without the benefit of calculator.
This presentation will review some major developments in technology
since the 1970s and share information on how technology advancement
has influenced calculus problem solving skills-both positively and
- PETER TAYLOR, Queen's University, Kingston, ON K7L 3N6
Introduction and some "Executive Class" Examples
I will briefly indicate a number of examples of exploratory calculus
problems we have used in our special MATH 121 "Executive Class".
I'm hoping that these example will spawn further ideas and reflections
in the workshop activity to come at the end of both the morning and
the afternoon sessions. Others are invited to contribute informally
at that time.