Similar Triangles Activity Report

Similar Triangles Activity Report
Introduction In this activity, I will be working with Carolyn Ulrich, a fellow geometry teacher, to
improve our students' achievement in our "Similar Triangles" unit. This application will occur at
Deer Valley High School in Glendale, Arizona; the website is: http://www.dvusd.org/Domain/42.
The mathematical level of geometry is the second–year math class taken by all sophomores and is
tested on the Arizona state standardized test. Mrs. Ulrich is our geometry level leader on our
campus, but she teaches four honors geometry classes and one regular geometry class, where I teach
three of the regular geometry classes on campus. In this activity, we have decided on three standards
to focus on when we instruct our students in this unit, which I will state in the next section. We will
work together in a collaborative inquiry which "involves identifying and agreeing on one problem or
area of student need." (Nelson et al., 2010, p. 36) We will meet throughout the week and discuss
what we did in class for our instructional practice, how we thought it went for each class, administer
the common assessment, and see how our students did on these three standards and compare results.
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205 (Carolyn's room) Standard 1:
G–SRT.2 – given 2 figures, use definitions of similarity in terms of transformations to decide if they
are similar (similarity of triangles)
Standard 2:
G–SRT.5 – use congruence/similarity for triangles to solve problems/prove
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Greek Mathematics Research Paper
1. Write a short (2 pages) essay on Greek Mathematics
The mathematicians of ancient Greece contributed to the Mathematic of the world, and its
applications vary on an intellectual basis, from geometry to engineering, astronomy to design.
Influenced by the Egypt mathematicians, Greek mathematicians made breakthroughs such as
Pythagoras' theory of right–angled triangles. Their Mathematic created the basic mathematical
building blocks, and being useful up to today for mathematicians and scientists. Talk about Geek
mathematic, people should know about their history, application and achievements such as: theorem
of Pythagoras, and Euclid, Approximation to the Value of Square Root of 2.
First, Greek mathematicians was influenced by civilizations around its such as Ionia (Turkey),
Mesopotamia (Iraq, Iran, and part of Syria and Turkey), Lydia (a region of western Asia Minor), and
especially Egypt. Thales and Pythagoras visited Egypt and learned new skills and knowledge.
Babylonian and Chaldean helped Greek mathematic in divide circles into 360 degrees. ... Show
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He was really good at Greek mathematical work, so he wanted to organize all this knowledge in a
book which was The Elements. It was famous and being the second best–selling book of all times.
The opening of Book I begins with different definitions on basic geometry.
The Square Root of 2, after the Pythagorean theorem was established, the following question was
put forth: If we had a square with each side a unit in length, and we also had a second square with
double the area of the first square, how would the side of the second square compare to the side of
the first square? This is the origin of the question regarding the square root of 2. The Pythagoreans
could not solve the puzzle, and they finally faced up to the reality that no ratio of two whole
numbers could express the value of the square root of 2, so they looked into

Greek Math Research Paper
Egyptian mathematics began as early as 6000 B.C. One thing the Egyptians are known for is taking
measurements with their body parts. For example, a cubit was the length from elbow to fingertips
and they often used the palm of their hand to measure land and buildings in early Egyptian times.
They also used their hands to develop a decimal numeric system based on our ten fingers. This was
thought to be the earliest developed base ten number system as early as 2700 B.C. The Egyptians
used hieroglyphic symbols to demonstrate their numbers. For example, a vertical line represented a
unit, a heel bone represented ten, a coil of rope represented hundreds, and so forth up to powers of a
million. However, they did not have a concept of place value, ... Show more content on
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Greek mathematics was also much more sophisticated than mathematics that had been developed by
other cultures whereas they show the use of deductive reasoning to derive conclusions from
definitions and axioms and proved them. Greek mathematics refers to the time between 600 B.C to
529 A.D. Most of the mathematical work done by the Greeks was geometry. It is said to have started
with Thales, a famous Greek mathematician, who was the first person to lay down the abstract
development of geometry. He established what is known as Thales Theorem. Thales used geometry
to calculate the height of pyramids and find the distance from ships to shore. However, some say
that Pythagoras deserves the true title for the birth of Greek mathematics. He is believed to have
coined the term mathematics, which is where the study of mathematics began. He established a
school and was also credited with the first proof of the Pythagorean theorem, but as we know, he
remains a controversial figure. Those these two were the start to Greek mathematics, it is obvious
that it was in no way limited to just these two men. Many more mathematical figures introduced
new ways of thinking as we will soon

Hieroglyphics and History of Mathematics
Hieroglyphics deal a lot with the history of math, because it was one of the earliest of maths. The
ancient Egyptians were the first civilization to practice the scientific arts. It is said that the Egyptians
introduced the earliest fully–developed base 10 numeration systems at least as early as 2700 B.C.
The word chemistry is derived from the word Alchemy which is the ancient name for Egypt. It was
between the third and first millennia B.C. It later then died in 400 AD. This was first used as legal
matters such as commerce, education, literature, and science. This type of math was mostly used by
Egyptians, but there numbering was different than ours today. Instead of them using numbers they
would use pictures to illustrate the numbers.
It is said that hieroglyphics were created by the Egyptian god Thoth. He is said to be the god of the
moon, magic, and writing. Hieroglyph comes from the root word hieros which is Greek meaning
sacred, and the root word glypho which means inscription. Hieroglyphs were first used by Clement
of Alexandria.
It was first used in ancient Egypt, as a decimal numbering system. But the decimal was non–
positional; it could deal with numbers of great scale. Egyptian used this method mostly in medicine
and geometry. The earliest known examples of hieroglyphs in Egypt have been dated to 3,400 BC.
The latest date in hieroglyphs was made on the gate post of a temple at Philae in 396 AD. The
Egyptians were really involved in medicine and applied

Euclid: A Ranking Of The Most Influential Person In History
Have you ever wondered what it means to be an influential person? One of the most significant
people throughout history was Euclid. While his birthplace is unknown, he was taught at an
Academy in Alexandria by the best mathematicians of the time. Thanks to the academy teachers that
taught and influenced him, Euclid is a very influential person whose works in the mathematics field
served as a basis for modern mathematics in many different ways. Michael Hart rates Euclid number
14 in his book, The 100: A Ranking of the Most Influential Persons in History, because of his
accomplishments in the mathematics field and his worldwide influence. His influence on the world
is spectacular and is notably used in the basis of modern mathematics.
Euclid ... Show more content on Helpwriting.net ...
This excerpt, "Euclid's textbook has probably had a greater influence on scientific thinking than any
other work." (Calinger), shows how he affected modern society. Euclid's book, Euclid's Elements
allowed the common man to understand the basics of geometry expanding human knowledge in the
math field. This allowed his work and knowledge of geometry to spread from person to person
through other people. It's influence is still shown from the quote, "Although Euclid's system no
longer satisfies modern requirements of logical rigor, its importance in influencing the direction and
method of the development of mathematics is undisputed." (Greek Mathematician Euclid). This
shows how even though the system is outdated to modern times, it is still influential in the
development of mathematics. The worldwide modern influence is still present to this day. This is
also stated in the quote, "his historical importance in the development of mathematics and in the
establishment of the logical framework necessary for the growth of modern science." (Hart, 78).
This confirms how Euclid's work in geometry allowed humankind to further their understanding of
mathematics and science. Without Euclid Earth would not be the same intellectual planet we know
today. His influence is also shown in other parts of the world, "Euclid was not translated into
Chinese until about

What Students Should Know?
What Students Should Know According to the learning progressions report, coming into third grade,
students know how to analyze, compare, and classify two–dimensional shapes by their properties.
When students do this, they relate and combine these classifications that they have made (The
Common Core Standards Writing Team, p. 13). Because the students have built a firm foundation of
several shape categories, these categories can be the "raw material" for thinking about the
relationships between classes. Students have learned that they can form larger, superordinate,
categories, such as the class of all shapes with four sides, or quadrilaterals, and recognize that it
includes other categories, such as squares, rectangles, rhombuses, ... Show more content on
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15). In fourth grade, students learn to represent angles that occur in various contexts as two rays,
explicitly including the reference line, which is a horizontal or vertical line when considering slope
or a "line of sight" in turn contexts. They understand the size of the angle as a rotation of a ray on
the reference line to a line depicting slope or as the "line of sight" in computer environments (The
Common Core Standards Writing Team, p. 15). In fourth grade students also might explore line
segments, lengths, perpendicularity, and parallelism on different types of grids, such as rectangular
and triangular (isometric) grids. Finally, students also learn how to reason about these above
mentioned concepts (The Common Core Standards Writing Team, p. 16). According to the learning
progressions report, coming into fifth grade, students should know how to represent angles that
occur in various contexts. They should also already understand the size of the angle as a rotation of a
ray. Finally, they should have developed explicit awareness of and vocabulary for many concepts
they have been developing, including points, lines, line segments, rays, angles (right, acute, obtuse),
and perpendicular and parallel lines (The Common Core Standards Writing Team, p. 17). In fifth
grade, students develop competencies in shape composition and decomposition, and especially the
special case of spatial structuring of rectangular

Math In The Criminal Justice System
In this day and age, math is utilized everywhere. Math can be found in books, television and
schools. However, above all, math is found in each employment. We may not trust that math is
applied all around, until we end up using it for the easiest things. Occupations require their
employees to know essential math for basic things, for example, running a money enlist and
numbering deals. Math is necessary in the world. Without math, the world would be very different.
Each occupation utilizes basic math to take care of fundamental issues and to solve basic problems.
A job that requires math is an educator or teacher. There are diverse math classes students can take
at school. In high school, understudies are learning Algebra 1, Geometry, Algebra ... Show more
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A sales associate is in charge of working with customers to help find what they want to ensure a
smooth transaction. Math is used in this occupation when the sales associate counts money during a
transaction. The sales associate makes sure the customer gives the correct amount of money and
counts the amount owed. When there is a discount on certain items in the store, the sales associate
must subtract the correct amount to have an accurate discounted price. Tasks such as counting and
keeping track of money in registers is math. Similarly, a store manager runs the store and makes
sure the store meets budget and sales goals. Working in a retail store uses the simplest math such as
adding and subtracting. Computing the total amount of sales, calculating the percentages of
discounts, and determining the amount of sales tax is basic math used every day by a sales associate
on the

Essay On Greek Geometry
The Greek are one of the founders of geometry, and had a big impact on math and the world today.
Greek geometry eventually passed into the hands of the great Islamic scholars, who translated it and
added to it. All around us we can observe some geometrical principles, which is probably what
started the ancients on their way to developing this field of study. The word geometry has its roots in
the Greek work geometrein, which means "earth measuring". There were many circumstances in
which primitive people were forced to take on geometric topics, man had to learn with situations
involving distance, bounding their land, and constructing walls and homes. For sure builders today
have to use geometry. In this study of Greek geometry, there were many more Greek mathematicians
and geometers who contributed to the history of geometry. Geometry is one of the most greatest
things mankind has ever thought of. Because of Greeks geometry has become a big part of our lives
as well. Some people might have jobs that include a lot geometry. People everyday use geometry
without even realizing. There were ... Show more content on Helpwriting.net ...
For engineers, having knowledge of geometry is very important for properly sizing and structuring
physical objects. Engineers use geometry to identify key shapes, such as triangles, cubes and
pyramids, and to find quantities like lengths, volumes, centroids, and movements of inertia, lines
and curves. Geometry is also used to reduce friction and determine the structure of soil. For builders
they have to learn how to measure their objects, and use shapes to build their buildings. Geometry is
used in astronomy to find the properties of stars and other objects in space. Also they need it to find
how many years does it take to get from one place to another and the distance between two or more
objects. They need to use equation formulas for basically

Using Geometry And Algebra, By Brianna Baker
Geometry and Algebra / By: Brianna Baker Five ways to use geometry in life is by: making my bed,
telling time on a clock, wrestling, driving, and doing geometry in math class. When making a bed
you use angles by horizontally pulling the covers up on to the rectangular mattress where they
belong. Telling time uses angles by the short hand moving every hour and the long hand moving
every minute. The hands are always either in an obtuse or acute angle. Wresting uses angles and if I
wanted to grab someone I would be predicting volume. Driving is obviously geometry making U–
turns and turning is geometry. Doing geometry in math class today is me moving my fingers up and
down to type an essay. ... Show more content on Helpwriting.net ...
Angles and measurements are important for the structure of the house. Being a house keeper or
cleaner would go along with how beds should be made to look like perfection. Now teaching school
takes degrees and work.Teaching sports is much easier than being a book teacher. Wrestling, football
and basketball all use volume and angles. Wrestling teaches specific movement and lengths of reach.
The game of football takes prediction of volume and of course you should now at least what angle
or way to throw. Basketball is both, from hitting the back board to how high to jump. Five ways to
use algebra in daily life is by: going to an algebra math class, predicting how much money you'll
get, counting, saving and spending money. When you take a class for algebra you'll count the days
left to go. For example, 10=100/x. If I worked for 20 days and got paid 9 dollars every hour than I
could find out how much money I would get. Working at a bank more en–likely wouldn't be fun.
keeping record of the progress or

Geometry And Geometry
My first article author was ( Tara Mastroeni). This article is basically about how the buildings are all
shapes and different sizes, and yet so similar at their core. For those wondering this occurs over and
over again in nature. It's is found in everything from the shape of the universe. The structure of
clouds, and the proportions of human body. Humans have incorporated it into everything from the
mathematics to artwork and music. That you or any person can make anything as long as you know
mathematics basically you can do anything. Some people say that "why do you need mathematics to
build a house" well that is the biggest part in build a home or building anything you need to build.
Why I say that building is the most biggest part is because say you want to measure each ... Show
Almost everything we build has to do something with geometry or mathematics if you know it or
not we use geometry in everyday life. The word geometry means, to measure the earth. Geometry is
the branch of math that is concerned with studying area, distance, volume, and other properties of
shapes and lines. Now that you know how much geometry is used in building and decorating homes
you yourself can build your own home with the help of me you will know what to do. Just make
yourself a plan and just do it. One more thing, in decorating these homes you will need a lot of
supplies, the basics you will need will be things like new things you will want to be in your house
that will make you feel comfortable. If you are building your own house make sure that you like it
and you feel comfortable in it because if you don't then you might not stay there or like living their.
This can go for just buying a house from someone and redecorating it the way you like. Most people
use a measuring tape to measure the blinds of the house, so they can go to the store and pick the
right size and the kind they want to put up in their

Geometry in Golf Essay
"Bringing it all Together: The Geometry of Golf"
Golf in Geometry?? No Way!
Geometry In The Game of Golf
For hundreds of years, golf has been an extremely popular and growing sport all around the world.
Looking where golf is now, it is growing rapidly from the young to the elder population. The first
round of gold was first played in the 15th century off the coast of Scotland, but it did not start to be
played until around 1755. The standard rules of golf were written by a group of Edinburgh golfers.
Today, people of the US, Scotland, and England, have been drawn to the game because it is fun,
challenging, and hardly any athletic ability at all is required for amateurs. In breaking down the
game, geometry plays a major ... Show more content on Helpwriting.net ...
In golf geometry is al around but it just takes one to stand back and look how line, angles, and
shapes are all around. Five geometry topics that are most prevalent in the game of golf include
circumference of circles, parallel lines, triangles, radius and diameter, and angle measurement. If
none of these geometric ideas were in golf, the game would not be here right now. The first
geometric topic involved in golf is the circumference of circles. For people who do not know the
objective of the game of golf, it is fairly simple. Put the ball in the hole. The circumference of the
hole is 13.35 inches, while the diameter is 4.25 inches. It is extremely important that the creators of
the course measure each distance from the tee box to the hole. Also, the diameter of a regulation golf
ball is 1.68 inches. The circumference is 5.28 inches. Though it is not clear exactly how the standard
measurement of the hole came up, it is obvious that it greatly affects the scores of golfers. If it was
made a little bit smaller, scores will rise greatly. If the hole were too be made a little bit bigger, and
scores would come down significantly.
Moreover, parallel lines are very important to the game of golf. In the golf swing, if a full swing is
taken, the club should always be parallel to the ground when the club is at the furthest point back. If
the club is not taken all the way back it takes away the distance and if

Essay On Geometry
Is Geometry the Most Fundamental Area of Math?
As the very name implies, Geometry means measuring earth ('Geo' meaning earth and
'metron'meaning measurement). Hence, one can understand how old this branch of Math is and what
importance it should hold among the branches of Math.
What is Geometry?
Geometry is the branch of Math which deals with shapes, sizes, figures and their various properties,
relations and measurements. Doing Geometry with seriousness helps a Math student develop good
mathematical abilities and a precise power of perception.
Origin of Geometry (How old is Geometry?)
Geometry was given importance right from the age of Greeks and most of its concepts were found
in measuring lengths, volumes and areas in their early culture. ... Show more content on
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The concepts of Geometry have given rise to Trigonometry with its angles, side angles, right angled
triangles and non–right triangles. Algebraic Geometry is also popular today with its concepts like
coordinates.
You have the emergence of Calculus from the aspects of Geometry. One can find the root of modern
integral Calculus in Archimedes' ingenuous techniques for calculating areas and volumes. You find
geometric figures like plane curves represented analytically in the form of functions and equations
leading to the emergence of infinitesimal Calculus. Today, you have Topology and differential
Geometry as well.
Overtones of Geometry in various areas of learning
Since Geometry is interlinked with Astronomy and is useful for calculating spatial distances, both
these subjects were learnt together in olden days. Not only that, Geometry has sprinkled its
influence upon various other areas like art, land survey, civil engineering and architecture. You can
find the overtones of Geometry in Science subjects like Physics also. Hence, Geometry has a vast
role to play in the contour of Math learning and makes for successful understanding of related topics
in Trigonometry and

The Ancient Inventions Of The Future
The Ancient inventions of the future Socrates was one of the greatest philosophers of the ancient
world. His teachings inspired young philosophers to go and explore the world. In addition, His
Knowledge even influenced boundless minds like Plato, who help make gigantic pushes in
geometry. Or Aristotle, who mad giant leaps in astronomy (also made a theory that the earth was
round in about 384 B. C.). therefore, a new era of philosophy was inaugurated and the course of
western civilization was decisively shaped. Modern mathematics, Medicine, and cartography was all
modernized during this age of philosophy. Without the Greeks, present ideals and the way we live
our life would not be the same. So it is important that we know what the Greeks invented and their
power over our civilization.
First of all, the use the same form of democracy that we use in America. They decided that a dictator
would have too much control and power. So they got a cabinet of over 500 people for democratic
voting. They voted on taxes all the way up to executions. It worked a lot like our jury system, with
people randomly chosen to serve. As it was said by Scott J. Cooper; "Demokratia, Greek for "power
of the people," was born in Athens in the 7th century BC. As the city–state's oligarchy exploited
citizens and created economic, political and social problems, Athenians were inspired by the
successful, semi–democratic model Sparta had adopted. They turned to lawmaker Solon, who tried
to help the

How Did Babylonians Contribute To Greek Geometry
Soon after civilizations emerged so did geometry. The early civilizations of the
Egyptians and Babylonians had a good grasp on mathematics and carried out calculations in
practical ways for building structures. The Egyptians and Babylonians even appeared to know the
Pythagorean Theorem, filled by tablets with impressive tablets of triplets. (Mlodinow,
2002) Despite the Egyptians accomplishments and Babylonians cleverness their contributions to
mathematics were limited to providing the Greek with a collection of concrete mathematical facts.
The discovery that mathematics is more than algorithms for calculating volumes of dirt or the
magnitude of taxes is credited to a lone Greek merchant– turned philosopher named
Thales.(Mlodinow,2002) Thales ... Show more content on Helpwriting.net ...
Central to Eudoxus' idea was the distinction between magnitude and numbers. A magnitude was not
a number but stood for entities such as line segments, angles, which could vary continuously.(Kline,
1990) Since no quantitative values were assigned to magnitudes, Eudoxus was then able to define a
ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking numbers
out of the equation, he avoided the trap of having to express an irrational number as a number.
Eudoxus' proportion theory enabled the Greek mathematicians to make tremendous progress in
geometry by supplying the necessary logical foundation for incommensurable ratios.(Kline,

Intermediate Grade Geometry
How important should geometry be in the primary grades, the intermediate grades, and the middle
school grades? Please support and explain your answer. Geometry helps us understand and describe
the world around us. Geometric concepts are used in architecture, engineering, astronomy, art,
navigation, sports, furniture design, toy making, road building–the list goes on and on. Children are
generally engaged in geometric thinking when they choose the shortest path to the playground, pack
food and drink containers into their lunch boxes, and grapple with how to maximize the number of
cutouts from a sheet of construction paper. For primary grades, you can identify geometric shapes in
picture books. For intermediate grades, the teacher can discuss

Advantages And Disadvantages Of Nx
Formerly known as UNIGRAPHICS, NX is an advanced high end CAD/CAM/CAE software
package. Owned by Siemens PLM Software, it is used for parametric design, direct solid and
surface modeling and simulation with respect to static, thermal, dynamic and manufacturing aspects.
NX design tools are superior in power, versatility, flexibility and productivity. Fast and intuitive
editing of the profiles has been enabled by incorporating the synchronous technology, thereby
making the job of the designer easy. It ensures improvement in efficiency by implementing tools
which facilitates easy–to–understand design changes. 2.1.1 FEATURES OF NXCAD: Some of the
important features of NX are as follows: 1. Feature–based modeling: The smallest building block in
a part model is known as a feature. A feature based approach for product design is being followed by
NX. It allows building a model incrementally, adding individual ... Show more content on
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Unequalled depth: Ansys provides an unequalled technical depth in any simulation domain whether
it is structural, analysis, fluids, thermal, electromagnetics, meshing or processing and data
management. It provides consistent technology solutions irrespective of being a casual user or an
experienced analyst. 2. Unequalled breadth: Ansys provides functionality across a diverse range of
disciplines ranging from structural analysis right up to electromagnetic, including fluid and thermal
domains. All these are efficiently supported by a complete set of analysis types and backed up by a
powerful set of meshing tools. 3. Adaptive architecture: In today's world of engineering, for the
overall design and development process, a software must have the ability to adapt to a variety of
CAD and PLM solutions. The software must have the ability to be customized to provide for the
inter–operability with other software's. These are the characteristics provided in the Ansys
simulation architecture, making it feasible to be used under any

Differences in Geometry Essay
Differences in Geometry
Geometry is the branch of mathematics that deals with the properties of space. Geometry is
classified between two separate branches, Euclidean and Non–Euclidean Geometry. Being based off
different postulates, theorems, and proofs, Euclidean Geometry deals mostly with two–dimensional
figures, while Demonstrative, Analytic, Descriptive, Conic, Spherical, Hyperbolic, are Non–
Euclidean, dealing with figures containing more than two–dimensions. The main difference between
Euclidean, and Non–Euclidean Geometry is the assumption of how many lines are parallel to
another. In Euclidean Geometry it is stated that there is one unique parallel line to a point not on that
line.
Euclidean Geometry has been around for ... Show more content on Helpwriting.net ...
Spherical Geometry is also the most commonly used Non–Euclidean geometry, being used by
astronomers, pilots, and ship captains. In Euclidean geometry it is stated that the sum of the angles
in a triangle are equal to 180. As for Spherical geometry it is stated that the sum of the
angles in a triangle are always greater than 180. When most people try and visualize a
triangle containing angle sums greater than 180 they say it's impossible. They're right, in
Euclidean geometry it is impossible, but as for Spherical geometry, it is possible. Think of the
triangle on a sphere, and then try and visualize it. See Appendix 1–1.
When thinking of the Non–Euclidean Spherical Geometry, we start of with a basic sphere. A sphere
is a set of points in three–dimensional space equidistant from a point called the center of the sphere.
The distance from the center to the points on the sphere is called the radius. See Appendix 1–2 to
visualize tangents, lines, and centers between the sphere, lines, and planes.
Unlike standard Euclidean Geometry, in Spherical Geometry, radians are used to replace degree
measures. It is usual for most people to measure angles and such with degrees, as for scientists,
engineers, and mathematicians, radians are used to substitute degree measures. The size of a radian
is determined by the requirement that there are 2pi radians in a circle. Thus 2pi radians equals 360
degrees. This means that 1 radian =

How Did Pythagoras Contribute To Geometry
Pythagoras was a Greek mathematician who was born around 569 BC in Samos, Ionia, which is in
Greece and died around 475 BC. Pythagoras is extremely important in the field of mathematics and
made many contributions to geometry. Not much is known about the early life of Pythagoras.
However, it is known that his father, Mnesarchus, was a merchant and that Pythagoras traveled
frequently with his father ("Pythagoras of Samos," n.d.). Thales and Anaximander were
philosophers who are believed to have influenced Pythagoras and his interest in mathematics
("Pythagoras of Samos," n.d.). In fact, Anaximander was interested in geometry. He lectured on
Miletus where Pythagoras attended his lectures ("Pythagoras of Samos," n.d.). Pythagoras is most

Ancient Greek Legacy Essay
Trip Griffin History – The Legacy of Ancient Greece
Ancient Greece's legacy contains numerous topics that influence our modern–day life including Art
and Architecture, Drama and History, War, Democracy, Science and Technology, and Philosophy.
My guess is that a lot of students will focus on Democracy. However, while it is quite important to
western culture, I have decided to focus on three examples of things that were very interesting to me
Mathematics and Physics, Comedy and Architecture.
First, Math and Physics. Math and science are the reasons we can understand our surrounds and how
on the largest and smallest scale, how stuff works. The Ancient Greek mathematicians used
geometry text compiled by Euclid (for example Euclidean geometry). They also used the
Pythagorean Theorem that was earlier discovered by the Chinese, but was used much more widely
in Ancient Greece. In scien¬ce, Archimedes of Syracuse, estimated the value of pi – the ratio of the
circumference of a circle to its diameter. As you can see, the Ancient Greeks contributed
significantly to the math and physics we use today and are learning in school.
While the Ancient Greeks wrote about Tragedy (Greek Tragedy) and Comedy, it is ... Show more
Greece is known for some amazing architecture including The Parthenon. The Parthenon was built
between 447 and 432 B.C. and was designed as a temple. Greek architects looked to nature to create
their beautiful buildings. They also started a more widespread and appeasing use of the arch and
dome. They also used math to create proportions and symmetry. I was fortunate to travel to Crete
when I was younger to see some of the amazing structures built by the Ancient Greeks. Other
significant pieces of Greek architecture include the first lighthouse in Alexandria comparable in size
to the Statue of Liberty. It was later destroyed by an earthquake, but set the standard model for all
future

Miller Shettleworth Essay
Journal of Experimental Psychology: Animal Behavior Processes 2007, Vol. 33, No. 3, 191–212
Copyright 2007 by the American Psychological Association 0097–7403/07/$12.00 DOI:
10.1037/0097–7403.33.3.191
Learning About Environmental Geometry: An Associative Model
Noam Y. Miller and Sara J. Shettleworth
University of Toronto
K. Cheng (1986) suggested that learning the geometry of enclosing surfaces takes place in a
geometric module blind to other spatial information. Failures to find blocking or overshadowing of
geometry learning by features near a goal seem consistent with this view. The authors present an
operant model in which learning spatial features competes with geometry learning, as in the
Rescorla–Wagner model. Relative total ... Show more content on Helpwriting.net ...
The signature phenomena of cue competition in conditioning are overshadowing and blocking. In
overshadowing (Pavlov, 1927), when two cues are redundant predictors of the same outcome, less is
learned about either than when it is the sole predictor of the outcome. In blocking (Kamin, 1969),
training with a single cue reduces (blocks) learning about a second, redundant cue added later.
Several studies have looked for blocking or overshadowing of geometric information by features
(for a review, see Cheng & Newcombe, 2005). Most studies have concluded that a predictive feature
near a goal does not block learning about the shape of an enclosure (e.g., Hayward, Good, & Pearce,
2004; Pearce et al., 2001; Wall et al., 2004). Moreover, in contrast with the expected competition
between cues, geometry is sometimes learned better in the presence than in the absence of
informative features. Pearce et al. (2001), for example, found that a beacon improved learning about
the geometry of a triangular water tank. Other researchers have come across hints of this same
phenomenon (e.g., Hayward et al., 2004; Hayward, McGregor, Good, & Pearce, 2003). Using a
geometrically unambiguous kite–shaped water tank, Graham, Good, McGregor, and Pearce (2006)
demonstrated in rats substantial potentiation of geometry learning by a feature. Kelly and Spetch
(2004a, 2004b) also found clear evidence of potentiation of geometry learning by a feature in an
operant task in which people and pigeons were

Geometry: Annotated Bibliography
¬
Geometry in Art¬¬
Samuel Burroughs
Farmingdale State University
MTH 107
Prof. Prof. D'Ambrosio
April 29, 2015
Mathematics and art have always been closely related: Golden ratio, symmetry, proportion and
geometry are elements in the art; not surprisingly, many great artists of history have been great
mathematicians; they have been supported in mathematics to express reality with an artistic
language. By definition geometry comes from the Greek: Ge = earth and Metron = measure. That is,
it is the branch of mathematics that studies the measurement of the Earth that is concerned with the
properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
(Geometry. (2015).
Geometry is present in art since prehistoric times. Primitive people showed an intuitive notion of
geometry in their own ... Show more content on Helpwriting.net ...
(1998). The Golden Ratio & Squaring the Circle in the Great Pyramid. Retrieved
April 29, 2015,https://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html)
Geometry and the Liberal Arts ( 1978) Daniel pedoe geometry. (2015). In Encyclopaedia Britannica.
Retrieved from http://academic.eb.com/EBchecked/topic/229851/geometry golden ratio. (2015). In
Encyclopaedia Britannica. Retrieved from
http://academic.eb.com/EBchecked/topic/237728/golden–ratio
Parveen, N. (n.d.). GOLDEN RATIO AND THE ANCIENT EGYPT. Retrieved March 28, 2015,
from http://jwilson.coe.uga.edu/emat6680/parveen/ancient_egypt.htm
Pierce, Rod. (5 Apr 2014). "Index by Year and Subject". Math Is Fun. Retrieved 28 Apr 2015 from
http://www.mathsisfun.com/links/index.html
Pythagorean theorem. (2015). In Encyclopaedia Britannica. Retrieved from
http://academic.eb.com/EBchecked/topic/485209/Pythagorean–theorem
Estetica de Las Proporciones En La Naturaleza y Las Artes (Spanish Edition) (Spanish) Paperback
use pre formatted date that complies with legal requirement from media matrix – August, 2000 by
Matila C. Ghyka

Summary Of ' The ' By Scott Foresman Addison Wesley
Charles, R. I. (2002). Math. Glenview, IL: Scott Foresman Addison Wesley. This book is a teacher
edition and it discusses how you can teach math lessons. This book is volume 1 and it has 6 chapters
those include: Numbers to 12 and Graphing, Addition and Subtraction Readiness, Addition and
Subtraction Concepts, Facts and Strategies to 12, Geometry and Fractions, and More Fact Strategies.
This books provides examples of how you can incorporate technology and assess students.
Contestable, J. W. (1995). Number power. a cooperative approach to mathematics and social
development. Menlo Park, CA: Innovative Learning Publications, Addison–Wesley Pub. Co. This
book has activities that focus on including math and social development together to ... Show more
Chicago, IL: Everyday Learning Corporation. This book is a teacher 's manual that correlates with
the math workbook that is written above. It provides additional math lessons and describes how
partner and small group cen benefit student learning. It also uses geometric shapes like that having
students the draw a circle or square around certain objects.
Breyfogle, L., & Lynch, C. (2010). Van Hiele Revisited. Mathematics Teaching in the Middle
School, 16. doi:10.1075/ps.5.3.02chi.audio.2f
This article provides a description, example, and a teacher activity of all the 5 Van Hiele levels. The
more experience children have with the levels the better they can develop and move on to the next
level. Teacher need to monitor student progress so they can provide adequate instruction to move on.
Mason, M. (1992). The van Hiele Levels of Geometric Understanding. Professional
Handbook for Teachers. doi:10.1107/s0108768104025947/bm5015sup1.cif
This article discusses frequently asked questions about the Van Hiele levels which is great for
teachers and parents. For example, it describes that students can't skip levels because they have to
understand the previous one in order to move on. Progress is determined by educational experiences
rather than age. Geometry Content taught at grade level:
Iowa Core. (2008). Retrieved April 07, 2017, from https://iowacore.gov/ This shows all the Iowa
Core standards and it states specifically what the

High School Students Struggle With Geometry
The four articles studied discussed the challenges and opportunities for developing high school
graduates competent in geometric thinking capable of deductive reasoning and proficient in writing
proofs. Underlying all four articles is the assumption that high school students struggle with
geometry. These articles were written fifteen to thirty years ago. Sixty years ago, the Van Hiele's
noted that their high school students struggled with geometry and they did extensive research on the
problem, proposing a system of education to remedy the problem. Unfortunately in our classrooms
today, high school students still struggle with geometry and writing proofs.
The article "Geometry: why is it so difficult?" answers that question by asserting that students are
not introduced to proofs in the elementary grades and that teachers do not follow the Van Hiele
model for teaching geometry.
If students were at the abstract relational level before entering high school, they would successfully
complete the next level, formal deduction, where students write proofs using givens, definitions, and
axioms. Since most students are not at the abstract relational level when entering high school, the
authors suggest that students in elementary school and middle school informally explain and justify
arguments. They also suggest that in all subject areas, teachers should encourage students to think
and reason. They also believe that ensuring that the student have a good diet rich in iron and

Geometry Of Geometry And Geometry
In the beginning there was Euclid. The geometry we studied in high school was based on the
writings of Euclid and rightly called Euclidean geometry. Euclidean geometry is based on basic
truths, axioms or postulates that are "obvious". Born in about 300 BC Euclid of Alexandria a Greek
mathematician and teacher wrote Elements. The book is one of the most influential and most
published books of all time. In his book the Elements Euclid included five axioms that he deduced
and which became the basis for the geometry we now call Euclidean geometry. In Greek Euclid is
Εὐκλείδης which means "renowned, glorious". This fits his work for he has been called the "father
of geometry" and his works continue to influence mathematical fields today. Elements was first set
in type in 1482 in Venice making it one of the earliest mathematical books to be printed following
the invention of the printing press. It is estimated by Carl Benjamin Boyer to be second only to the
Bible in the number of editions published,[7] with the number reaching well over one thousand.[8]
For centuries the quadrivium was included in the curriculum of all university students and
knowledge of at least part of Euclid 's Elements was required of all students. When the content
became part of other textbooks, during the 20th century, it ceased to be considered something all
educated people had to read.[9]
The five axioms or postulates that Euclid presented were basically:
1. A straight line segment can be drawn

How Did Plato Contribute To Geometry
Plato and Archimedes: Two Great Mathematicians of All Time
Plato is one of the greatest mathematicians and teachers of geometry to ever live. To some, he is
known as the "maker of math." He himself made an academy that stressed mathematics as a way of
understanding reality. He founded this academy in 387 BCE. According to one website, "... he was
convinced that geometry was the key to unlocking the secrets to the universe." Plato had many
contributions in mathematics and geometry that helped in the past and still help in the present. One
thing that Plato did, that he probably best known for, is his identification of 5 3D symmetrical
shapes. The 5 shapes were: the tetrahedron, the octahedron, the cube, the icosahedron, and the
dodecahedron. The tetrahedron, which Plato represented as fire, is made of 4 triangles. The
octahedron, which Plato represented as air, is made of 8 triangles. The icosahedron, which is what
Plato represents as water, is made of 20 triangles. The cube, represented as earth according to Plato,
is made of 6 squares. The dodecahedron, which Plato describes it as "the god used for ... Show more
Most Greek's avoided the concept of actual infinity. Even mathematicians like Euclid said that there
were more prime numbers than any given finite number. The same website states, "Archimedes,
however, in the "Archimedes Palimpsest", went further than any other Greek mathematician when,
on compared two infinitely large sets, he noted that they had an equal number of members, thus for
the first time considering actual infinity, a concept not seriously considered again until Georg Cantor
in the 19th Century." That proves that Archimedes was very smart and very exact in his
measurements because he was the only one really to even think and use infinite numbers in a long
period of time. That also proves that he discovered many things and though outside the

Annual Islamic Symposium On The Arts And Sciences
Siddarth Kumar H Block Mathematics News Article Annual Islamic Symposium on The Arts and
Sciences Inside Dover–Sherborn Regional High School In Room 214 the Annual Islamic
Symposium on The Arts and Sciences has just concluded. The symposium was one of the most
important events of our time, where notable scholars and key figures met in the "House of Wisdom"
in order to hold panel discussions and present displays on the advancements in the fields of Art and
Science through the 15th century. Mathematics took center stage at the symposium with Al–
Khwarizmi speaking about history of his field during the Arab empire and discussing major
contributors to the field. Arabic Mathematics derived from the simple system of finger reckoning,
using one 's fingers in order to do basic forms of arithmetic. After some time "Finger reckoning
started to disappear with the introduction of Hindu arithmetic, the base for the current scheme of
numeration and calculation" (Esposito 184). Hindu numerals were used for these basic forms of
arithmetic more efficiently than Finger reckoning. This numeric system was easier for calculations,
rather than using Roman numerals, which is why it is considered a great achievement–moved the
possibilities of mathematics ahead. It was a base ten counting system that originated in India and
Al–Khwarizmi was the first person outside of India to rework this system, giving us the Arabic
numerals, which the numbers we use today are derived from. The abacus was the

The Importance Of Geometry
Problem statement.
Geometry is a requisite skill to be mastered (Copley, 2000). Geometry is the branch of mathematics
that deals with point, line, plane, space, spatial figures and the relationships among these(Kösa,
2016). It is an essential component of mathematics, and plays a crucial role in fill the gap between
mathematics and science. geometry is very important in architecture and design, in engineering and
in various aspects of life.(Abdullah & Zakaria, 2013) geometry is the aspect of mathematics that
involves shape, size, position, guidance, and movement and classifies and describes the physical
world we live in (Copley, 2000). Geometry is an interesting area of mathematics to teach (Keith
Jones, 2002)
The study of geometry helps students to develop the skills of critical thinking visualisation,
perspective, and intuition, solving problem, conjecturing, deductive reasoning, proof and logical
argument. We can be used to help students make sense of other branches of ... Show more content
on Helpwriting.net ...
The mass failure in mathematics examinations is real and the trend of student's performance has
been on the decline.(Adolphus, 2011)
Lack performance is not only results in child having a low confidence, but also causes significant
anxiety to the parents (Karande and Kulkarni, (2005). Students' attainment in geometry is of
considerable international interest (Poon & Leung, 2016).
So we should determination the major problems related with secondary school geometry. The
factors include; learners' negative attitude, lack of prerequisite concepts, inability to apply it,
conceptual difficulties, abstract nature of three dimensions , underutilization of instructional
resources, deficiency in problem solving skills, language indigence , teachers still use ineffective
and traditional mode to concept presentation (Ali et al., 2014; Origa Japheth,

Comparing Euclidean, Spherical Geometry, And
When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are
many similarities and differences among them. For example, what may be true for Euclidean
Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where
something is true for one or two geometries but not the other geometry. However, sometimes a
property is true for all three geometries. These points bring us to the purpose of this paper. This
paper is an opportunity for me to demonstrate my growing understanding about Euclidean
Geometry, Spherical Geometry, and Hyperbolic Geometry. The first issue that I will focus on is the
definition of a straight line on all of these surfaces. For a Euclidean plane the definition ... Show
In my homework I used two different proofs to prove the Vertical Angle Theorem on a Euclidean
plane and a sphere. The first idea I used was looking at the Vertical Angle Theorem using angle as
measure. The second idea I used was looking at the Vertical Angle Theorem using angle as rotation.
I have provided my homework so that one can see my reasoning behind both of these proofs. I
found that they worked on a Euclidean plane and a sphere. Although I did not have to say if my
proofs worked on a hyperbolic plane, I can say that they would because we can look at a hyperbolic
plane locally. From Chapter 6 in our textbook Experiencing Geometry by Henderson and Taimina,
we formulated a summary of the properties of geodesics on the plane, spheres, and hyperbolic
planes. I feel this is a good homework assignment to mention in this paper. For the first part of the
problem we were to explain why for every geodesic on the plane, sphere, and hyperbolic plane there
is a reflection of the whole space through the geodesic. For the second part of the problem I showed
that every geodesic on the plane, sphere, and hyperbolic plane can be extended indefinitely (in the
sense that the bug can walk straight ahead indefinitely along any geodesic). The third part asked to
show that for every pair of distinct points on the plane, sphere, and hyperbolic plane there is a (not
necessarily unique) geodesic containing them. In the fourth part of the problem I

How Did The Egyptians Influence Greek Math
Ancient Greek mathematicians contributed enormously to fundamental math that built up from
practical math such as geometry, engineering, astronomy and the astonishing academic contributions
to worldly influences. Greek mathematicians would the one of the earliest civilizations to transform
mathematics into rational thoughts when viewing all the concepts in the world. From ancient
mathematicians such as the Egyptians and Babylonians, they both viewed their calculations through
reasoning and using repeated observations to seek solutions to their equations. There was no real
framework of their proof being certain since geometric considerations played a second role in
arithmetic formulas. Greek mathematicians were influenced by the Egyptians and ... Show more
They were interested in proving that certain mathematical ideas were true and spent a lot of time
using geometry to prove their theories. Due to this, the Greeks were all influenced on the idea of
proof and they used logical stages to prove or disprove their theories and its solutions. Also to
distinguish the difference between what can work and what cannot. It can heavily influence on
means to convince someone or oneself that something is true through proved arguments based on
reason. In mathematics, a proof is a deductive argument for a mathematical statement and nobody
will ever find a counterexample, nor ever gainsay that particular mathematical fact (Krantz). That is
why math is based on deductive reasoning and through this mathematicians are reassured on their
absolute and proven theories. This also gave the building blocks to the mathematician Euclid and his
famous work, the Elements, which proved geometry from deductive reasoning to prove common
notions and postulates. Through the Elements, Euclid organized and presented the basics of
mathematical knowledge with results that were presented in a formally logical order. His logical
framework for geometry was very concise and even if one accepts the consensus view, it is still
reasonable to seek some sort of the explanation of the success of the practice (Avigad). Through
this, every statement demanded a

Philosophy Of Geometry
BRIEF PHILOSOPHY for 21st CENTURY MATHEMATICS EDUCATION Geometry is one of the
best classes your son or daughter can take to prepare them for their future. This course lays the
foundation for using logic, expressing reason in a practical progression, solving problems, applying
critical thinking skills, translating concrete ideas into abstract representations, and gaining spatial
reasoning skills while improving both their collaborative learner and independent learner strategies.
In a nutshell, it is my goal to make your kid THINK!
CLASS STRUCTURE & EXPECTATIONS
I set up my classroom to encourage collaborative learning. Students will often work in groups, so
they have opportunities to learn from each other. There are several activities ... Show more content
For instructions about signing up for Remind click here.
Learn more about Remind here.
My husband, Austin, and I spent part of our summer in Maui, Hawaii, on our honeymoon in 2013.
On a hike to visit a waterfall, we came across this AWESOME Banyan Tree. I could not help but
wonder how long that branch was, so I did what any normal person without any measuring tools
would do; I laid on it! If I am 5'3", about how long do you think that particular branch is?
About Me
This year is my eighth at WCHS and my ninth year teaching mathematics. I have taught Geometry
and Algebra I all nine years. I love spending time with my friends and family. My husband and I live
in Washington, and we are proud to reside in this strong and resilient community. This year is my
third as the Mathletes coach and my first year as a freshman class adviser. I am looking forward to
upcoming Homecoming activities and Boo Bash, as well as preparing for the Regional ICTM
(Illinois Council of Teachers of Mathematics) Math contest. Mrs. Cox and I are co–teaching Algebra
Block for our third year together, and we look forward to sharing our enthusiasm about math with
our class!

Geometry Scholarship Essay
Geometry, why did she take it a year early when she knew too well that she struggled in math? She
was pressured by her parents, teachers recommended her, and her friends expected her to take on the
challenge of geometry as a freshman. It was the first week of high school full of excitement and
confusion. She knew from the start that she would struggle, but could she pull through? Was a
passing grade within her reach? She always had a horrified look on her face during fourth period, for
she had taken notes, done the homework with ease, attended after school tutoring but constantly
drew a blank when exams were announced and taken. Exams read 'D' after 'D', week after week.
Geometry was the only class she struggled with, all A's in every single other academic class, but
geometry never seemed to get better. Sleepless nights not just studying but drowning herself in the
thought that she may not pass the class. Constantly crying herself to sleep because she knew that if
that one 'C' was unattainable she'd receive the first failing grade in her life, have to face her enraged
parents, irritated A.V.I.D teachers, forced to re–take the course, and be placed in a blended–learning
online class. ... Show more content on Helpwriting.net ...
Finals filled her mind ninety–five percent of the time, studying and studying was all she did for
hours every day. The only class that truly worried her was geometry. Was all the time she spent
getting tutored, staying after class asking questions, and studying the topics night and day worth it?
Was the passing grade she yearned for in reach at this point in the year? She felt confident about her
final exam, and the pending final grade tortured her during the winter break, it was the only thing
she could think about. When reports cards arrived not only was she disappointed in herself, but her
work. She knew she could have done better, but could she

Geometry Standards
During my level two's I was in a first grade classroom and I had a wonderful experience teaching at
Orchard Hill Elementary, I grew to love teaching first grade. My mother was also a preschool
teacher and my aunt was a kindergarten teacher, so I have always had a passion for teaching lower
elementary. I think first grade would be the perfect classroom to teach, because that is when students
are starting to become less dependent on help from the teacher. Students are still at the age where
they love going to school and are excited to learn. Also, I can empower them to keep having a love
of learning for years to come. Students in first grade are starting to develop their critical thinking
and problem solving skills, so this is very important when it comes to math. I plan on staying in
Iowa to teach, so I am going to be discussing how geometry can be taught in a first grade classroom.
Then, I will discuss how to implement the standards in a first grade classroom. Then, I will discuss
briefly how the Iowa Core and the Common Core compare in geometry standards and also describe
the three Iowa Core standards for geometry. As well as, explaining the Van Hiele levels and what
level first graders should be at. Overall, this paper will be explaining how teachers can apply
geometry content in there everyday, first grade classroom. ... Show more content on Helpwriting.net
...
The first standard for first grade is 1.G.A.1: Distinguish between defining attributes (e.g., triangles
are closed and three–sided) versus non–defining attributes (e.g., color, orientation, overall size);
build and draw shapes to possess defining attributes (Iowa Core, 2008). When learning this standard,
students will begin to understand the specific characteristics for any give shape. They will learn how
to draw such figures and be able to define their

Conic Sections in Taxicab Geometry
In this essay the conic sections in taxicab geometry will be researched. The area of mathematics
used is geometry. I have chosen this topic because it seemed interesting to me. I have never heard
for this topic before, but then our math teacher presented us mathematic web page and taxicab
geometry was one of the topics discussed there. I looked at the topic before and it encounter
problems, which seemed interesting to explore. I started with a basic example, just to compare
Euclidean and taxicab distance and after that I went further into the world of taxicab geometry. I
explored the conic sections (circle, ellipse, parabola and hyperbola) of taxicab geometry. All
pictures, except figure 12, were drawn by me in the program called Geogebra. ... Show more content
All of them are distant from the origin or 5 units (kilometers) and that is where the Euclidian and
taxi distance match each other. To find other points, we should move among x–axis and then up and
down as far as possible. We have enough fuel for five kilometers in one–way, which means that:
|x|+|y|=5
When solving the equation we get: y=5–x for x,y>0 y=x–5 for x>0; y≤0 y=–x–5 for x,y≤0 y=x+5
for x≤0;y>0
So when all the functions are drawn we get the final picture of taxi circle.
Figure 4: Picture showing a taxicab circle of radius 5.
CIRCLE
A circle is a set of all points that are given distance, called radius, usually denoted by r, away from
the center.
An equation of Euclidean circle:
〖(x–h)〗^2+〖(y–k)〗^2=r^2
Figure 5: An Euclidean circle
Figure 6: A taxicab circle
CONIC SECTIONS OF TAXICAB GEOMETRY
In the following part of an essay, the conic sections in taxicab distance are researched, as they must

vary from the Euclidean distance as well as circle of taxicab distance from circle of Euclidean
distance.
WHAT ARE CONIC SECTIONS the conic section is the locus of a point, such that its distance from
focus is in the constant ratio to its distance from directrix ratio, called e; if e<1, the conic section is
an ellipse, if e =1 a conic section is a parabola and if e >1 the conic section is a

Rubaiyat Of Omar Khayyam : Poem : Mathematics And Mathematics
As a mathematician, he is most notable for his work on the classification and solution of cubic
equations, where he provided geometric solutions by the intersection of conics.[5][6] As an
astronomer, he composed a calendar which proved to be a more accurate computation of time than
that proposed five centuries later by Pope Gregory XIII.[7]:659[8] Omar was born in Nishapur, in
northeastern Iran. He spent most of his life near the court of the Karakhanid and Seljuq rulers in the
period which witnessed the First Crusade. There is a tradition of attributing poetry to Omar
Khayyam, written in the form of quatrains (rubāʿiyāt ‫)رباعیات‬. This poetry became widely known to
the English–reading world due to the translation by Edward FitzGerald ... Show more content on
Helpwriting.net ...
In 1076 Khayyam was invited to Isfahan by the vizier and political figure Nizam al–Mulk to take
advantage of the libraries and centers in learning there. His years in Isfahan were productive. It was
at this time that he began to study the work of Greek mathematicians Euclid and Apollonius much
more closely. But after the death of Malik–Shah and his vizier (presumably by the Assassins' sect),
Omar had fallen from favour at court, and as a result, he soon set out on his pilgrimage to Mecca. A
possible ulterior motive for his pilgrimage reported by Al–Qifti, is that he was attacked by the clergy
for his apparent skepticism. So he decided to perform his pilgrimage as a way of demonstrating his
faith and freeing himself from all suspicion of unorthodoxy.[4]:29 He was then invited by the new
Sultan Sanjar to Marv, possibly to work as a court astrologer.[1] He was later allowed to return to
Nishapur owing to his declining health. Upon his return, he seemed to have lived the life of a
recluse.[13]:99 Khayyam died in 1131, and is buried in the Khayyam Garden. Mathematics[edit]
"Cubic equation and intersection of conic sections" the first page of two–chaptered manuscript kept
in Tehran University. Khayyam was famous during his life as a

Euclidean Geometry Compare And Contrast
When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are
many similarities and differences among them. For example, what may be true for Euclidean
Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where
something is true for one or two geometries but not the other geometry. However, sometimes a
property is true for all three geometries. These points bring us to the purpose of this paper. This
paper is an opportunity for me to demonstrate my growing understanding about Euclidean
Geometry, Spherical Geometry, and Hyperbolic Geometry.
The first issue that I will focus on is the definition of a straight line on all of these surfaces. For a ...
Show more content on Helpwriting.net ...
These symmetries include: reflection–in–the–line symmetry, reflection–perpendicular–to–the–line
symmetry, half–turn symmetry, rigid–motion–along–itself symmetry, central–symmetry, and self–
symmetry. If a line on a hyperbolic plane satisfies these conditions then we can say that it is straight.
I have included my homework of my definition of a straight line on a hyperbolic plane so that one
can see why these conditions must be satisfied.
The next issue that I will address for these three geometries is the definition of an angle on all three
surfaces. The definition that I will give applies to all three surfaces. There are at least three different
perspectives from which we can define "angle". These include: a dynamic notion of angle–angle as
movement, angles as measure, and angles as a geometric shape. A dynamic notion of angle involves
an action which may include a rotation, a turning point, or a change in direction between two lines.
Angles as measure may be thought of as the length of a circular arc or the ratio between areas of
circular sectors. When thinking of an angle as a geometric shape an angle may be seen as the
delineation of space by two intersecting lines. I have provided my homework assignment on my
definition of an angle so that one can see the reasoning of my definition for all three surfaces.
However, my homework assignment does not ask to define an angle on a hyperbolic plane. This is
because a region on a hyperbolic plane

Who Is Leonhard Euler?
Leonhard Euler was born on April 15, 1707, in Basel, Switzerland, Leonhard Euler was one of
math's most pioneering thinkers, establishing a career as an academy scholar and contributing
greatly to the fields of geometry, trigonometry and calculus. He released hundreds of articles and
publications during his lifetime, and continued to publish after losing his sight.
Euler showed an early aptitude and propensity for mathematics, and thus, after studying with Johan
Bernoulli, he attended the University of Basel and earned his master's during his teens. Euler served
in the navy before joining the St. Petersburg Academy as a professor of physics and later heading its
mathematics division. In the mid–1740s, Euler was appointed the mathematics director

Eudoxus Major Accomplishments
Eudoxus of Cnidus, born in 408 B.C. and died 355 B.C in Cnidus, Asia Minor, was well–known for
his multitude of great achievements; both in astronomy and mathematics. However, focusing on his
mathematical accomplishments, Eudoxus has made a significant mark in history due to his
numerous contributions to mathematics. Working along with many famous philosophers and
mathematicians such as Archytas, Philiston, and Plato, Eudoxus was able to broaden his knowledge
and thus help lead modern–day mathematicians to a greater comprehension of the universe.
Eudoxus, son of Aeschines, as a young man traveled throughout Greece and Egypt with the hopes of
expanding his intellect of the world. One of his first voyages consisted of a journey to Tarentum. ...
Show more content on Helpwriting.net ...
However, Pythagoreans "believed that all phenomenon can be reduced to whole numbers or their
ratios." (Lecture 8. Eudoxus, Avoided a Fundamental Conflict) In that Hippasus' discoveries
contradicted the Pythagoreans' conjecture, they murdered him. As time continued, more and more
irrational numbers began to appear; making it hard to ignore the fact that not all phenomenon is
reducible to whole numbers or their ratios. For this reason, the Greeks quickly realized that a
solution to irrational numbers was imperative. Fortunately, Eudoxus was able to present a new
theory of proportions which did not involve numbers. "Instead, he studied geometrical objects such
as line segments, angles, etc., whiling avoiding giving numerical values to lengths of line segments,
sizes of angles, and other magnitudes." (Lecture 8. Eudoxus, Avoided a Fundamental Conflict) By
using geometry to evade irrational numbers, a mathematical crisis had been covered. Although
Greeks could not tolerate irrational numbers, they accepted "irrational geometric quantities such as
the diagonal of the unit square" (Lecture 8. Eudoxus, Avoided a Fundamental Conflict), or square
root

Application Of Computational Geometry On View
Application of Computational Geometry on View
Planning
Name: Pravakar Roy
Student ID:4927267
Graduate Student
Department of Computer Science
University of Minnesota, Twin Cities
April 27, 2015
Abstract
View planning is a crucial part of building vision system for autonomous robots or critical coverage
problems. In computational geometry the problem of covering/guarding a region is known as the art
gallery theorem. The version where static guards are re– placed by mobile guards is known as the
watchman route theorem. Both these classic theorem adopts the straight line notion of visibility
which is impractical for real world sensors. In this work we review the existing techniques for
solving the classical ver– sion of these problems and also discuss the modi cations needed to
handle real world sensors. We also discuss some recent methods that takes physical visibility
constraints into account.
1 Introduction
Producing autonomous robots that can operate without any sort of human intervention is one of the
ultimate goals in robotics [42]. Such robots should be capable of exploring their environment,
interpret the surroundings and act accordingly. Vision is clearly one of the most critical capability
such a robot must have to act autonomously. When exploring an unknown environment, a robot is
rst required to extract information about the surroundings.
Following this exploration, it should be able to carry out speci c task such as nding potentially
hazardous materials in the

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