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Topic: Pascals triangle

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Fibonacci series

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Sierpinski triangle

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Blaise Pascal

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	Generating Pascals Triangle
		Pascal's Triangle is a well known mathematical pattern.
		According to Yunze He, "Pascal's" triangle was first developed during the Song Dynasty by a mathematician named Hue Yang.
		The simplest view of Pascal's Triangle is that it may be generated by affixing a one a either end of the new row and then generating all numbers in between by by adding together the two numbers above it.
	www.geocities.com /CapeCanaveral/Launchpad/5577/musings/Pascals.html (515 words)

	Pascals triangle - Hutchinson encyclopedia article about Pascals triangle (Site not responding. Last check: 2007-10-20)
		In Pascal's triangle, each number is the sum of the two numbers immediately above it, left and right – for example, 2 is the sum of 1 and 1, and 4 is the sum of 3 and 1.
		It is named after French mathematician Blaise Pascal, who used it in his study of probability.
		When plotted at equal distances along a horizontal axis, the numbers in the rows give the binomial probability distribution (with equal probability of success and failure) of an event, such as the result of tossing a coin.
	encyclopedia.farlex.com /Pascals+triangle (210 words)

	All You Ever Wanted to Know About Pascal's Triangle and more
		Pascal's Triangle was originally developed by the ancient Chinese, but Blaise Pascal was the first person to discover the importance of all of the patterns it contained.
		A number in the triangle can also be found by nCr (n Choose r) where n is the number of the row and r is the element in that row.
		When all the odd numbers (numbers not divisible by 2) in Pascal's Triangle are filled in (fl) and the rest (the evens) are left blank (white), the recursive Sierpinski Triangle fractal is revealed (see figure at near right), showing yet another pattern in Pascal's Triangle.
	ptri1.tripod.com (1016 words)

	Pascals Triangle Fibonacci Numbers
		Pascal's Triangle Blase..numbers (1 4 6 4 1) are the fourth row of numbers in Pascal Triangle (Ladja, 5).
		To..numbers (1 4 6 4 1) are the fourth row of numbers in Pascal Triangle (Ladja, 5).
		The numbers are not in triangular..a numerical triangle, or an arithmetic triangle.
	www.fibonacci-market-scanner.com /fibonacci-pages/Pascals-Triangle-Fibonacci-Numbers.htm (917 words)

	[No title]
		Pascals father had a very special view on education, and therefore he decided to teach his son by himself.
		Pascal started using all his sparetime trying to learn geometry, and as the great natural genius he was, he soon started to discover many properties of the figures.
		Pascal was the second person in history to invent a mechanical calculator, as Schickard manufactured one in 1624.
	home.online.no /~sim-bakk/pascal/who.htm (1017 words)

	Math Forum: Ask Dr. Math FAQ: Pascal's Triangle
		Pascal's Triangle is an arithmetical triangle you can use for some neat things in mathematics.
		The other main area where Pascal's Triangle shows up is in Probability, where it can be used to find Combinations.
		In about 1654 Blaise Pascal started to investigate the chances of getting different values for rolls of the dice, and his discussions with Pierre de Fermat are usually considered to have laid the foundation for the theory of probability.
	mathforum.org /dr.math/faq/faq.pascal.triangle.html (751 words)

	Fibonacci Sequence Pascals Triangle
		fibonacci and pascal essays on the golden section, the Fibonacci sequence, Pascal's triangle, logarithms, the..com/cgi-brookscole/course_pr..
		Comprehensive..between Pascal's triangle and the Fibonacci sequence interesting, followed by http..resource links and more..fibonacci and pascals triangle..
		Pascal's triangle related to patterns in Pascal's triangle.
	www.fibonacci-market-scanner.com /fibonacci-pages/Fibonacci-Sequence-Pascals-Triangle.htm (888 words)

	Pascal's triangle
		Although named after Blaise Pascal, who studied it, this arithmetic triangle has been known about since the twelfth century and has a variety of other names.
		The Chinese triangle appears again in 1303 on the front of Chu Shi-Chieh's Ssu Yuan Yü Chien (Precious Mirror of the Four Elements), a book in which Chu says the triangle was known in China more than two centuries before his time.
		In addition, the shallow diagonals of the triangle sum to give the numbers in the Fibonacci sequence.
	www.daviddarling.info /encyclopedia/P/Pascals_triangle.html (294 words)

	Free Term Papers on Pascals Triangle
		Pascal’s Triangle is one of the strangest triangles around.
		Pascal’s Triangle is many numbers in the form of a triangle.
		In spite of the fact that Blaise Pascal did not actually discover the triangle, he was the first to find all of the patterns within the triangle itself.
	www.freefortermpapers.com /show_essay/3919.html (423 words)

	KryssTal : Pascal's Triangle
		In fact, the triangle was known to both the Chinese and the Arabs for several hundred years previously.
		It is not a geometrical triangle but a triangle of numbers.
		From Pascal's Triangle, there are total of eight objects so you look at the line beginning with 1, 8, etc. You want to select six so you count along from zero, until you count six.
	www.krysstal.com /binomial.html (2031 words)

	Sierpinski Gasket (Site not responding. Last check: 2007-10-20)
		While it is easy to understand that the points all lie within the triangle specified by the 3 vertices it is not so easy to imagine that there are "large" areas where points will never be drawn.
		If the entry in pascals triangle is odd then it is part of the gasket otherwise it is not part of the gasket.
		Instead of removing the central third of a triangle, the central square piece is removed from a square sliced into thirds horizontally and vertically.
	astronomy.swin.edu.au /~pbourke/fractals/gasket (1697 words)

	Pascal's Triangle and the Fibonacci Series
		Pascal's Triangle, developed by the French Mathematician Blaise Pascal, is formed by starting with an apex of 1.
		Every number below in the triangle is the sum of the two numbers diagonally above it to the left and the right, with positions outside the triangle counting as zero.
		Pascal's triangle has many unusual properties and a variety of uses:
	goldennumber.net /pascal.htm (245 words)

	Pascals Triangle
		Pascal’s Triangle Blasé Pacal was born in France in 1623.
		Pascal's triangle is a triangluar arrangement of rows.
		This combination of 1 and 1 is the firs row of Pascal's Triangle.
	www.freeessays.cc /db/30/mdg32.shtml (777 words)

	BBC - h2g2 - Pascals Triangle
		In 1653, a french mathematician named Blaise Pascal described a triangular arrangement of numbers corresponding to the probabilities involved in flipping coins, or equivalently the number of ways to choose n objects from a group of m indistinguishable objects.
		One way of seeing some of these patterns is to pick a number x and color all numbers in the triangle that are evenly divisible by x with one color, and all the other numbers in the triangle with a second color.
		To see as much of the pattern as possible, you need to be able to see as many rows of the triangle as possible, but coloring a large number of rows like this by hand is very boring and time consuming.
	www.bbc.co.uk /dna/ww2/A280306 (324 words)

	Chinese Mathematics : Rebecca and Tommy
		The so called 'Pascal' triangle was known in China as early as 1261.
		In '1261 the triangle appears to a depth of six in Yang Hui and to a depth of eight in Zhu Shijiei (as in diagram 6) in 1303.
		It was therefore credited to Pascal due to the ingenious use he made of it in probabilities.
	www.roma.unisa.edu.au /07305/pascal.htm (457 words)

	[No title]
		From: wodzak@cs.missouri.edu (Michael Wodzak) Newsgroups: sci.math Subject: pascal's triangle Date: 17 Apr 1995 14:15:13 GMT A couple of months ago I posted this problem, which was answered (I am told by colleagues) almost instantly, by someone in Harvard.
		>v1 is the k_th row of pascal's triangle followed by an infinite tail of zeros.
		Subject: Re: pascal's triangle The Tarry-Escott problem: given a positive integer n, find two sets of integers a_1,..., a_r and b_1,..., b_r, with r as small as possible, such that sum (a_j)^k = sum (b_j)^k for k = 1, 2,..., n.
	www.math.niu.edu /~rusin/known-math/94/multigrades (1364 words)

	[No title]
		Starting off the activity with a hands-on exploration, the students cut acute triangles out of stiff paper and draw the centroid by constructing line segments from the vertexes to the midpoints of the opposite sides.
		With Geometer's Sketchpad they are able to measure the area of each small triangle and drag the vertex of the large triangle to observe the always equal areas.
		Pascals Triangle is one of the most interesting and useful mathematical phenomenon.
	www.stolaf.edu /people/wallace/Courses/MathEd/Reviews/Reviews98/Janareviews (4166 words)

	Pascals Triangle, Mathematics, Free Essays @ ChuckIII College Resources
		When Pascal was 19 he invented the first calculating machine that actually worked.
		Because of this connection, the entries in Pascal's Triangle are called the binomial coefficients.There's a pretty simple formula for figuring out the binomial coefficients (Dr. Math, 4): n!
		If you then multiply 1331 x 11 you get 14641 which is the 4th line in Pascal's Triangle, but if you then multiply 14641 x 11 you do not get the 5th line numbers.
	www.chuckiii.com /Reports/Mathematics/Pascals_Triangle.shtml (749 words)

	[No title]
		On the yop of Pascal's Triangle is the number 1, that is the zeroth row.
		A number in the triangle can also be found by nCr where n is the number of the row and r is the element in that row.
		For example, in row 3, 1 is the zeroth element, 3 is the first element, the next 3 is the second element, and the last 1 is the 3rd element.
	home.online.no /~sim-bakk/pascal/triangle.htm (716 words)

	Blaise Pascal
		At 16 he wrote a paper on conic sections which was thought to be the most powerful and valuable contribution that had been made to mathematical science since the days of Archimedes (287-212 BC).
		Invented Pascals Triangle to determine the probability of certain outcomes.
		Because of this near death experience, from age 31 til he died at 39 Blaise Pascal's desire was to point men's thoughts to his savior.
	www.godcreatedthat.com /BlaisePascal.html (447 words)

	Nov 13 index (Site not responding. Last check: 2007-10-20)
		This paper presents a 3-dimensional version of Pascal's triangle, which I have named 'Pascal's Pyramid'.
		We will see that the numbers in the Pascal's pyramid are linked to the numbers in Pascal's triangle.
		If we picture the entire Pascal's Pyramid as a solid, each of the surfaces around the apex consists of a Pascal's triangle.
	www.marksr.demon.co.uk /M500_Pascal's_Pyramid.htm (442 words)

	Leadership Program in Discrete Mathematics
		The first (red) diagonal of Pascal's triangle indicates the numbers of new gifts given on the consecutive days.
		The second (green) diagonal of Pascal's triangle indicates the combined gifts given on the consecutive days.
		Continue the pattern and see if you can find the cell in Pascal's triangle that shows the total number of gifts given over the twelve-day celebration.
	dimacs.rutgers.edu /~judyann/LP/lessons/12.days.pascal.html (399 words)

	Classroom Activities Using Number Patterns
		Pascal's triangle, named after the seventeeth-century French mathematician Blaise Pascal, turns up in a number of mathematical problems.
		The numbers of paths to the fifth row intersections are 1 4 6 4 1.
		Tell them it comes up in a number of mathematical contexts, the two most common areas being the probability of coin tosses, and the expansion of binomials in high school algebra.
	www.dpgraph.com /janine/mathpage/patterns.html (2107 words)

	Pascal's Triangle (Site not responding. Last check: 2007-10-20)
		Pascal's Triangle displays the structure and interrelationships of Number.
		So there is a relationship between the 5th level of the triangle and the number of ways that "5" can be partitioned.
		This seems to correspond with the 6th level of Pascal's Triangle1,6,15,20,15,6,1.
	www.virtuescience.com /zztriangle.html (351 words)

	metalesson_11_27 (Site not responding. Last check: 2007-10-20)
		Previous to today the only relationship that I remembered in Pascals triangle was the relationship to expanding out perfect squares.
		I also think that it is interesting to use Pascals triangle with middle school and high school students since I dont really remember learning Pascals triangle until sometime in college.
		I was surprised at how much people knew about Pascal's triangle considering I didnt really know anything about it coming in to the lesson.
	www.mste.uiuc.edu /courses/ci303fa01/students/mulford/Metalessons/metalesson_11_27.html (290 words)

	Jim Frankenfield - Science Page
		Pascals triangle is known in Italy as Tartaglias Triangle.
		Italian algebraist Niccolo Fontana Tartaglia lived a century earlier than Blaise Pascal and is credited with the first general formula for solving cubic polynomials.
		Pascals (Tartaglias) Triangle is perhaps best known for providing the coefficients in a binomial epansion, but it also has many other fascinating properties.
	www.snowman-jim.org /science/tartaglia.php (102 words)

	Pascals Triangle (Site not responding. Last check: 2007-10-20)
		To form the "hockey stick", and understand the basis of this neat pattern, first draw a diagonal line downwards from one of the number 1's on the triangle.
		Look at the triangle with the numbers filled at the bottom of the page and try to find a "hockey stick" for your self.
		Here are some examples like below; the numbers that are being added up are the ones in the diagonal and the bottom part of the "hockey stick" or the blade of the stick is the number that the diagonal equals.
	www.ga.k12.pa.us /academics/us/Math/Geometry/stwk00/sloanelikemen/paterns.htm (558 words)

	[No title]
		Especially with the birthday example today, it was q> difficult at first but gave me a new look at something I definitely q> wouldn't have thought of.
		Pascals Triangle was obviously rushed at the end so q> I can't mark my understanding as good, yet.
		I didn't quite q> get the significance of the Pascal's Triangle, but other than that, q> everything was good.
	www.cs.cornell.edu /Courses/cs100m/2001sp/Examples/feed-02-01.txt (1120 words)

	Pascal's Triangle (Site not responding. Last check: 2007-10-20)
		You may choose to lead the students in short discussions on multiples and integer multiplication, remainders and Euclidean division, and / or Pascal's triangle.
		Have the students try the computer version of the Coloring Remainders activity to investigate the patterns of the remainders in Pascal's triangle.
		Have the students try coloring by hand, using copies of the paper version of Pascal's triangle.
	www.shodor.org /interactivate/lessons/frac6.html (673 words)

	Articles - Sound (Site not responding. Last check: 2007-10-20)
		Amplitude is the maximal displacement of particles of matter that is obtained in compressions, where the particles of matter move towards each other and pressure increases the most and in rarefactions, where the pressure lessens the most.
		While the pressure can be measured in pascals, the amplitude is more often referred to as sound pressure level and measured in decibels, or dBSPL, sometimes written as dBspl or dB(SPL).
		In fact, most sound waves consist of multiple overtones or harmonics and any sound can be thought of as being composed of sine waves (see additive synthesis).
	www.edscamera.com /articles/Sound (1031 words)

	Pascals Triangle (Site not responding. Last check: 2007-10-20)
		Blaise Pascal was born in 1623-1662, in Clermont- Ferrand, France, but lived mostly in Paris.
		Besides the triangle, he also invented a calculating machine (1647), and later the barometer and the syringe.
		At the top of Blaise Pascal's Triangle is the number 1.
	www.germantownacademy.org /academics/US/Math/Geometry/stwk00/sloanelikemen/background.htm (267 words)

	Investigations into Pascal's Triangle (Site not responding. Last check: 2007-10-20)
		The entries of Pascal's triangle are found by adding the single or two numbers immediately above that entry.
		In the above example, the numbers in the sixth layer can be obtained directly from the first six layers of Pascal's triangle by multiplying successively each row by the entries in the sixth row.
		As a conclusion, it was found that the even entries far outweigh the odd entries in Pascal's Triangle.
	www.irishscientist.ie /2000/contents.asp?contentxml=237as.xml&contentxsl=insight3.xsl (398 words)

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