Holicong Middle School

“The purpose of life is to conjecture and prove.” - Paul Erdos 

 “Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true?  Where does the proof use the hypothesis?” - Paul Halmos

One of the most important things I can do as a math teacher is to convey to my students what it is like to be a mathematician.  Too often people have the idea that mathematicians are people who open up a textbook and solve the problems listed on the pages.

In reality, mathematicians are people who play.  They ask questions that have no known answers.  They are excited by the possibility of solving a problem that no one has ever been able to tackle.  And, with a lot of hard work and heavy thinking, they may even be lucky enough to find the answer.

The goal of this page is to publish the original ideas of my students.  The papers you see here represent the students’ own original conjectures along with their proofs.  Hopefully, as a result of these papers, they will better understand what it is like to be a mathematician.

And maybe, just maybe, they will even see how much fun it can be!

ORIGINAL CONJECTURES WITH PROOFS:

An extension of Theorem 11.5 to Concave Quadrilaterals

Submitted by:  Pops (2009)

A well known theorem in Euclidean Geometry states: If the diagonals of a quadrilateral are perpendicular, then the area of the quadrilateral is half the product of the lengths of the diagonals  (Theorem 11.5, Larson, Boswell and Stiff.)  In this paper, the author extends this theorem to certain types of concave quadrilaterals.

An area formula for a rhombus using an interior angle and side length

Submitted by:  George (2009)

In this paper, the author takes the traditional area formula for a rhombus and rewrites it, replacing diagonal lengths with an interior angle and side length.  What is unique is the author’s use of several trigonometric identities ... identities learned independently by the author.

Strengthening the Perpendicular Bisector Converse Theorem

Submitted by:  John (2010)

In this paper, the author tackles a weakness of the following theorem:  If point C is equidistant from the endpoints of a segment, then C lies on the perpendicular bisector of the segment (Theorem 5.2 Perpendicular Bisector Converse, Larson, Boswell and Stiff.)  The weakness is that, while the theorem guarantees that the perpendicular bisector will pass through point C, it does not guarantee that any segment passing through point C will be the perpendicular bisector.  The author wanted to find a condition where the segment passing through point C must be the perpendicular bisector.  He found that by adding an addition point D that is also equidistant from the endpoints of the segment, the line that passes through both of these points must be the perpendicular bisector.

RESEARCH PAPERS

A googol

Submitted by:  Paige, age 5 (7/22/09)

For those of you who would like to know what “a googol” looks like, my daughter wrote it out for you in crayon.  She wanted to ... who am I to say no!

A Brief History of the Quadratic Formula

Submitted by:  Pops (2009)

In this paper, the author gives us a brief overview of the Quadratic Formula.  Topics include its history, derivation and application.  This is a must read for anyone who needs a quick orientation to this amazing formula.