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The student then places it on the playing field. The system allows a chosen playing card to be dragged by means of a mouse to the playing field and, if properly placed, to "stick" in place on the playing field. (Improperly placed cards "snap" back to their original file position.) After each card has been correctly placed, a line between properly placed cards is generated connecting proper statements and reasons to each other and the GIVEN or CONCLUSION displays the completed proof (Herbst, 2002).
In working with geometric proofs, it is important for the student and teacher alike to approach this new and intimidating subject with an open mind. Even though students may have never experienced any type of logic or reasoning prior to the introduction of proofs, if presented correctly, this new way of approaching math can be both fun and enlightening. Teachers should keep this in mind when…
Discovering Geometry: A Guide for Parents. 2008, Key Curriculum Press. Retrieved October 19, 2009 at http://www.keymath.com/documents/dg4/GP/DG4_GP_02.pdf
Herbst, Patricio G. Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century, Educational Studies in Mathematics, Vol. 49, No. 3 (2002), pp. 283-312,
The Golden section has a special relationship to the Fibonacci sequence. This is a mathematical sequence in which the first two numbers being 0 and 1, each subsequent number is a sum of the previous two numbers: 0, 1,1, 2 (1+1), 3 (2+1), 5 (3+2), etc.
Like the Golden section, the Fibonacci numbers are used to understand the way trees branch, leaves occur, fruit ripens, etc. -- it is a set of numbers that explains nature's patterns.
Subdividing shapes has no effect on their ratio or relationship to Fibonacci.
Chapter 4 -- Root Rectangles- the idea of the root angel reduction allows the Golden section to become more vital in several aspects of modern life. Not only does this impact modern design of furniture, technology, and appliances; it has a far larger and more robust meaning as we begin to understand the roots of organic chemistry and the structure of…
In 2003, a study showed that based on psychometric data, the Golden mean appears in the chronological cycle of brain waves. This was empirically confirmed in 2008.
In 2010, the journal Science reported that the golden ratio is present at the atomic level.
For a designer, the importance of the ratio cannot be overstated; from the Volkswagen Beetle to the Gutenberg Bible, the Golden ratio is everywhere; natural and man-made.
This will not only introduce elementary students to geometry, but also begin the complicated thinking associated with algebraic concepts. Using the formula to plug in the known degrees and then find the x is the beginning of much more abstract algebraic thinking.
Circles rule our lives and have rules of their own! Each circle measures to 350 degrees, and with this knowledge we can begin to find unknown angles!
If a circle measures 360, that means that a half circle measures half -- 180 degrees. In a half circle, there are many different angle combinations. But, we know that they all equal out to 180 degrees.
Knowing this, we can find the great unknown!
Well, we know that the total of the two angles equals 180 degrees. Therefore, angle 1 = angle 2 = 180 degrees.
Let's just plug the numbers into the equation.
63 + x = 180.…
National Council of Teachers of Mathematics [NCTM]. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.
Fractal Geometry is a somewhat new branch of mathematics that was developed in 1980 by enoit . Mandelbrot, a research mathematician in I..M.'s Thomas Day Watson laboratory in New York. Mandelbrot was experimenting with the theories of Gaston Julia, a French mathematician when he discovered the fractal set was discovered.
Julia dedicated his life to the study of the iteration of polynomials and rational functions. Around the 1920s, Julia published a paper on the iteration of a rational function, which brought him to fame. However, after his death, he was all but forgotten...until the 1970's when Mandelbrot, who was inspired by Julia's work, revived his work.
y using computer graphics, Mandelbrot was able to show the first pictures of the most beautiful fractals known today.
Mandelbrot, who is now Professor of Mathematics at Yale, made the discovery of fractal geometry by going against establishment and academic mathematics -- going beyond…
Mandelbrot, Benoit B. The Fractal Geometry of Nature W.H. Freeman and Company, 1977.
Crilly, R.A. Fractals and Chaos. Springer-Verlag, 1991.
Dictionary of Scientists, Oxford University Press, Market House Books Ltd., 1999
Natural Sciences and Geometry in Metaphysical Poetry
Love in metaphysical poetry: Donne and Marvell
"Metaphysical texts, primarily characterized through the conflation of traditional form with seditious linguistic techniques such as satire, irony, wit, parody and rhetoric, generate a microcosmic emphasis in many of the texts" even while the authors ultimately address 'macro' concerns of religion and man's place in the universe (Uddin 45). In poems such as John Donne's "The Flea" and "A Valediction Forbidding Mourning" and Andrew Marvell's "The Definition of Love," subjects such as the poet's adoration for his beloved take on a much higher significance than the personal sphere within the context of the poem. Metaphysical poetry embodies what is often considered a paradox: it is, on one hand, intensely emotional, but it is also, on the other hand, quite explicit in its suggestion of universality. "Introspection, being 'a careful examination of one's own thoughts, impressions and…
Donne, John. " The Flea." Poetry Foundation.
http://www.poetryfoundation.org/poem/175764 [16 Jan 2013]
Donne, John. "A Valediction Forbidding Mourning."
http://www.luminarium.org/sevenlit/donne/mourning.php [16 Jan 2013]
MATH - Measurement, Geometry, Representation
A standard unit of measurement offers a point of reference by which items of weight, length, or capacity can be delineated. It is a quantifiable semantic that aids every individual comprehend the relation of the object with the measurement. For instance, volume can be expressed in metrics such as gallons, ounces, and pints. On the other hand, a non-standard unit of measurement is something that might fluctuate or change in terms of weight or length. The item that was measured is a door as this is an object that can be found in everyday life. Through the use of standard units of measurement, I found that the door is equivalent to 2 meters or 200 centimeters in length. This particular standard measure was used for the reason that it is possible to utilize it using items including tape measures. On the other hand,…
Mathematics -- to the Moon & Back
Once upon a time, Alexander, a young man from Athens fell in love with a local girl, Adrianna, whose beauty was far greater than any other young woman he had ever seen. Alexander was so smitten with Adrianna that he promised her the moon. Being an astute girl, Adrianna told Alexander that she wasn't at all sure that he could deliver the moon, but he could begin to convince her that he was intelligent and clever by measuring the distance from the earth to the moon. Alexander had long heard the stories about his Greek ancestors who were experts in mathematics and astronomy, so he sought out some wise elders to learn more.
Alexander spent some time with two elders, one of whom told him he knew how to measure the size of the earth (which, Alexander mused, was bound to impress, Adrianna),…
This letter is in response to your question: Why are students in elementary school learning geometry when they do not yet know the basic facts and should be spending their time working on them instead?
There are two parts to the answer. The first is concerned with the learning of math facts. It is an ongoing process for students in the elementary grades. It begins with the development of number sense, which is a child's facility and flexibility in using and manipulating numbers (Chard, Baker, Clarke, Jungjohann, Davis, and Smolkowski, 2008, p. 12). Some students develop number sense in preschool or informally in familial settings before kindergarten; other children do not begin to develop number sense until their formal schooling begins, whether because of opportunity or because of developmental readiness. Developing number sense takes time. It does not happen quickly and it does not happen because…
Chard, D.J., Baker, S.K., Clarke, B., Jungjohann, K., Davis, K., and Smolkowski, K. (2008).
Preventing early mathematics difficulties: The feasibility of a rigorous kindergarten mathematics curriculum. Learning Disability Quarterly 31(1), pp. 11-20.
Common core standards adoption by state. (2012). ASCD. Retrieved from http://www.ascd.org/common-core-state-standards/common-core-state-standards - adoption-map.aspx
Cooke, B.D., and Buccholz, D. (2005). Mathematical communication in the classroom: A teacher makes a difference. Early Childhood Education Journal 32(6), pp. 365-369).
Note the distinct similarities.
An examination of Escher's Circle Limit III can thus tell us much about distance in hyperbolic geometry. In both Escher's woodcut and the Poincare disk, the images showcased appear smaller as one's eye moves toward the edge of the circle. However, this is an illusion created by our traditional, Euclidean perceptions. Because of the way that distance is measured in a hyperbolic space, all of the objects shown in the circle are actually the same size. As we follow the backbones of the fish in Escher's representation, we can see, then, that the lines separating one fish from the next are actually all the same distance even though they appear to grow shorter. This is because, as already noted, the hyperbolic space stretches to infinity at its edges. There is no end. Therefore, the perception that the lines are getting smaller toward the edges is, in…
Corbitt, Mary Kay. "Geometry." World Book Multimedia Encyclopedia. World Book, Inc., 2003.
Dunham, Douglas. "A Tale Both Shocking and Hyperbolic." Math Horizons Apr. 2003: 22-26.
Ernst, Bruno. The Magic Mirror of M.C. Escher. NY: Barnes and Noble Books, 1994.
Granger, Tim. "Math Is Art." Teaching Children Mathematics 7.1 (Sept. 2000): 10.
relearn several mathematical concepts and learn how to instruct other about them. It also became necessary to learn the different components of educating students on math based upon their current knowledge and abilities and how the teacher will evaluate the students to make that determination. Not only did I learn how to teach the subject, but I was also instructed on how to submit and fulfill standards. In short, this class taught me how to be an effective and efficient math teacher for students from kindergarten up to the eighth grade. This class had good moments, difficult moments, and has influenced both what concepts I will teach my students and how I will teach them when the time comes.
It is hard determining which of the components learned in this class were the most important. Each mathematical concept will be necessary when entering the teaching profession. Certainly it was useful…
Billstein, R., Libeskind, S., & Lott, J.W. (2010). A Problem Solving Approach to Mathematics
for Elementary School Teachers (10th ed.). Boston, MA: Addison-Wesley.
National and State Subject Matter Content Standards for Math
According to the California standards for high school students, the geometry curriculum contains six critical components: "to establish criteria for congruence of triangles based on rigid motions; establish criteria for similarity of triangles based on dilations and proportional reasoning; informally develop explanations of circumference, area, and volume formulas; apply the Pythagorean Theorem to the coordinate plan; prove basic geometric theorems; and extend work with probability" (Common Core Standards, California Department of Education: 69). The elucidated standards are often quite specific in terms of how students are asked to apply basic concepts such as measuring angles; understanding the different properties of parallel lines; and manipulating various polygons. Not only must the students prove theorems but they must also be able to construct such shapes using a variety of methods in a hands-on fashion (Common Core Standards, 2013, California Department of Education: 70).…
Common Core Standards. California Department of Education. ca.gov. [21 Oct 2013] http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf
Common Core Standards. Official Website. [21 Oct 2013]
The system also has to undergo thousands of cycles and vibrations and needs to be able to stand up to the same reliability standards as the rest of the components on the bike.
Conventional and Proven ear Suspension Designs
Given all of the previously mentioned considerations, the design itself is important in making sure the rider and the manufacturer are getting the most out of the system.
The Fox acing Homepage (2011) has some excellent examples of both the strut style rear suspension as well as the shock with spring and strut combination system. The latter is typically reserved for use on higher-end advanced bikes since these systems are costlier and requires more maintenance. To be more specific, the Van C product represents the higher-end strut and spring combination while the Float design is a basic, oil dampened design for use on more entry-level designs. The Van C model is…
Bu, Yan; Tian Huang, Zhongxia Xiang, Xiaofan Wu and Chun Chen. (2010). "Optimal design of mountain bicycle based on biomechanics." Transactions of Tianjin University,
Volume 16, Number 1, 45-49.
DT Swiss Homepage. (2011). Accessed Jan. 5 at:
When fully loaded the weight distribution is 40% on the front axle and 60% on the rear axle. Given the likely adhesion conditions, the powertrain will drive all axles.
Suspension geometry design and assessment
When the vehicle is cornering, each wheel must go through a turning circle. The outer turning circle, is to our main subject of interest. This calculation is never precise because when a vehicle is cornering the perpendiculars via the centres of all wheels never intersect at the curve centre point (Ackermann condition). Additionally, while the vehicle is moving, certain dynamic forces will always arise that will eventually affect the cornering manoeuvre (MAN,2000).
The formula used.
Vehicle Model T31, 19.314 FC
Wheelbase lkt = 5000 mm
Front axle Model V9-82L
Tyres 315/80 . 22.5
Wheel 22.5 x 9.00
Track width s = 2058 mm
Scrub radius r0 = 58 mm
Inner steer angle…
Catapillar (2001). 769D-Off-Highway Truck. http://xml.catmms.com/servlet/ImageServlet?imageId=C199012
Ivanov V, Shyrokau B, Augsburg K, Vantsevich V (2010)System Fusion in Off-Road Vehicle Dynamics Control09/2010; in proceeding of: Joint 9th Asia-Pacific ISTVS Conference, at Sapporo, Japan
MAN (2009).vehicle calculations
Rafael, M, a. Lozano, J. Cervantes, V. Mucino, C.S. Lopez-Cajun (2009).A method for powertrain selection of heavy-duty vehicles with fuel savings. International Journal of Heavy Vehicle Systems
raster graphics, wire-frame and 3D modeling performance, and refresh rates of their screens. What began to occur in the company's culture as a result of this focus on graphics performance and CPU acceleration was a bifurcation or splitting of product lines. At the high end Apple was gradually turning into a workstation company that could easily challenge Sun Microsystems or Silicon Graphics for supremacy of graphically-based calculations. At the low-end, the company was pursuing an aggressive strategy of dominating special-purpose laptops.
This strategy was entirely predicated on the core metrics of price/performance on hardware defining a culture that put pricing above all else, paradoxically nearly driving the company out of business during this period. The focus on metrics that were meant to purely define the Apple competitive advantage made the company descend into pricing wars with competitors whose business models were much more attuned to pricing competition. The metrics the…
Berling, Robert J. (1993). The emerging approach to business strategy: Building a relationship advantage. Business Horizons, 36(4), 16. Accessed from: http://www.berlingassociates.com/features/horizon.pdf
Keidel, Robert . THE GEOMETRY OF STRATEGY New York: Routledge, May 2010
Sakakibara, Kiyonori, Lindholm, Chris, & Ainamo, Antti. (1995). Product development strategies in emerging markets: The case of personal digital assistants. Business Strategy Review, 6(4), 23. Accessed from http://www.soc.utu.fi/laitokset/iasm/SakakibaraEtAlPDAs1995.pdf
Frank Gehry has become a leading architect noted for his innovative structures using industrial materials in new ways and with a certain deconstructivist approach to architecture. Philip Johnson, the dean of American architecture and a power since the 1930s, more recently joined with other architects who have been shattering all the rules, leaving behind symmetry and classic geometry in favor of distorted designs, twisted beams, and skewed angles. Johnson in 1988 showcased this style in a program at the Museum of Modern Art, and he called the show "Deconstructivist Architecture." Among the designers following this approach are Frank Gehry of California or ernard Tschumi from France and Switzerland. Johnson says of this new architecture that it evokes "the pleasures of unease." These ideas have been utilized directly by Johnson in his design for the Canadian roadcasting Corporation building in Toronto. Today, Gehry is probably the foremost proponent if this approach.…
Arnold, Dana. Art History: A Very Short Introduction. Oxford: Oxford University Press, 2004.
Ballantyne, Andrew. Architecture: A Very Short Introduction. Oxford: Oxford University Press, 2002.
Bletter, Rosemarie Haag. "Frank Gehry's Spatial Reconstructions." In The Architecture of Frank Gehry. New York: Rizzoli, 1986.
Celant, Germano. Frank Gehry: Buildings and Projects. New York: Rizzoli, 1985.
Colleen -- but then again, when you're dealing with food services, every day's a long day. As she made her way toward the stairs and away from the brooding purgatory that is the HUB (name of cafeteria), shutting off the lights behind her like a row of fluorescent dominoes, the clock on the wall read "10:45." The sound of the door shutting at the top of the stairwell signaled the end of another day at the HUB.
Actually, perhaps this was not true. Just as the door was shutting above, the lights down below flickered on once again to reveal a ghostly line of customers stretching from the "Pizza Hut" station to the cash register. Near the end of the line, Mohandas Gandhi stood with a cup of tea and a veggie wrap balanced on his tray. Martin Luther King stood next to him, his tray empty except for a…
SIC (School of the rt Institute of Chicago) personal statement
One of the most exciting aspects of studying the field of design and architecture is its collaborative aspect. No building is constructed alone; rather the structure that is produced is the result of the combined effort of designers, architects, sponsors, and ideally the community where the building is going to be located. This is why I am so eager to become part of the SIC -- (School of the rt Institute of Chicago) architecture program with a concentration in interior design. The spirit of the school is a community where I know I can feel at home in the next four years. Both inside and outside of the classroom, the people with whom I work with at SIC will shape my vision as a professional throughout the duration of my career, long after I have left the institution. The school…
As an international student who received an undergraduate degree in interior design in Saudi Arabia, I am eager to attend a school that is global in focus, yet also grounded in the independent spirit of the United States. I seek a fresh perspective on the world of architecture and I believe that I could be an invaluable asset in the classroom, sharing with my fellow students the worldview of someone who has grown up outside of the United States. The culture of the industry is different where I worked, as are the logistical challenges of climate and functionality. I seek to learn from my fellow classmates and teachers and see things from their perspective, as I hope they will learn from mine.
In Saudi Arabia, there is great wealth, which has supported the architectural industry. I have worked as an architectural assistant and have come to understand the degree to which financial backing is essential to realize the vision of every architect, as I have seen some potentially great projects flounder for want of backing. Learning how to make a project financially feasible yet still suit the vision of the original designer was one of the core challenges I witnessed in my work.
Attending SAIC would not be my first time living abroad. My desire to become an architect was first formulated when I was training at a workshop in architectural geometry in Vienna during the summer of 2011. My ultimate goal in getting a degree from SAIC would be to open up my own architectural firm with a global focus and an emphasis on cutting edge design. I still require more knowledge of the field to realize my dream, however. Given that my undergraduate degree was interior design, I seek out a sounder theoretical and practical base of academic knowledge in the field of architecture. I also desire to make contacts within this very difficult and rewarding industry. And finally, I seek a 'safe space' to try out experimental designs that I did not have the opportunity to propose when I was merely working as an assistant. Attending SAIC would give me the courage to dream and reach for the sky, as high as the tallest buildings I could imagine yet also provide my career and knowledge base with a sound framework and foundation. It would build my knowledge as an architect -- from the ground up.
Lesson for Children With Learning Disabilities
Developing a Lesson for Children with Learning Disabilities
Learning disability is a term misused severally. In essence, it applies to students who have different learning challenges. Most people associate learning disability to the development of a child, thus assuming that it is a short-term condition and disappears as the person matures. The accepted definition, provided by the National Adult Literacy and Learning Disability Center states that; learning disability is generic and refers to a composite group of disorders that become evident in the person; through observing that they have challenges in the acquisition and use of speaking, listening, reading, reasoning and execution of mathematical concepts, as well as, understanding social skills. As teachers process the learning procedure in class, they encounter various children with varied challenges, which constitute the learning disorders (Aster & Shalev, 2007). Thus, they have the obligation to accommodate those children…
Aster, M.G. v., M.D., & Shalev, R.S., M.D. (2007). Number development and developmental dyscalculia. Developmental Medicine and Child Neurology, 49(11), 868-73. Retrieved
Canizares, D.C., Crespo, V.R., & Alemany, E.G. (2012). Symbolic and non-symbolic number magnitude processing in children with developmental dyscalculia. The Spanish Journal
of Psychology, 15(3), 952-66. Retrieved from http://search.proquest.com/docview/1439791245?accountid=458
misconception in people that having an aneurysm means bleeding in the brain. An aneurysm is in fact a balloon-like swelling in a blood vessel that can affect any large vessel in your body; these larger vessels being arteries. Aneurysms pose a risk to health from the potential for rupture, clotting, or dissecting. It is the pressure of the blood passing through a weak part of the blood vessel that forces it to bulge outwards, forming a sort of a blister. If the sac that is formed extends the artery too far, the vessel may burst, causing death by bleeding. upture of an aneurysm in the brain causes stroke, and rupture of an aneurysm in the abdomen causes shock. (THIJ, 2001)
Aneurysms are the cause of many deaths because they are usually silent until a medical emergency occurs. "One author has referred to an AAA as a "U-boat" in the belly,…
Lieber, B.B, and Wakhloo, A.K. (1996) Optimization of Stents for cerebrovascular disorders using Computational Fluid Dynamics and Particle Image Velocimetry. http://www.mae.buffalo.edu/research/summaries/1996 / (Accessed April 7, 2002)
Minyard. Andrea N, MD, and Parker, Joseph C. JR., MD, (1997) Intracranial Saccular (Berry) Aneurysm: A Brief Overview. http://www.sma.org/smj/97july2.htm , Louisville, Ky. (Accessed April 7, 2002)
Petito CK (1993) Cerebrovascular diseases. Principles and Practice of Neuropathology. St. Louis, Mosby-Year Book Inc., pp 436-458
Stehbens WE (1995) Aneurysms. Vascular Pathology. London, Chapman and Hall Medical, pp 379-400
Euclid -- 323-285 B.C was a historical figure who taught at Alexandria in Egypt. There are three hypotheses revolving around Euclid's life. The first is that he wrote his magnum opus the Elements as also contributed a lot of other works. Another interesting hypothesis is that Euclid was a member of a group of mathematicians working at Alexandria with each one contributing to writing the 'complete works of Euclid', with the group engaged in writing books under the name of Euclid also after his demise. The third hypothesis is that Euclid of Megara lived roughly 100 years prior to Euclid of Alexandria. A team of mathematicians wrote the complete works of Euclid and took Euclid's name from Euclid of Megara. The proof surrounding the first hypothesis is significant that he wrote his magnum opus the Elements as also contributed a lot of other works. Scanty evidence is there that deny…
Dietz, Elizabeth. "Euclid 323-285 B.C. Biography" Retrieved from http://www.albertson.edu/math/History/edietz/Classical/biography.htm
Accessed on 8 August, 2005
Dietz, Elizabeth. "Euclid 323-285 B.C: Discoveries" Retrieved from http://www.albertson.edu/math/History/edietz/Classical/discoveries.htm
Accessed on 8 August, 2005
This study focuses on the rise and significance STEP-NC as the most efficient model to transfer knowledge and communication on different CAD and CAM structures to improve the product design and overall project management. The paper will be divided into six chapters: 1) ntroduction chapter which will include the statement of the problem, significance of the problem, purpose and scope of the study, the relevant definitions, the assumptions of the study as well as its limitations; 2) Literature Review chapter which will present an analysis of all the prior studies done on similar topics; 3) chapter variables and hypothesis creation where all the variables (dependent and independent) will be listed; 4) methodology chapter which will include data collection processes and the qualitative procedures used; 5) results and presentation of data chapter which will include all statistical calculations and the regression analysis; 6) conclusion chapter which will includes discussion…
In this study, while both can be very useful, we will use the deductive approach and form hypothesis/research questions. The deductive approach will prove to be more useful as it will help us determine all the necessary technological changes that are occurring and support our decision of why choosing Step NC was the right form of technology. Furthermore, it will also help us solidify and strengthen our hypothesis as the study and data collection process moves into the latter stages. This will perhaps be most advantageous for this study as in reality, it is probably not very challenging to arrange the necessary grounds of reasoning and incorporate it into a single minded approach or aim of the study that consistently works its way from general ideas to specific observations. The fact of the matter is that even if this approach presents a lot of limitations and constraints, we might still be able to conclude some definite trends and patterns in the information that can be eventually helpful in developing up newer theories (Trochim, 2006) and help us form better recommendations for future research on the given topic.
Research Horizon and timeline
When it comes to the design of any research, time typically plays the part of
He made three main contributions to the theory of numbers: congruence theory, studies on the separation of the circle into equal parts, and theory of quadratic forms.
Algebra and Analysis
Up to Gauss's time, no one had been able to prove that every algebraic equation has at least one root. Gauss offered three proofs. And he modified the definition of a prime number.
We discussed briefly his assistance to astronomers in relocating Ceres. His success in this effort spurred him to develop the mathematical methods he used further. In 1809 his Theoria motus corporum coelestium used the method of least squares to determine the orbits of celestial bodies from observational data. In arguing his method, Gauss invented the Gaussian law of error, or, as we know it today, the normal distribution.
Mathematicians, for centuries, had been attempting to prove Euclid's postulate concerning parallels (the sum of…
Bell, E. (1986). Men of mathematics. New York: Simon Schuster.
Dunnington, G., Gray, J., & Dohse, F. (2004). Carl Frederich Gauss: titan of science.
Washington D.C.: MAA.
Encyclopedia of World Biography. (2005). Karl Friedrich Gauss biography. Retrieved September 29, 2009, from bookrags.com (Encyclopedia of World Biography): http://www.bookrags.com/biography/karl-friedrich-gauss/
Euclid's Fifth Postulate
Philosophical and Logical Problems Contained in Euclid's Fifth Postulate
Euclid gave the world much of the information it has on planar geometry in his five postulates. hile the first four are relatively easy to understand, the fifth one is very difficult in relation to the others. It is this fifth postulate that many people feel can never be proven. There are those that say it is simply incorrect, those that say it's both true and false, and others that say there is no possible way to prove it, and Euclid himself may have realized that the task was impossible. His fifth postulate states:
If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles."
Bennett, Andrew G. The Axiomatic Method. 2000. Math 572 Home. 2 December 2002. http://www.math.ksu.edu/math572/notes/824.html .
Bogomolny, Alexander. The fifth postulate: attempts to prove. 2002. Cut the Knot. 2 December 2002. http://www.cut-the-knot.com/triangle/pythpar/Attempts.shtml .
Parallel lines and planes. 2002. Connecting Geometry. 2 December 2002. http://www.k12.hi.us/~csanders/ch_07Parallels.html .
eview two separate Internet resources. In a separate document, write two or three paragraphs, per source, evaluating the resource by at least three of the following standards: age-appropriateness, the accuracy of the content, usability of the resource in the classroom, and the accommodation of different learning styles. Cite the resources you evaluated. http://www.scienceu.com/geometry/classroom/buildicosa/index.html
This resource is very good for my lesson plan because it focuses on engaging students in the actual properties of a polygon. It is age appropriate at the 8th grade level for several reasons. First it is a very sophisticated project in that the instructions for constructing this polygon is complex, therefore it takes high cognitive skills in order to build such a project. It is ambitious enough that the entire class is engaged and that at this grade level students will have enough knowledge to be both challenged in its construction while still feeling that they…
Resources section of the template in APA format.
Aristotle used mathematics in many of his other studies, as well. Another writer notes, "Aristotle used mathematics to try to 'see' the invisible patterns of sound that we recognize as music. Aristotle also used mathematics to try to describe the invisible structure of a dramatic performance" (Devlin 75-76). Aristotle used mathematics as a tool to enhance his other studies, and saw the value of creating and understanding theories of mathematics in everyday life and philosophy.
During his life, Aristotle also worked with theories developed by Eudoxus and others, and helped develop the theories of physics and some geometric theories, as well. Two authors quote Aristotle on mathematics. He writes, "These are in a way the converse of geometry. While geometry investigates physical lines but not qua physical, optics investigates mathematical lines, but qua physical, not qua mathematical" (O'Conner and obinson). He also commented on infinity, and did not believe that…
Devlin, Keith E. The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are like Gossip. 1st ed. New York: Basic Books, 2000.
Lane, David. "Plato and Aristotle." The University of Virginia's College at Wise. 2007. 18 June 2007. http://www.mcs.uvawise.edu/dbl5h/history/plato.php
O'Connor, John J. And Edmund F. Robertson. "Aristotle on Physics and Mathematics." Saint Andrews University. 2006. 18 June 2007. http://www-history.mcs.st-andrews.ac.uk/Extras/Aristotle_physics_maths.html
Robinson, Timothy a. Aristotle in Outline. Indianapolis: Hackett, 1995.
Instead, spatial reasoning appears to be based on environmental inputs and old-fashioned cognitive development.
Why this should come as such a surprise to some researchers is uncertain. Core knowledge theorists claim that infants almost immediately express certain types of knowledge. But this suggestion assumes two things: one, that it is possible to measure infant cognition at the moment of birth; and two, that infants are incapable of learning before they are born. On the matter of the former point, it seems apparent that logistical and ethical concerns would make it exceedingly difficult, if not outright impossible, to test infant cognition immediately after birth. With regards to the second issue, we already have evidence that infants are capable of basic learning while still in the womb. Though developmentally unfinished, the basic sensory organs that the fetus develops permit it to learn information about its environment. Lecuyer (2006) reminds us that it…
Dehaene, S., Izard, V., Pica, P., and Spelke, E.S. (2006). Core knowledge of geometry in an Amazonian indigene group. Science, 311, pp. 381-384.
Haith, M.M. (1998). Who put the cog in infant cognition? Infant Behavior and Development, 21(2), pp. 167-179.
Hespos, S.J. And Spelke, E.S. (2004, July 22). Conceptual precursors to language. Nature, 430, pp. 453-456.
Hofsten, C., Feng, Q., and Spelke, E.S. (2000). Object representation and predictive action in infancy. Developmental Science, 3(2), pp. 193-205.
Finally, efficiency is te fourt area of te evaluation. Efficiency is an important area to consider because it allows students te optimum level of education and understanding witout eiter oversimplifying to te level tat tey lose interest, or overburdening so tat students feel tere is too muc material to cover. Efficiency measures ow muc necessary and unnecessary information is included witin te lesson, and quality materials contain very little unnecessary information. Anoter aspect of efficiency is weter or not a sufficient number of examples and practice problems were included witin te lesson itself. Finally, efficiency implies tat te lessons provided witin te material will actually elp tem in understanding te material.
B) te objective for Mat 8t grade level tat I will pursue is understanding of polygons witin geometry and ow tey function.
ttp://www.scienceu.com/geometry/classroom/buildicosa/index.tml would not ultimately use tis as part of my lesson plan because of several reasons.…
http://www.scienceu.com/geometry/classroom/buildicosa/index.html would not ultimately use this as part of my lesson plan because of several reasons. While the procedures of this project is very unique in that it allows students to interact with the project, it does a poor job of conveying the actual objective of the course material. Since building such a polygon requires a significant amount of time, rather than being a tool to facilitate learning, the building and hands on nature of this project actually becomes the bulk of the educational experience. Thus students have a great time during the construction but do not gain an understanding of the course material. Thus this is an inefficient use of class time and does not meet the design and efficiency standards of the evaluation. http://www.scienceu.com/geometry/articles/tiling/index.html
This following exercise also focuses on the importance of geometric design and formulation of shapes. However, it lacks a specific focus and attempts to teach too much in a given timeperiod. There are many different things to analyze when concerning tiles within geometry therefore this lesson plan is a little too complicated. However, it does allow students to interact with its model and develop an interest in learning more about the geometric problems presented in this analysis. Overall I would recommend this project. http://www.scienceu.com/geometry/activities/tetrapuzzles/index.html
This exercise is the best of the three reviewed because it meets almost all of the criteria of the material evaluation. Not only does it present a simple and clear concept, how a cube is made and the geometric properties of this polygon, but more importantly it does not take a significant amount of time. It emphasizes the point but does not belabor it and thus allows students to understand and interact without making the exercise redundant and too complicated. This will arouse interest, but clearly emphasize the lesson plan.
Euclid of Alexandria: 325 .C. ~ 265 .C. (?)
The dates are not exact as little is known about Euclid's life. It is generally believed that he studied under the students of Plato and it is known that he established a school of mathematics and taught at the library in Alexandria. His most well-known work is The Elements, which is a wonderfully organized development of the plane and solid geometry, geometric algebra, theory of proportions, number theory, and the theory of irrational numbers known then. The work is divided into 13 books and contains 465 propositions. eginning in ook I with 5 postulates, 5 common notions, and 23 definitions, Euclid develops the basic properties of plane geometry from the construction of an equilateral triangle in Proposition 1 to his beautiful proof of the Theorem of Pythagoras in Proposition 47 (the book closes with a proof of the converse of the…
Gillispie, Charles C. ed. The Dictionary of Scientific
Biography, 16 vols. 2 supps. New York: Charles Scribner's
Sons, 1970-1990. S.v. "Euclid: Life and Works" by Ivor
The Jansenists were condemned by the pope in 1653 and 1713. Characteristic beliefs of the school included "the idea of the total sinfulness of humanity, predestination, and the need for Christians to rely upon a faith in God which cannot be validated through human reason. Jansenism often, but it continued to have a strong following among those who tended to reject papal authority, but not strong moral beliefs" ("Jansenism," About.com, 2008).
After his final conversion, Pascal moved to the Jansenist monastery in Port Royal. He had already convinced his younger sister to move to the nunnery in the same location. It was there he penned the work that would contain his famous wager, the famous Pensees. He continued to live at the monastery until his death in 1662, worn out, it was said, "from study and overwork," although later historians think that tuberculosis stomach cancer was the likely culprit (Ball…
Ball, Rouse. "Blaise Pascal (1623-1662)." From a Short Account of the History of Mathematics. 4th edition, 1908. Excerpt available on 7 Apr 2008 at http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html
Blaise Pascal." Island of Freedom. 7 Apr 008. http://www.island-of-freedom.com/PASCAL.htm
Blaise Pascal." Oregon State University. 7 Apr 008. http://oregonstate.edu/instruct/phl302/philosophers/pascal.html
Hajek, Alan. "Pascal's Wager." The Stanford Internet Encyclopedia of Philosophy. First Published Sat May 2, 1998; substantive revision Tue Feb 17, 2004. 8 Apr 2008. http://plato.stanford.edu/entries/pascal-wager/#4
An Eigenface representation (Carts-Power, pg. 127, 2005) created using primary "components" (Carts-Power, pg. 127, 2005) of the covariance matrix of a training set of facial images (Carts-Power, pg. 127, 2005). his method converts the facial data into eigenvectors projected into Eigenspace (a subspace), (Carts-Power, pg. 127, 2005) allowing copious "data compression because surprisingly few Eigenvector terms are needed to give a fair likeness of most faces. he method of catches the imagination because the vectors form images that look like strange, bland human faces. he projections into Eigenspace are compared and the nearest neighbors are assumed to be matches." (Carts-Power, pg. 127, 2005)
he differences in the algorithms are reflective in the output of the resulting match or non-match of real facial features against the biometric database or artificial intelligence generated via algorithm. he variances generated by either the Eigenspace or the PCA will vary according to the…
Thus, finding a principal subspace where the data exist reduces the noise. Besides, when the number of parameters is larger, as compared to the number of data pints, the estimation of those parameters becomes very difficult and often leads to over-learning. Over learning ICA typically produces estimates of the independent components that have a single spike or bump and are practically zero everywhere else
. This is because in the space of source signals of unit variance, nongaussianity is more or less maximized by such spike/bump signals." (Acharya, Panda, 2008)
The use of differing algorithms can provide
Imagining architecture as the structure upon which meaning grows and contributes to the phenomenon of a place is particularly helpful when investigating Holl's Linked Hybrid, because the design expresses a desire to meld the objective, concrete of the building itself to the experience of the residents living and moving within.
Construction on Linked Hybrid began in 2003 and completed in 2009, when Holl's design won the Council on Tall Buildings and Urban Habitat's award for Best Tall Building (CTBUH 2009). Part of a slew of new developments born out of Beijing's revitalization as a result of its hosting of the 2008 Olympic games, Linked Hybrid is a mixed-use development consisting of "a ring of eight 21-story towers, linked at the 20th floor by gentling sloping public sky bridges, lined with galleries, cafes, restaurants, bars and shops" (Busari 2008). Each tower is rectangular, with some towers being additionally linked at the…
Busari, Stephanie. CNN, "Beijing embraces Brave New World of buildings." Last modified June
24, 2008. Accessed November 6, 2011.
Council on Tall Buildings and Urban Habitat, "2009 Awards." Last modified October 2009.
Then, by beginning with the idea that there may or may not be a chair present at all, one can begin building on those truths that remain to establish more truths and eventually establish the presence of the chair.
Descartes uses such reasoning not only to establish the presence of those things that can be verified by the use of the senses, but also to establish the existence of God and the immortality of the soul. Descartes begins with the premise that neither mountains nor valleys may exist, but that if they do exist, then "a necessary attribute of a mountain is that it be adjacent to a valley" (Burnham and Fieser). Descartes acknowledges that the same could be said of the existence of God:
In the same way, even though the concept of supremely perfect being necessarily possesses certain attributes, it doesn't follow that this being exists. It only…
Burnham, Douglas and James Fieser. "Rene Descartes." The Internet Encyclopedia of Philosophy. 2001. University of Tennessee at Martin. 4 Mar. 2005 http://www.utm.edu/research/iep/d/descarte.htm .
Chew, Robin. "Rene Descartes: Philosopher." Lucidcafe. 2005. Lucidcafe. 4 Mar. 2005 http://www.lucidcafe.com/lucidcafe/library/96mar/descartes.html.
Descartes, Rene. "Meditations." Eds. David B. Manley and Charles S. Taylor. Descartes'
Meditations. 1996. Wright State University. 4 Mar. 2005 http://www.wright.edu/cola/descartes/ .
Hypatia of Alexandria, daughter of Theon. Specifically, it will examine the life of Hypatia, especially her mathematical accomplishments. Hypatia was the first female mathematician that left a record that historians can trace. She was a philosopher, mathematician, and teacher who lived in Alexandria, Egypt from about 350 to 415 A.D. She was the daughter of Theon, a renowned mathematician and head of the library in Alexandria.
Historians do not agree on the year Hypatia was born. Some estimate it at around 355, while others place it as late as 370. What is known of Hypatia is that she was extremely influential in mathematics and philosophical thought. Hypatia was born in Alexandria and most historians believe she spent her entire life there. Some historians believe Hypatia studied mathematics in Athens, and then traveled through Europe (Coffin, 1998, p. 94), while others believe her father taught her most of what she knew…
Coffin, L.K. (1998). Hypatia. In Notable women in mathematics: A biographical dictionary, Morrow, C. & Perl, T. (Eds.) (pp. 94-96). Westport, CT: Greenwood Press.
Osen, L.M. (1974). Women in mathematics. Cambridge, MA: MIT Press.
Russell, N. (2000). Cyril of Alexandria. London: Routledge.
Williams, Robyn. (1997). Ockham's razor. Retrieved from the ABCNet.au Web site: http://www.abc.net.au/rn/science/ockham/or030897.htm 8 Aug. 2005.
[I also had my students write how they would say it out loud when naming it. Example: "Line AB or line segment AB is perpendicular to line segment CD."] Below is information on how students should label rays, lines, etc.
1. ay - the endpoint letter first, then a second point with a line ending in an arrow over the two letters, pointing to the right.
2. Point - a dot and then the point's letter.
3. Line - Two points on the line with a line with arrows in both directions above the letters.
4. Segment - the two endpoint letters of the segment with a line, no arrows, above the two letters
5. Intersecting - (AB x BC) the AB and BC would have a line or a line with arrows above them to show what figures they were. The x stands for intersects.
6. Parallel - (AB…
Baiker, K. And J. Robinson. (2004). Origami Math: Easy-to-Make Reproducible Activities that
Build Concepts, Skills, and Vocabulary in Geometry, Fractions, measurement, and More.
Minneapolis: Scholastic Books.
Bedford, M. (2007). Memorization: The Neglected Key to Learning. Efficacy Institute. Retrieved from: http://www.efficacy.org/Resources/TheEIPointofView / tabid/233/ctl/ArticleView / mid/678/articleId/84/Memorization-the-Neglected-Key-to-Learning.aspx
Tom Shulich ("ColtishHum")
A comparative study on the theme of fascination with and repulsion from Otherness in Song of Kali by Dan Simmons and in the City of Joy by Dominique Lapierre
In this chapter, I examine similarities and differences between The City of Joy by Dominique Lapierre (1985) and Song of Kali by Dan Simmons (1985) with regard to the themes of the Western journalistic observer of the Oriental Other, and the fascination-repulsion that inspires the Occidental spatial imaginary of Calcutta. By comparing and contrasting these two popular novels, both describing white men's journey into the space of the Other, the chapter seeks to achieve a two-fold objective: (a) to provide insight into the authors with respect to alterity (otherness), and (b) to examine the discursive practices of these novels in terms of contrasting spatial metaphors of Calcutta as "The City of Dreadful Night" or "The City of…
Barbiani, E. (2005). Kalighat, the home of goddess Kali: The place where Calcutta is imagined twice: A visual investigation into the dark metropolis. Sociological Research Online, 10 (1). Retrieved from http://www.socresonline.org.uk/10/1/barbiani.html
Barbiani, E. (2002). Kali e Calcutta: immagini della dea, immagini della metropoli. Urbino: University of Urbino.
Cameron, J. (1987). An Indian summer. New York, NY: Penguin Travel Library.
Douglas, M. (1966). Purity and danger: An analysis of concepts of pollution and taboo. New York, NY: Routledge & K. Paul.
Menorah and Its Symbolism to the Jewish Community
The menorah, originally a seven-branched candelabrum used in the Temple, is one of the oldest symbols used by the Jewish faith. In contrast to the ancient menorah of Exodus is the Chanukkah menorah with eight candles, which is used today. The use of eight candles celebrates the miracle that a small amount of oil lasted for eight days.
Today's nine-branched menorah is used to celebrate Chanukkah, the festival of lights which occurs near the winter solstice. A ninth candle, the shamesh, is used to light the other eight, one night at a time, for the eight days of Chanukkah.
The Symbolism of the Menorah
It has been said that the menorah is a symbol of the nation, in this case meaning the nation of Israel. The term "nation" is used in the classical sense, meaning a group of people with a shared…
Allen, Mike, et al. "Subject: Question 11.9.5: Symbols: What is a Menorah?" Internet FAQ Consortium. 10 March 2002. http://www.faqs.org/faqs/judaism/FAQ/05-Worship/index.html
Calter, Paul. "Number Symbolism: Menorah." Geometry in Art and Architecture Unit 4: Dartmouth College. 1998. http://www.dartmouth.edu/~matc/math5.geometry/unit4/unit4.html
Hirsch, S.R. "The Menorah." The Hope. 6 March 2001. http://www.thehope.org/menorah.htm
Huberman, Ida. "The Seven-Branched Menorah: An Evolving Jewish Symbol." Jewish Heritage Online Magazine. 2001. http://www.jhom.com/topics/seven/menorah.html
lives of Archimedes and Carl Friedrich Gauss, two of the greatest mathematicians of all time, through a point by point comparison of their childhood and education, mathematical contributions and the influence their work has on the science of mathematics.
Childhood and Education
Archimedes (287 BC to 212 BC) lived most of his life in Syracuse, Greece. This son of an astronomer and mathematician was born into a distinguished family and was able to comfortably devote his life to mathematical research.
Carl Friedrich Gauss (1777-1855) was born into a humble German family. His early mathematical promise marked him as a prodigy and eventually earned him admission to university.
Major Mathematical Ideas
The mathematical work of Archimedes centered on the theoretical, particularly geometry. His greatest mathematical contribution involved measuring areas and segments of plane and conic sections.
Gauss's work centered on number theory. Unlike Archimedes, Gauss also used ventured into applied mathematics…
Archimedes," in Guide to the History of Calculus. Retrieved 30 November 2002 from http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/archimedes.htm
Bell, E.T. Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincare. New York and London: Simon and Schuster, 1965.
Boyer, Carl B. A History of Mathematics, 2nd ed. New York: John Wiley and Sons, 1991.
Gauss," in Guide to the History of Calculus. Retrieved 30 November 2002 from http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/gauss.htm
Education: Teaching Math to Students ith Disabilities
orking with students with disabilities (SD) can be quite challenging, especially for teachers working on a full-time basis. Almost every classroom today has one or more students dealing with either an emotional, educational, or physical disability; and teachers are likely to find themselves looking for resources or information that would enable them teach all their students in the most effective way. There are numerous special-education websites from which teachers and instructors can obtain information or lessons on teaching their respective subjects. Five websites available to the math special education teacher have been discussed in the subsequent sections of this text.
Teachers Helping Teachers: http://www.pacificnet.net/~mandel/
This online resource provides teaching information for all teachers, with a 'Special Education' segment that provides a number of activities meant specifically for instilling basic conceptual skills in learners with special needs. The activities are submitted by…
Oldham County Schools. "Instructional Resources for Math." Oldham County Schools, n.d. Web. 17 August 2014 http://www.oldham.k12.ky.us/files/intervention_resources/Math/Instructional_Resources_for_Math.pdf
Starr, Linda. "Teaching Special Kids: Online Resources for Teachers." Education World, 2010. Web. 17 August 2014 from http://www.educationworld.com/a_curr/curr139.shtml
Staircase ramps which are comprised of steep and narrow steps that lead up one face of the pyramid were more in use at that time with evidence found at the Sinki, Meidum, Giza, Abu Ghurob, and Lisht pyramids respectively (Heizer).
A third ramp variation was the spiral ramp, found in use during the nineteenth dynasty and was, as its name suggests, comprised of a ramp covering all faces of the pyramids leading towards the top. Reversing ramps zigzag up one face of a pyramid at a time and would not be used in the construction of step pyramids, while lastly interior ramps that have been found within the pyramids of Sahura, Nyuserra, Neferifijata, Abusir, and Pepi II (Heizer, Shaw).
Ancient Greek architecture exists mainly in surviving temples that survive in large numbers even today and is tied into Roman and Hellenistic periods which borrowed heavily from the Greeks.…
Ackerman, J.S. "Architectural Practice in the Italian Renaissance." Journal of the Society of Architectural Historians (1954): 3-11.
Alchermes, Joseph. "Spolia in Roman Cities of the Late Empire: Legislative Rationales and Architectural Reuse." Dumbarton Oaks Paper (1994): 167-178.
Allen, Rob. "Variations of the Arch: Post -- and lintel, Corbelled Arch, Arch, Vault, Cross-Vault Module." 11 August 2009. Civilization Collection. 5 April 2010 .
Anderson, James. "Anachronism in the Roman Architecture of Gaul: The Date of the Maison Carree at Nimes." Journal of the Society of Architectural Historians (2001): 68-79.
Students will work together in groups of at least four to answer the questions on the exercise. Then, they will be required to present their findings to the class in a short, five-minute group presentation.
3. In order to familiarize students with the concepts and properties of triangles, quadrilaterals and other polygons, students will access the website, (http://www.explorelearning.com). Under the Mathematics Gizmos section of this website, students will select the following: Grade 9-12 > Go > Geometry. Students will be required to perform all of the interactive activities in the Triangles and Quadrilateral and Polygons sections. Students will be required to write a brief explanation of understanding for each exercise.
4. Students will pick up where they left off in the first semester by continuing to use the website (http://www.explorelearning.com) for the entire second semester. The will follow the similar path that they did during the third unit: Grade 9-12…
Monroe, Kara, Wilson, Margaret Mary, Bergman, Kathleen and Marisa Nadolny.
(2009). High School Math Made Simple (2009/2010 ed.). New Jersey: TutaPoint,
Source: Hockett 1940:264
This land surveying method proved to be highly accurate, a feature that was in sharp contrast to the methods that had been used in some American colonies such as Virginia that allowed the use of so-called "indiscriminate locations," a practice that caused an enormous amount of land boundary disputes (Hockett 1940). hile the land surveying method used pursuant to the Land Survey Ordinance of 1785 was partially based on techniques that had evolved in New England, the origins of some of the features included in the legislation remain unclear (Hockett 1940). Notwithstanding this lack of historical precision concerning the origins of the features contained in the Land Survey Ordinance of 1785, the land surveying methods it set forth were so efficient and effective that the same techniques were applied to the rest of the country as westward expansion continued, eventually dividing all of the public lands in…
Allen, John L. North American Exploration, Vol. 3. Lincoln, NE: University of Nebraska Press,
Ariel, Avraham and Nora Ariel Berger. Plotting the Globe: Stories of Meridians, Parallels, and the International Date Line. Westport, CT: Praeger, 2006.
Black's Law Dictionary. St. Paul, MN: West Publishing Co., 1991.
6). Pi is, therefore, on the level of philosophical discourse because many other mathematical problems elucidated by the ancients have since been solved. Arndt et al. claim that pi is "possibly the one topic within mathematics that has survived the longest," (6). Initial pi explorations may have been prehistoric. Ancient Egyptians and Mesopotamians later developed systems of writing and mathematics that enabled rigorous investigations into crucial problems. In 1650 BCE, ancient Egyptian scribe Ahmes recorded what are likely the first formulas for pi. The formulas are written on what is referred to as the Egyptian hind Papyrus (Eymard, Lafon & Wilson).
The Ahmes formulas relate the circle to the square, foreshadowing further investigations into pi by the Greeks. The Egyptians were therefore the first to record attempts to "square the circle," or relate the area of a square to that of a circle in search of a constant variable that…
Arndt, Jorg, Haenel, Christoph, Lischka, Catriona & Lischka, David. Translated by Catriona Lischka, David Lischka. Springer, 2001
Beckman, Petr. A History of Pi. Macmillan, 1971.
Berggren, Lennart, Borwein, Jonathan M. & Borwein, Peter B. Pi, A source book. Springer, 2004.
Blatner, David. The Joy of Pi. Walker, 1999.
The actual construction was the work of ast (Villa ast). Similar to his previous creation, classicism is captured within the "fluted pillars" and "lateral projections." Numerous ornaments, such as pearl, egg-and-dart, and leaf moldings, are incorporated. Notable sculptures include one by Anton Hanak, above the tall windows on the right side of the house. Hoffmann's geometric motifs are also detected through the verticals and latticework. The furnishings also bear geometric grid patterns. Specific features include square flowers and lozenge patterns with complementary colors of white and black (white and gold is used as well). An overall impression of lightness is also achieved, with high stairwells, freestanding marble columns, and decorative glasswork. Notably, the design of the garden was intended to give off an exclusive impression. The terraces (some semi-cylindrical, some not) and ground level disparities instigate a conservative sense. In contrast, freedom is also employed with the rich modulations of…
The reason for its existence can be affiliated with the world of art and legend. Unicorns also have the same two forms as the horse and the number four. When the word "unicorn" is mentioned, no doubt the image of a single horned horse comes to mind. This confirms that the unicorn has an archetypical form. A form within the mind and imagination that can consistently exist within the minds of people. To argue that a unicorn does not exist would be to disprove the very definition. While it is true that there is currently not a living form of unicorn, the fact that the image can be conjured means that it must exist. Should an image be non-configurable, then one could argue that the unicorn does not exist in any form. The second from that the unicorn takes, and the one that most people image is that from art.…
Boche-ski, I.M. "Ancient Formal Logic." Amsterdam: North-Holland Publishing Company 1951.
Chappell, V. "Aristotle's Conception of Matter." Journal of Philosophy 70: 679 -- 696. 1973.
Dancy, R., 1986, 'Aristotle and Existence', in Knuuttilla & Hintikka 1986, pp. 49 -- 80.
GE.R. Lloyd. "Early Greek Science: Thales to Aristotle." Norton and Company. 1974.
Sensory experiences are nor reliable for making any statements, since people often mistake one thing for another. (Descartes talks about mirages). Knowledge based on reasoning is not always trustworthy, because people often make mistakes. (adding numbers is a classical example). Finally, knowledge is deemed by Descartes to be illusory, since it may come from dreams or insanity or from demons able to deceive men by making them believe that they are experiencing the real world, when are they are in fact not doing so. (the metaphysical approach in Descartes work is can be easily recognized here).
Following this analysis of existent forms of knowledge, Descartes concludes that certainty can be found in his intuition that, even if deceived, if he thinks he must exist: "Cogito ergo sum." The thought ("cogito") is a self-evident truth that gives certain knowledge of a particular thing's existence, i.e. one's self, but only the existence…
9. Dicker G, Descartes: An Analytical and Historical Introduction," Oxford, 1993
10. Flage D.E., Bonnen C.A., Descartes and Method: The Search for a Method in the Meditations," Routledge, 1999
Brians P., Gallwey M., Hughes D., Hussain, a., Law R., Myers M., Neville M., Schlesinger R., Spitzer a, Swan S. "Reading About the World," Volume 2, published by Harcourt Brace Custom Books. - excerpts from Descartes' works
It also set up a conflict between labour and capital, a variation of the old conflict between peasants and nobility. Because it was based on a competitive "free" market, capitalism inherently sought labour-saving and time-saving devices by which it might increase efficiency and productivity. In other words, manufacturing and production processes were sped up through specialisation (division), automation, mechanisation, routinisation, and other alienating forms of production in which the human being was less a personality at work and more a replaceable cog in a much larger system. This changed the way construction products were made. The concept of capitalism itself envisioned the mass production system and then made it a reality.
Furthermore, with the rise of the factory and the mechanisation of labour, farming began a decline and people flocked to the cities to find other types of work. Added to this there were advances in medicine which meant that…
O'Conner, P. (2003). Woe is I: The grammarphobe's guide to better English in plain English. New York: Riverhead Books
Leadership-Level Implementation of Strategic Plan
Good leadership is a key aspect of project management. Leading a project requires working with the manager and other staff drawn from the project's functional areas. It is not accurate to say that a leader only influences the subordinates under him. esponsibilities of a leader can go either vertically or horizontally. An effective leader will not only lead the subordinates under him or her, but also all the people involved in the project including those who are his seniors. A leadership model referred to as 3D model has been fronted by various players and promotes team leadership, self-leadership as well as teamwork that is leadership oriented. Effective leadership takes the ability to spot opportunities to improve a project and also execute on the project improvement. Besides the existing good personal traits, the style of leadership can be modeled through experience, training as well as dedication.…
Abou-Zeid, E.S. (2005). A culturally aware model of inter-organizational knowledge transfer. Knowledge management research & practice, 3(3), 146-155.
Ale Ebrahim, N., Ahmed, S., & Taha, Z. (2009). Virtual teams: a literature review. Australian Journal of Basic and Applied Sciences, 3(3), 2653-2669.
Bidgoli, H. (2013). MIS 3. Boston, Mass: Course Technology/Cengage Learning.
Caligiuri, P. (2006). Developing global leaders. Human Resource Management Review, 16(2), 219-228.
classroom assessment, a teacher determines his or her current point within the instructional sequence of a unit of study and identifies the student academic learning goals to measure.
"Select one class, a content area, and a unit of study to work with as you complete this performance task. Respond to the prompts below about the unit of study and its assessment."
Content Area: Math:
Grade level: 5 Content area: Mathematics Subject matter: _Graphs, Functions and Equations
"List the state-adopted academic content standards or state-adopted framework you will cover in this unit."
Graphs, Function Probability and statistics, and Equation: Statistics, Data Analysis, and Probability:
1.1: Arranges the raw data to plot graph and interprets the meaning of the data to produce information from the graph.
1.2: Understands the strategy to produce pair correctly .
Functions and Equations:
1.1: Uses the information collected from the equation or graph to answer…
If she is familiar with lossfeldt's epochal Art Forms in Nature, she may be a student of Ernst Haeckel's similarly titled investigation of the quasi-geometric, quasi-organic microbial univeral, or of the "wonder cabinet" juxtaposition of nature with art that spawned the visions of illustrators like Albertus Seba, Maria Sibylla Merian, and, more recently, Leo Lionni's arresting Parallel otany.
Lionni's hybrid plant forms sometimes demonstrate something of the quality of a nightmare. Gouthro's are more dreamlike in conception, and even their thorns and other occasional sharp edges are more likely to produce shudders of whimsy (erotic or otherwise) than outright menace. These are not the carnivorous plants through which the erotic "Venus" of nature traps and digests her animal prey. Instead, Gouthro is a self-described "optimist" (Gouthro 1) who takes deliberate care to balance the "edgy" elements in her garden by incorporating fairy-tale associations into her pieces:
I often include found-object…
Gouthro, Carol. Artist's Statement. N.D. http://www.carolgouthro.com/statement.pdf
Turner, Anderson. Surface Decoration: Finishing Techniques. American Ceramics Society, 2008. Print.
Wagonfeld, Judy. "Futuristic Artifacts." Ceramics Monthly, Aug/Sep 2007, pp. 47-49. Print.
But the of the efficiency of the design requires further proof. It is necessary to ensure that the all of the rays "directed to the virtual elliptic receiver at the exit aperture are reflected by the concentrator to some point on this receiver" (Garcia-Botella, et al.). The explanation of the math for this process is explained in a long section of the report from Garcia-Botella, et al.;
"Three properties of one-sheet hyperbolic concentrator geometry are useful: (1)
All meridional sections of a one-sheet hyperbolic concentrator are hyperbolas, (2)
all cross sections of a onesheet hyperbolic concentrator are ellipses and (3) the tangent plane, at any point P. Of a one-sheet hyperbolic concentrator, is defined by the bisector of the angle FPF0, where F. And F0 are the foci of the hyperbola in the meridional plane (Fig. 2), and the tangent line to the elliptic cross section at P. All the…
Ali, Imhamed M. Saleh, Tapas K. Mallick, Peter a. Kew, Tadhg S. O'Donovan, and K.S.
Reddy. "Optical Performance Evaluation of a 2-D and 3-D Novel Hyperboloid
Solar Concentrator." World Renewable Energy Congress, 2010. Print.
Chaves, Julio. Introduction to Non-Imaging Optics. New York: CRC Press, 2008. Print.
This image has lasted for nearly three thousand years but may now be in need of renewal. "God" may be longing for release from His immolation in the structure of our beliefs. To use a gardening metaphor, God has become pot-bound, fixed and constricted by the anthropomorphic, gender-biased, paternalistic image that we have projected onto Him. As Teilhard de Chardin suggested, we need to formulate a new image of God that is related to the phenomenal discoveries science has made about the new dimensions of the universe.
What have we done to God? The old image we have inherited from the Iron Age portrays God creating the Earth from a distance; God as something transcendent to, different from, creation and ourselves; God as male; God as fearful Judge, God as both punishing and loving Father. We have divided life into two - spirit and nature - and have lost the…
Edinger, E. (1985). Anatomy of the psyche: Alchemical symbolism in psychotherapy. La Salle, IL: Open Court
Edinger, E. (1996). The new god-image: A study of Jung's key letters concerning the evolution of the western god-image. Wilmette, IL; Chiron publications.
Goodchild, V. (2001). Eros and chaos: the sacred mysteries and dark shadows of love. York Beach, ME Nicolas-Hays, Inc.
Goodchild. V. (2006). Psychoid, psychophysical, P-subtle! Alchemy and a New Worldview. In Spring: A journal of archetype and culture, 74, "Alchemy." New Orleans, LA: Spring Journal Inc.
He looks at thee methods: histoy (melding infomation about the divese geogaphical oigins of algeba with the poblems themselves), multiple epesentations (using notation, naative, geometic, gaphical, and othe epesentations togethe to build undestanding), and the object concept of function (teaching functions without genealizing about how taits of an individual elate to taits of a goup). The aticle seves to offe some inventive solutions to a common poblem in math education: How to make mateial elevant and compelling to a beadth of students.
Matinez, a.A. (2010). Tiangle sacifice to the gods. 1-11.
The aticle looks at Pythagoas, paticulaly the mythology suounding his life and his most famous discovey, the Pythagoean theoem. It calls into question the histoical evidence on which mathematics teaches base thei teaching of this theoy. The autho points out how vey little is known about Pythagoas and how he has been canonized by the math discipline because his…
references the impact that Newton's work had on mechanical applications. Lastly, the piece points out how Newton used the thought patterns associated with calculus in what appears to the modern reader as a work of geometry (with respect to his book "The Principia"). In this way, the article functions as a reminder of how scientific discoveries are created, which is by building upon the theories of others and by giving weight to the importance to mathematical principles.
He died four years before Peurbach's matriculation, leaving the University without an astronomy lecturer. However, his library and instruments were probably accessible to Peurbach.
While it is known that Peurbach travelled throughout Europe between the years 1448 and 1453, there is no record of the precise dates. At the time, he also had an international reputation as an astronomer of note, despite the fact that he had not publications at the time. He did however lecture in Germany, France and Italy.
After lecturing at Bologna and Padua, these universities offered him permanent appointments as lecturer, but Peurbach turned these down. During his travels he also met the leading Italian astronomer of the time, Giovanni Bianchini, in Ferrara. Bianchini also offered Peurbach a post at an Italian university. Peurbach however remained unwilling to be tied to any specific institution of learning and turned down the offer. In 1453, Peurbach returned to…
McFarlane, Thomas J. (2004). Nicholas of Cusa and the Infinite. http://www.integralscience.org/cusa.html
O'Connor, JJ & Robertson, EF (2006). Georg Peurbach. http://www-groups.dcs.st-and.ac.uk/history/Biographies/Peurbach.html
O'Connor, JJ & Robertson, EF (1996). Nicolas of Cusa. http://www-groups.dcs.st-and.ac.uk/history/Biographies/Cusa.html
This specific example is also indicative of some of the general ways in which the building was modified and updated. In his restoration of Castelvecchio, Carlo Scarpa uses the basic geometric designs and patterns of the original medieval castle, but accentuates, develops, and emphasizes these geometric expressions in a very modern way. The concrete beam in the example above compliments the angularity of the room at large, but is almost an exaggeration of it. The beam itself s composed of three slabs of concrete at right angles to each other, forming three sides of a square or a sort of sideways "c," with the bottom open to the floor. Not only does the beams itself represent an intrusion of modern geometric appreciation into the castle, but even the construction of the beam itself reflects Scarpa's extreme devotion to geometry and geometric expression. This amplifies the geometry that is such an…
The interior architecture of the Querini Stampalia Foundation also provides a connection to the more historical details of Venetian and Italian architecture while at the same time not tying itself to the restrictions of an historic reproduction. Something as simple as a staircase has become, under Scarpa's careful design and guidance, something of strange geometric beauty that almost crosses the line into sculpture. The odd split in the stones that make up the stairs, and the spaces left in the faces of the stairs, look almost Moorish in their design. They could also be seen to draw from far more ancient sources, like the Romans who occupied Italy long before the Moors were ever heard of in Spain. The regular geometry of the stairs makes them appear both very strong and also simple and easy to build, which would definitely have been favored qualities of older civilizations working with other…
Therefore, the most different variable in the study is the change in treatment, i.e. The doubling of class time.
The following definitions are provided to ensure uniformity and understanding throughout this study. All definitions, not otherwise noted, have been developed by the researcher:
AYP -- Adequate Yearly Progress refers to the state-stipulated percentage of students by subject (math/English) by demographic (race/socio-economic strata) that must pass the HSPA. Schools that do not meet or surpass AYP are subject to sanctions. These may differ by state.
Class time -- The prescribed time during which a single class is conducted, i.e. one period. In this case, one period prior to the doubling of class time is initially equal to 42 minutes and subsequently equal to 43 minutes.
Doubling of class time -- Increasing class time from 42 minutes to 84 minutes plus the consumed passing time of 4 minutes for a total of…
We do not know if Grammatica is portraying something real or merely representing an image from the stage. This too, is symbolic of its period. The mannerist and baroque eras reflected a time when art was being made to serve the propagandistic purposes of a Church under siege. Like the headless warrior on the stone table, it also was under attack by evil forces. These forces needed to be fought by any means necessary. All were participants in this battle -- and so the figure of the lowly older woman following the direction of the more upper class Judith. Individual initiative is replaced by group action. Impulse is substituted for carefully-planned covert attack. Judith's pointing finger is indicative of a place to which the two must go, a prearranged location that apparently is known only to Judith. Again, secrecy is more important than openness, if secrecy means victory and ultimate…
Some of the mathematics of the book are shown to correlate to certain political aspects of the book, making the work perhaps more profound than Abbott ever intended (McCubbins & Schwartz, 1985). Certainly, the entire novel pushes for freedom, justice, and equality, both by satirizing certain social institutions and beliefs and by promoting the free and rigorous use of logical examination as a way of discovering, learning, and truly knowing things about the world we live in. he alternative that Flatland shows is a world full of people that do not really listen to or respect each other, and they often show as little regard for the realities of physical and theoretical truth.
hough this book is almost one hundred and thirty years old, it is still very useful today. It can be read as an introductory text to certain mathematical and philosophical concepts, a historical document showing the opinions…
The last portion of the novel, after the Sphere ridicules the Square and leaves him, is again highly political, and deals with the justice system in the basically totalitarian state that the Square and his family live in. In order to maintain complete control over the citizens, the government of Flatland outlaws the use of color or "chromatic expression," and also ends up outlawing the discussion or mention of dimensions beyond the second, which carries the death penalty for some.
The Square is not killed, but he is imprisoned for continuing to discuss his ideas about other dimensions (Abbott, 1884). This mirrors the persecution that many scientific figures suffered at the hands of various governments and religious institutions (particularly the Catholic Church) for spreading knowledge that they had verified through repeated observation or mental exercises. In fact, both the mathematics and the politics that appear in Flatland have continued to influence political thinking and evaluation. Some of the mathematics of the book are shown to correlate to certain political aspects of the book, making the work perhaps more profound than Abbott ever intended (McCubbins & Schwartz, 1985). Certainly, the entire novel pushes for freedom, justice, and equality, both by satirizing certain social institutions and beliefs and by promoting the free and rigorous use of logical examination as a way of discovering, learning, and truly knowing things about the world we live in. The alternative that Flatland shows is a world full of people that do not really listen to or respect each other, and they often show as little regard for the realities of physical and theoretical truth.
Though this book is almost one hundred and thirty years old, it is still very useful today. It can be read as an introductory text to certain mathematical and philosophical concepts, a historical document showing the opinions and mores of an incredibly restrictive society and the response that this restrictiveness inspired in certain individuals, and a manifesto for social justice. It is still meaningful in all of these ways, and the popularity that the book has enjoyed over the past century is a good indicator that it will remain popular for centuries to come.
Thus, the invention of perspective by the artists of the Renaissance reflected the emergence of science and the mathematical ordering of man's observations of the physical world.
The manifestation of perspective can clearly be observed in the paintings of many Renaissance artists. For instance, da Vinci's masterpiece the Last Supper, rendered between 1495 and 1498 as a wall fresco, portrays the figure of Jesus Christ sitting in the center of the picture with his body framed by a central window in the background and a curved pediment, the only curve in the architectural framework serving as a halo, arching above his head which serves as the focal point for all the perspective lines/axis in the composition, a system not invented by da Vinci but one copied from earlier master painters.
Another earlier example is Christ Delivering the Keys of the Kingdom to St. Peter by Perugino, rendered as a wall…
It is now recognized that individuals learn in different ways -- they perceive and process information in various ways. The learning styles theory suggests that the way that children acquire information has more to do with whether the educational experience is slanted toward their specific style of learning than their intelligence.
The foundation of the learning styles methodology is based in the classification of psychological types. The research demonstrates that, due to heredity factors, upbringing, and present circumstantial demands, different students have an inclination to both perceive and process information differently. These different ways of learning consist of: 1) concrete or abstract perceivers, where concrete perceivers acquire information through direct experience of doing, sensing, and feeling, and abstract perceivers, instead accept new ideas through analyzing, observing and thinking; 2) active or reflective processors -- active processors understand a new experience by immediately utilizing new information, and reflective processors analyze an…
Bruner, J. (1973). Going Beyond the Information Given. New York: Norton.
Dewey, J. (1910) How We Think. Boston: Heath.
Dryden, G. And Vos, (1999) Jeannette. The Learning Revolution. Austin, TX: Jalmar
Gardner, Howard (1983) Frames of Mind: The theory of multiple intelligences, New York: Basic Books.