MAT Exam Expository Papers

Amicable Pairs

Lexi Wichelt — Fri, 21 May 2010 13:50:03 PDT

The ancient Greeks are often credited with making many new discoveries in the area of mathematics. Euclid, Aristotle, and Pythagoras are three such famous Greek mathematicians. One of their discoveries was the idea of an amicable pair. An Amicable pair is a pair of two whole numbers, each of which is the sum of the proper whole number divisors of the other.

The Art Gallery Question

Vicki Sorensen — Fri, 21 May 2010 13:45:39 PDT

MAT question Suppose you have an arbitrary room in an art gallery with v corners, and you wanted to set up a security system consisting of cameras placed at some of the corners so that each point in the room can be seen by one of the cameras. How many cameras do we need? (See the example at right for a possible room with an interesting shape.)

Fractals and the Collage Theorem

Sandra S. Snyder — Fri, 21 May 2010 13:41:01 PDT

The idea of fractals is relatively new, but their roots date back to 19th century mathematics. A fractal is a mathematically generated pattern that is reproducible at any magnification or reduction and the reproduction looks just like the original, or at least has a similar structure. Georg Cantor (1845-1918) founded set theory and introduced the concept of infinite numbers with his discovery of cardinal numbers. He gave examples of subsets of the real line with unusual properties. These Cantor sets are now recognized as fractals, with the most famous being the Cantor Square.

Taxicab Geometry

Kyle Lannin Poore — Fri, 21 May 2010 13:36:59 PDT

Taxicab geometry was founded by a gentleman named Hermann Minkowski. Mr. Minkowski was one of the developers in “non-Euclidean” geometry, which led into Einstein’s theory of relativity. Minkowski and Einstein worked together a lot on this idea Mr. Minkowski wanted people to know that the side angle side axiom does not always hold true for all geometries. He wanted to prove this in the case that you can not always use the hypotenuse to find the shortest way from one spot to another. The best way to think of his idea is to think of a taxicab going from one place to another, thus the name taxicab geometry.

Triangulation

Jim Pfeiffer — Fri, 21 May 2010 13:31:51 PDT

Map making has been a scientific endeavor for mankind since the beginning of recorded human history (ca. 5000 years ago) and it is today more sophisticated than ever before. In terms of trigonometric functions there is evidence that dates back to Babylonian times that angles and distances from points on a triangle were utilized in measurement with significant amounts of work in this field done by ancient Greeks, Indians, as well as Arabic mathematicians. For example the ancient Egyptians utilized the trigonometric functions for surveying properties in order to determine how much of their land had washed away when the Nile River would flood, additionally, the near perfect squareness and the north-south orientation of the Great Pyramid of Giza built c. 2700 BC, affirm the ancient Egyptians skills in surveying.

Order of Operations and RPN

Greg Vanderbeek — Fri, 21 May 2010 13:29:08 PDT

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 — Fri, 21 May 2010 13:25:39 PDT

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.

The Four Numbers Game

Tina Thompson — Fri, 21 May 2010 13:22:36 PDT

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 if multiples of the original numbers are used to create a new game. Fifth, we show that all Four Numbers Games will come to an end. Sixth, we offer an investigation of the general form and some special cases and how they can help predict a more accurate end. Finally, there will be an example of how this problem could be used in a sixth grade classroom.

Perimeter and Area of Inscribed and Circumscribed Polygons

Lindsey Thompson — Fri, 21 May 2010 13:19:00 PDT

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 (see Appendix A). Next, I needed to find the perimeter of this inscribed triangle. I used a ruler to measure the distance of one of the sides, then multiplied that value times 3, the number of sides of the triangle. This method will work for finding the perimeter of all regular polygons. The only change would be substituting three with the number of sides of the polygon being studied. A general formula would be: Perimeter of a regular polygon with n sides equals the length of one side times n.

Pythagorean Triples

Diane Swartzlander — Fri, 21 May 2010 12:56:34 PDT

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 study. The Pythagorean Theorem for which Pythagoras is given credit states: In a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. There is no evidence that Pythagoras himself worked on or proved this theorem, rather the earliest mention of his name with the theorem occurred five centuries after his death. Other accomplishments attributed to Pythagoras are include a system of analyzing music based on proportional intervals of one through four, the number system based on ten, the identification of square numbers and square roots, knowledge that the earth is round, and that all planets have an orbit that travels around one central point.

Simple Statements, Large Numbers

Shana Streeks — Fri, 21 May 2010 12:48:24 PDT

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 number as an approximation or as an exact amount; for example 531,441. This number could be called “half a million” but it is also the specific solution to the question “how many ways are there to color the 12 numerals on a clock face if you have three different colored markers?” (Contributed by Maria Pierce). You would compute this number as follows: you have 3 choices of color for each hour number 1,2,3,…,12. So there are 3 choices for 1; followed by 3 choices for 2; followed by 3 choices for 3; etc. This would give you 312= 531,441 number of choices.

The Volume of a Platonic Solid

Cindy Steinkruger — Fri, 21 May 2010 12:45:07 PDT

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 of pure electromagnetic energy) , and the icosahedron was water.

Hyperbolic Geometry

Christina L. Sheets — Fri, 21 May 2010 12:41:41 PDT

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 theorems. Although many of his results had been stated by earlier Greek mathematicians, Euclid was the first to show how everything fit together to form a deductive and logical system.

Master of Arts in Teaching (MAT)

Josh Severin — Fri, 21 May 2010 12:28:39 PDT

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 discuss many different uses of the number zero in mathematics and in the world. I am also going to discuss the origins of the number zero.

Experimentation with Two Formulas by Ramanujan

Daniel Schaben — Fri, 21 May 2010 12:25:18 PDT

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.

Spherical Geometry

Linda Moore — Fri, 21 May 2010 12:22:20 PDT

Spherical geometry was studied in ancient times as a subset of Euclidian three-dimensional space. It was a logical outcome as the earth is a sphere. The word geometry literally means the measure of the earth. However, the undefined terms, axioms and postulates of Euclidian geometry take on a new meaning when studied on a sphere.

How to Graphically Interpret the Complex Roots of a Qudratic Equation

Carmen Melliger — Fri, 21 May 2010 11:36:12 PDT

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.

Ethnomathematics

Chad Larson — Fri, 21 May 2010 11:31:11 PDT

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 world such an interesting place to live, study, and explore. One might think that mathematics has no place in a discussion of different cultures and worldly studies, citing that mathematics is purely numbers and the manipulation that occurs between them. However, adopting this train of thought would be closing the door and the mind on a topic that is bridging cultural gaps from around the globe. This relatively new field of study is called Ethnomathematics.

Evaluating Polynomials

Thomas J. Harrington — Fri, 21 May 2010 11:26:29 PDT

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 as such.) Write this number of multiplications as a function of n. b) Use mathematical induction to prove that your answer is correct. c) Find another way to evaluate this polynomial by doing the operations in a different order so that fewer multiplications are needed. Hint: Think of ways to intermix addition and multiplication and experiment with polynomials of lower degree. Write the number of multiplications as a new function of n. The best algorithm will use only n multiplications. Explain the algorithm you will use. d) How many multiplications does it take to evaluate the two-variable polynomial, ΣΣ = = n i n j i j ij a x y 0 0 when the operations are performed as indicated? Write this number of multiplications as another function of n. e) Use mathematical induction to prove that your answer is correct. f) Find another way to evaluate the two-variable polynomial by doing the operations in a different order so that fewer multiplications are required. Write down the associated function of n. Do you think that this is the most efficient algorithm? If not hunt for a better algorithm.

The Polygon Game

Kyla Hall — Fri, 21 May 2010 11:21:04 PDT

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 side sums are equal, i.e. for which you have equal side sums. The most general problem we might state is, “Find all solutions to The Polygon Game.”