Using PROJECTS to Invigorate Honors Calculus
Dan Kemp
Fishback Honors College at South Dakota State University
Honors Calculus at SDSU
• Three semester sequence
• Higher placement scores required
• Multiple majors enrolled
Honors Calculus distinctions
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Common content covered at the professor’s discretion
Textbook chosen by professor
Meets one additional day per week
Use of a TI-nspire CX CAS calculator required in Calc I & II
Sage Math used in Calc III
Homework graded by an Honors TA
What makes Calculus ‘Honorable’?
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Emphasis on concepts as opposed to techniques, but
Formal proofs are not emphasized
Group work on most class days
Calculus I & II students are introduced to the History of
Mathematics via readings and projects
• Calculus III students read about the Philosophy of
Mathematics
• Projects done in groups outside of class
What are Projects?
• Substantial mathematics related to course material
• Content not normally covered in the course
• Mathematical level
challenging
• Work done in assigned
groups of students
• Project reports are
turned in as word
processed documents
including mathematics
and diagrams
• Projects written by the
professor
• The best projects have
an historical basis
RESEARCH POSTER PRESENTATION DESIGN © 2012
www.PosterPresentations.com
A Calculus II Project
Why Projects?
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Projects force students to work cooperatively
Projects can involve students with mathematics they
would not normally be exposed to.
Writing mathematics is hard – getting early practice is
beneficial to students
Honors students enjoy projects!
• In 1656 the Englishman John Wallis
(1616-1703) published
Arithmetica infinitorum
• There he asserted the formula that
now bears his name:
Projects and History
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“It appears to me that if one wishes to make progress in
mathematics, one should study the masters and not the
pupils”. – Neils Henrick Abel
Other disciplines have freshmen studying ‘Masters’ such as
Shakespeare, Renoir, Beethoven etc., but that does not
happen in beginning college mathematics courses.
Many important mathematics topics have interesting
historical roots
A Calculus I Project
• Circa 200 CE a Greek mathematician, Heron, proved a
formula for the area of a triangle in terms of is side lengths
that now bears his name.
• 600 years later an Indian
mathematician, Brahmagupta,
stated a similar formula for
cyclic quadrilaterals:
□=
3∙3∙5∙5∙7∙7 𝑒𝑡𝑐
continued indefinitely
2∙4∙4∙6∙6∙8∙8 𝑒𝑡𝑐
• The ‘box’ symbol represents a number that satisfies
‘the circle to the square of its diameter is as 1 is to
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𝜋
□’,
that is, □ = in current notation
The historical interest is that the symbol we call ‘pi’ had
not yet been invented and consequently Wallis created
a (bad) symbol
The project is completed using mathematical induction
to help show that:
𝜋
2 2 4 4
2𝑛
2𝑛
= lim ∙ ∙ ∙ ⋯
∙
2 𝑛→∞ 1 3 3 5 2𝑛 − 1 2𝑛 + 1
• This result is then used in a later project that also has
historical origins
A Calculus III Project
𝐴=
𝑠−𝑎 𝑠−𝑏 𝑠−𝑐 𝑠−𝑑
where the semiperimeter, 𝑠, is
𝑎+𝑏+𝑐+𝑑
𝑠=
2
• The proof the student groups
work through involves algebra and trigonometry
• Benefits of this first project include
 Socialization of freshman students
 Review of basic pre-calculus skills
 Working with interesting new material
 Introduction to word processing mathematics
• In 1687 Isaac Newton published his Principia in which
he proved, based upon his Law of Universal
Gravitational Attraction, that Kepler’s laws of motion
for planets were correct
• This was a watershed result that empowered others to
use the new Calculus to explain the world around them
• Unfortunately, using calculus to establish Kepler’s laws
is not a part of the standard calculus curriculum
• Using modern vector calculus to prove that if Newton’s
law of gravitation holds then a planet must move in an
elliptical path is a great Honors Calculus III project