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2007

Science and Mathematics Education

MAT Exam Expository Papers

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The Polygon Game, Kyla Hall Dec 2007

The Polygon Game, Kyla Hall

MAT Exam Expository Papers

The Polygon Game ‐ Take a regular, n‐sided polygon (i.e. a regular n‐gon) and the set of numbers, {1, 2, 3, …, (2n‐2), (2n‐1), 2n}. Place a dot at each vertex of the polygon and at the midpoint of each side of the polygon. Take the numbers and place one number beside each dot. A side sum is the sum of the number assigned to any midpoint plus the numbers assigned to the vertex on either side of the midpoint. A solution to the game is any polygon with numbers assigned to each dot for which all ...


Extending A Finite Sequence, Jessica Fricke Jul 2007

Extending A Finite Sequence, Jessica Fricke

MAT Exam Expository Papers

One of the most common mistakes in school mathematics is to list a few terms of a sequence and ask “What term comes next?” For example, a teacher may write: 1 2 4 … and ask what term comes next? The best answer is “almost anything could be the next term.” But people often find that answer unsatisfying. It helps, instead, to give an example. Perhaps, one might respond by saying: “The next number might be 8, but then again, it might be 7.” This answer might, of course, result in an angry teacher, so you need to be prepared with ...


Evaluating Polynomials, Thomas J. Harrington Jul 2007

Evaluating Polynomials, Thomas J. Harrington

MAT Exam Expository Papers

Computers use algorithms to evaluate polynomials. This paper will study the efficiency of various algorithms for evaluating polynomials. We do this by counting the number of basic operations needed; since multiplication takes much more time to perform on a computer, we will count only multiplications. This paper addresses the following: a) How many multiplications does it take to evaluate the one-variable polynomial, Σ= + + + + = n i i i n n a a x a x a x a x 0 2 0 1 2 ... when the operations are performed as indicated? (Remember that powers are repeated multiplications and must be counted ...


Ethnomathematics, Chad Larson Jul 2007

Ethnomathematics, Chad Larson

MAT Exam Expository Papers

When asked to think about a foreign country the first thing that comes to my mind is the language barrier and the customs that accompany that specific country. The culture of the citizens and how it differs from my culture are also things which peak my interest. Things which I view as “normal” may seem very odd to someone who lives thousands of miles away, and likewise, traditions that have been past down from generations of people from distant lands may seem peculiar to me. These customs and cultures of which I speak are also the things that make this ...


How To Graphically Interpret The Complex Roots Of A Quadratic Equation, Carmen Melliger Jul 2007

How To Graphically Interpret The Complex Roots Of A Quadratic Equation, Carmen Melliger

MAT Exam Expository Papers

As a secondary math teacher I have taught my students to find the roots of a quadratic equation in several ways. One of these ways is to graphically look at the quadratic and see were it crosses the x-axis. For example, the equation of y = x2 – x – 2, as shown in Figure 1, has roots at x = -1 and x = 2. These are the two places in which the sketched graph crosses the x-axis.


Experimentation With Two Formulas By Ramanujan, Daniel Schaben Jul 2007

Experimentation With Two Formulas By Ramanujan, Daniel Schaben

MAT Exam Expository Papers

Srinivasa Ramanujan was a brilliant mathematician, considered by George Hardy to be in the same class as Euler, Gauss, and Jacobi. His short life, marred by illness and tragic educational events, was unique in the history of mathematics. Mathematical discoveries are still being gleaned from his personal notebooks. Paper was a hard commodity to come by so his notebooks were a cluttered mix of pen over pencil mathematical hieroglyphics. The following highlights Ramanujan’s life in connection with Hardy, his work with ellipses, and his work with the partition function.


Master Of Arts In Teaching (Mat), Josh Severin Jul 2007

Master Of Arts In Teaching (Mat), Josh Severin

MAT Exam Expository Papers

The number zero is a very powerful tool in mathematics that has many different applications and rules. An interesting fact about the number zero is that according to our calendar (the Gregorian calendar), there is no “year zero” in our history. There is also no “zeroth” century as time is recorded from centuries B.C. to the 1st century A.D. However, certain calendars do have a year zero. In the astronomical year numbering system year zero is defined as year 1 BC. Buddhist and Hindu lunar calendars also have a year zero. In this paper I am going to ...


Hyperbolic Geometry, Christina L. Sheets Jul 2007

Hyperbolic Geometry, Christina L. Sheets

MAT Exam Expository Papers

In general, when one refers to geometry, he or she is referring to Euclidean geometry. Euclidean geometry is the geometry with which most people are familiar. It is the geometry taught in elementary and secondary school. Euclidean geometry can be attributed to the Greek mathematician Euclid of Alexandria. His work entitled The Elements was the first to systematically discuss geometry. Since approximately 600 B.C., mathematicians have used logical reasoning to deduce mathematical ideas, and Euclid was no exception. In his book, he started by assuming a small set of axioms and definitions, and was able to prove many other ...


The Volume Of A Platonic Solid, Cindy Steinkruger Jul 2007

The Volume Of A Platonic Solid, Cindy Steinkruger

MAT Exam Expository Papers

A regular tetrahedron and a regular octahedron are two of the five known Platonic Solids. These five “special” polyhedra look the same from any vertex, their faces are made of the same regular shape, and every edge is identical. The earliest known description of them as a group is found in Plato’s Timaeus, thus the name Platonic Solids. Plato theorized the classical elements were constructed from the regular solids. The tetrahedron was considered representative of fire, the hexahedron or cube represented earth, the octahedron stood for gas or air, the dodecahedron represented vacuum or ether (which is made up ...


Simple Statements, Large Numbers, Shana Streeks Jul 2007

Simple Statements, Large Numbers, Shana Streeks

MAT Exam Expository Papers

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, as defined by Wikipedia (2007). Large numbers typically refer to large positive integers, or more generally, large positive real numbers, but may also be used in other contexts. Very large numbers often occur in fields such as mathematics, cosmology, and cryptography. Sometimes people refer to numbers as being “astronomically large”. However, it is easy to mathematically define numbers that are much larger than those even in astronomy. We are familiar with the large magnitudes, such as million or billion. In mathematics, we may know a ...


Pythagorean Triples, Diane Swartzlander Jul 2007

Pythagorean Triples, Diane Swartzlander

MAT Exam Expository Papers

Who was Pythagoras after which the Pythagorean Theorem is named? Pythagoras was born between 580-572 BC and died between 500-490 BC. Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality. Very little is known about Pythagoras because none of his writings have survived. Many of his accomplishments may actually have been the work of his colleagues and students. Pythagoras established a secret cult called the Pythagoreans. His cult was open to both females and males and they lived a structured life consisting of religious teaching, common meals, exercise, reading and philosophical ...


Jean Baptiste Joseph Fourier, Gary Eisenhauer Jul 2007

Jean Baptiste Joseph Fourier, Gary Eisenhauer

MAT Exam Expository Papers

Jean Baptiste Joseph Fourier was born in Auxerre, France on March 21, 1768. He was the ninth of twelve children from his father’s second marriage. When he was nine, his mother died. The following year, his father, a tailor, also passed.


Order Of Operations And Rpn, Greg Vanderbeek Jul 2007

Order Of Operations And Rpn, Greg Vanderbeek

MAT Exam Expository Papers

There is not a wealth of information regarding the history of the notations and procedures associated with what is now called the “order of operations”. There is evidence that some agreed upon order existed from the beginning of mathematical study. The grammar used in the earliest mathematical writings, before mathematical notation existed, supports the notion of computational order (Peterson, 2000). It is clear that one person did not invent the rules but rather current practices have grown gradually over several centuries and are still evolving.


Distance, Rate, Time And Beyond, Janet Timoney Jul 2007

Distance, Rate, Time And Beyond, Janet Timoney

MAT Exam Expository Papers

In middle school mathematics, students learn to use the formula “distance equals rate times time,” usually expressed as d = r × t. Why not consider the formula distance = velocity × time? Does the term velocity mean something different than the term rate? We could also consider the variations of these formulas: distance ÷ time = rate, or distance ÷ rate = time. We can examine the definitions of these words and words which are very similar. After looking at the definitions of these words, maybe we will have a better understanding of how to use the formulas and of the meaning behind them.


Perimeter And Area Of Inscribed And Circumscribed Polygons, Lindsey Thompson Jul 2007

Perimeter And Area Of Inscribed And Circumscribed Polygons, Lindsey Thompson

MAT Exam Expository Papers

This paper looks at comparing the perimeter and area of inscribed and circumscribed regular polygons. All constructions will be made with circles of radius equal to 1 unit. To begin this exploration, I created a circle with a radius of 1(for my purposes I used 1 inch as my unit of measure). I chose my first construction to contain the most basic regular polygon, an equilateral triangle. A regular polygon implies that all sides of the figure are equal and all interior angles of the figure are congruent. My first construction shows an equilateral triangle inscribed in a circle ...


The Four Numbers Game, Tina Thompson Jul 2007

The Four Numbers Game, Tina Thompson

MAT Exam Expository Papers

The Four Numbers Game is a fun way to work with subtraction and ordering of numbers. While trying to find an end to a game that is played with whole numbers, there are several items that will be investigated along the way. First, we offer an introduction to how the game is played. Second, rotations and reflections of a square will be presented which will create a generalized form. Third, we explain how even and odd number combinations will always end in even numbers within four subtraction rounds. Fourth, we argue that the length of the game does not change ...


Testing Naval Artillery And Other Things, Tricia Buchanan Jul 2007

Testing Naval Artillery And Other Things, Tricia Buchanan

MAT Exam Expository Papers

In WWII a tremendous amount of artillery shells were made to support the war efforts. There were problems with the artillery shells sent to the battlefield; the main problem was their lack of ability to blow things up. In other words, they were duds! While one may think that dud shells were the proverbial rare case, in my paper I hope to show you that instead it unfortunately seemed more the norm. The reasons behind this are varied but in this paper I will focus on the testing practices of the artillery shells and some of the issues that occurred ...


A Monte Carlo Simulation Of The Birthday Problem, Stacey Aldag Jul 2007

A Monte Carlo Simulation Of The Birthday Problem, Stacey Aldag

MAT Exam Expository Papers

Question, how many people would you need in a group in order for there to be a 50-50 chance that at least two people will share a birthday? Answer, 23 people. But how can this be? There are 365 days in a year and half of that would be 182, so why wouldn’t you need at least 182 people to have a 50-50 chance? Strangely enough the answer to this question is only 23 people are necessary to have a 50% chance at least two people in the group will share a birthday. This situation, where the answer is ...