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…...

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References

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 living organisms. One…...

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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.
Handout

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.

The first step is to isolate the…...

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References

National Council of Teachers of Mathematics [NCTM]. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.

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45

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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 Einstein's theories to…...

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Bibliography

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 feelings'…...

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References

Donne, John. " The Flea." Poetry Foundation.

  [16 Jan 2013]http://www.poetryfoundation.org/poem/175764 

Donne, John. "A Valediction Forbidding Mourning."

  [16 Jan 2013]http://www.luminarium.org/sevenlit/donne/mourning.php 

MATH - Measurement, Geometry, Representation Part A
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, with respect…...

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), and another…...

Introduction
Intersection theory, a fundamental aspect of algebraic geometry, serves as a bridge between various mathematical disciplines, offering insights into the geometric properties of different mathematical spaces. This essay provides an overview of intersection theory, exploring its origins, fundamental principles, applications, and the influence it exerts on other mathematical fields.

Origins and Historical Context

Intersection theory originated from the need to understand and quantify the intersections of geometric shapes in various dimensions. Historically, mathematicians like Bernhard Riemann and André Weil contributed significantly to its development. Weil, in particular, was instrumental in laying the groundwork for modern intersection theory through his work on algebraic surfaces and their intersection numbers. His contributions, along with those of other 20th-century mathematicians, transformed intersection theory into a major branch of algebraic geometry.

Fundamental Principles

At its core, intersection theory is concerned with the study of how subspaces of a given space intersect with each other. It involves calculating intersection numbers…...

Teach Geometry
Dear Parent,

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 a child…...

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References

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   adoption-map.aspxhttp://www.ascd.org/common-core-state-standards/common-core-state-standards -

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…...

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Works Cited

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…...

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Works Cited:

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).

Deeper…...

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Works Cited

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]

 http://www.corestandards.org/Math/Content/HSG/introduction

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 also an…...

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References

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:

 http://www.dtswiss.com/Products/Suspension/DT-Swiss-Shocks.aspx .

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

Steering design

Turning circle

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 ?i = 50.0°

Outer steer angle ?a = 30°30' = 30.5°

1. Distance between…...

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References

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…...

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Reference

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

Intersection theory, in abstract terms, is a complex mathematical framework used in algebraic geometry to understand the intricate "meetings" between shapes within a mathematical space. While its roots and applications stem from the world of shapes and equations, it can be viewed through a more metaphorical lens to gain insights into human experiences. Here's how:

Understanding Intersection:

• Imagine two overlapping circles: Their intersection represents shared experiences, common ground, or areas of mutual understanding between two individuals. Intersection theory, in this metaphor, provides tools to quantify and analyze these overlaps, their complexities, and how they influence the overall experience.

How it helps:

• Empathy and Perspective: By studying the "intersections" of lived....

Outline for an Essay on Intersection Theory

I. Introduction

Begin with a compelling hook or question that captures the reader's attention.
Define intersection theory and explain its significance in algebraic geometry.
State the thesis statement, which should articulate the main argument or purpose of the essay.

II. Background and Historical Context

Provide a brief overview of the historical development of intersection theory.
Discuss the contributions of key mathematicians, such as Bézout, Euler, and Poincaré.
Explain the role of intersection theory in resolving classical geometric problems.

III. Fundamental Concepts

Define the basic concepts of intersection theory, such as:
Intersection number
Cycle
Homology and cohomology....

1. The Fibonacci sequence and its applications in physics
2. Chaos theory and its implications for understanding complex systems in physics
3. The role of symmetry in modern physics
4. Fractal geometry and its applications in modeling natural phenomena
5. The use of wave equations in describing physical processes
6. The concept of infinity in calculus and its significance for physics
7. The mathematical foundations of quantum mechanics
8. Differential equations and their role in modeling physical systems
9. The geometry of spacetime in general relativity
10. The role of group theory in understanding the fundamental forces of nature
11. The applications of calculus in solving problems in classical mechanics
12.....

One interesting topic that has been in the news recently is the use of abstract mathematical concepts in physics, specifically in the field of quantum physics.

There has been a lot of research and debate on the role of abstract mathematical structures in understanding and describing the behavior of particles at the quantum level. For example, the use of complex numbers, matrices, and abstract algebraic structures has been essential in developing the mathematical framework of quantum mechanics.

One recent development in this area is the use of category theory, a branch of mathematics that studies abstract structures and relationships between different....