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Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
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Spherical Geometry, Linda Moore
Spherical Geometry, Linda Moore
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
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.
Symmetry Of Scale Expository Paper, Darla R. Kelberlau-Berks
Symmetry Of Scale Expository Paper, Darla R. Kelberlau-Berks
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
A checkerboard, a beehive, a brick wall and a mud flat dried in the sun all have something in common. They are all examples of tilings or tessellations. Although you may think of mosaics or other pieces of artwork when you hear these words, in actuality you should also think of mathematics and science. I will describe in more detail the mathematics involved with tessellations and tilings, and discuss specific tilings such as the Pinwheel Tiling and the Penrose Tiles.
Heron, Brahmagupta, Pythagoras, And The Law Of Cosines, Kristin K. Johnson
Heron, Brahmagupta, Pythagoras, And The Law Of Cosines, Kristin K. Johnson
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
The formula for the area of a triangle can be developed by making an exact copy of the triangle and rotating it 180°. Then join it to the given triangle along one side to obtain a parallelogram as shown above. To form a rectangle, cut off a small triangle along the right and join it at the other side of the parallelogram. Because the area of the rectangle is the product of base (b) and height (h), the area of the given triangle must be 1⁄2bh.
The Art Gallery Question, Vicki Sorensen
The Art Gallery Question, Vicki Sorensen
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
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.)
The Vigenére Cipher Expository Paper, Virginia L. Clark
The Vigenére Cipher Expository Paper, Virginia L. Clark
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
French diplomat and cryptographer Blaise de Vigenére (1523-1596), developed the Vigenére Cipher in 16th century France in the mid-1580s. Vigenére was on the court of Henry III of France. Vigenére developed a polyalphabetic coding system in which one letter of plain text may be encrypted as different letters rather than one plain text letter represented as one cipher text letter throughout the encoded message.
The Game Of Nim, Dean J. Davis
The Game Of Nim, Dean J. Davis
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
The game of Nim is possibly one of the most frustrating games I have ever played. Just when I started to feel that I had figured the strategy out, my brother, who is a computer programmer, blew me out of the water. I should have known better than to take on a computer wizard.
Area And Perimeter Of Polygons, Bryan Engelker
Area And Perimeter Of Polygons, Bryan Engelker
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
A student comes to class excited. She tells you she has figured out a theory you never told the class. She says she has discovered that as the perimeter of a closed figure increases, the area also increases. She shows you two pictures to prove what she is doing. The first picture is of a 4 by 4 square. Of course, its perimeter is 16 and its area is 16. The second picture is of a 4 by 8 rectangle. Here the perimeter is 24 and the area is 32. What do you say to the student?
Just What Do You “Mean”?, Myrna L. Bornemeier
Just What Do You “Mean”?, Myrna L. Bornemeier
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
In Ancient Greece the Pythagoreans were interested in three means. The means were the arithmetic, geometric, and harmonic. The arithmetic mean played an important role in the observations of Galileo. Along with the arithmetic mean, the geometric and harmonic mean (formerly known as the subcontrary mean) are said to be instrumental in the development of the musical scale. As we explore the three Pythagorean Means we will discover their unique qualities and mathematical uses for helping us solve problems.
Farey Sequences, Ford Circles And Pick's Theorem, Julane Amen
Farey Sequences, Ford Circles And Pick's Theorem, Julane Amen
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
One of the ongoing themes through the Math in the Middle coursework has been the idea of identifying patterns. From our first course, Math as a Second Language, patterns have been useful to explain phenomena and determine future values. Some patterns are numerical but can be described using algebra. Some are visual or geometric and also can be described using numbers and symbols. Many of these patterns have resurfaced in different forms and at different times in new and interesting ways. It has been a humbling experience to see the interconnectedness of seemingly unconnected ideas. Pick’s Theorem, Farey Sequences and …
Fractals And The Chaos Game, Stacie Lefler
Fractals And The Chaos Game, Stacie Lefler
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
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.
Taylor Polynomials, Doug Glasshoff
Taylor Polynomials, Doug Glasshoff
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
Before the age of calculators, studying functions such as sin x, cos x , ex , and ln x was quite time consuming. The graphs of these functions are important when studying their characteristics. James Gregory, a Scottish mathematician in the 17th century, made an important discovery about these functions. Using calculus, he wrote a series of terms to approximate very closely the graph of the curve. His main focus was with the function ln x ; he was able to calculate any positive value of x using a polynomial series. Brook Taylor, an English mathematician, generalized the Maclaurin series, …
Comparing Infinite Sets, Julie M. Kreizel
Comparing Infinite Sets, Julie M. Kreizel
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
I have been assigned to explore the theorem stating that there is no largest (infinite) set as established and proven by Georg Cantor. To do this I need to start by defining what it means to say that a set is infinite. This can be quite difficult because the tendency might be to say that a set is infinite if it is not finite, and I don’t believe that grants us the clarity of definition we are looking for. When trying to understand the size of a given set, the number of objects (elements) in the set, we may not …
Triangulation, Jim Pfeiffer
Triangulation, Jim Pfeiffer
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
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 …
Taxicab Geometry, Kyle Lannin Poore
Taxicab Geometry, Kyle Lannin Poore
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
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 …
Fractals And The Collage Theorem, Sandra S. Snyder
Fractals And The Collage Theorem, Sandra S. Snyder
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
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.
The Exponential Function, Shawn A. Mousel
The Exponential Function, Shawn A. Mousel
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range (Lial & Miller, 1975). An exponential function is a function with the basic form f (x) = ax , where a (a fixed base that is a real, positive number) is greater than zero and not equal to 1. The exponential function is not to be confused with the polynomial functions, such as x2. …
Conic Sections Expository Paper, Delise Andrews
Conic Sections Expository Paper, Delise Andrews
Department of Mathematics: Master's of Arts in Teaching, Exam Expository Papers
Conic sections, or conics, include the various geometric figures created by the intersection of a plane with a cone. It is important to note that the definition of a cone includes the surface generated by a straight line that moves so that it always intersects the circumference of a given circle and passes through a given point not on the plane of the circle. The point, called the vertex of the cone, divides the cone into two “halves” called nappes.