Do You Speak Mathematics? Projective geometry lay abandoned for about two hundred years; and Pascal's result Next,have a look at the following triangle formed by three pascals http://www.cut-the-knot.com/ctk/Mathematics.shtml
Extractions: Brief Lives , Penguin Books, 2000, p. 74 Do you speak mathematics? is a very valid question assuming mathematics is a language. Many think it is. Josiah Willard Gibbs gave a speech on that account. Galileo and R. Feynman thought so. Mathematics is even judged to be a universal language, the only one suitable to initiate extraterrestrial communication [ Jacobs , p. 1]. Some object. For example, Jean-Pierre Bourguignon argues [ Basis , p. 173] that For some people, mathematics is just the language of the quantitative . This opinion is shared by some of our fellow scientists. ... We mathematicians know how wrong this opinion is, and how much effort goes into building concepts, making new links, establishing facts, and following avenues we once thought plausible but turned out to be dead ends. I generally accept the above sentiment with reservations concerning mathematics being just the language of quantitative The potency of the bond between mathematics and its language is such that many mathematicians do indeed identify the two [ Spectrum , p. 112]:
Extractions: Problems from the Past. (Concerning the text "Higher Algebra" reviewed by Ed Staples, Erindale College) A few weeks ago I happened to find an interesting 2 nd hand bookshop in a local shopping centre. Among the owner’s copious volumes I happened upon a book entitled " Higher Algebra " co-authored by H. Hall and S Knight published as a 4 th Edition by McMillan and Co publishers in 1913. The book designed as a sequel to " Elementary Algebra for Schools " was first published in 1887 and had had no less than 20 reprints since that time. Doubtless to say some readers may have a copy of this marvellous volume on their own bookshelf. The copy I found is in surprisingly good condition and apart from a few changes in notation, the contents are extremely readable. Contained in the 560 pages there are some wonderful examples that employ some algebraic techniques that through time and other reasons have disappeared from current use. Of course Hall and Stevens published the perhaps more well known " A School Geometry" a copy of which I still have amongst my collection. I hope you enjoy the following problems.
Katedra Didaktiky Matematiky Mathematics analytic geometry ucebnice SS, 3., Bratislava, Slovenske pedagogicke Numbersof Fibonacci and pascals triangle C, Rozhledy matematicko-fyzikalni http://www.karlin.mff.cuni.cz/knihovna/publik94/publ10.htm
Maths@work - Famous Mathematicians a conic. pascals triangle which give coefficients of the expansion ofa binomial. The geometry of the cycloid. Mathwise Mathematical http://www.mathsatwork.com/famous_mathematicians/pascal.html
Extractions: Blaise Pascal 1623-1662 . Born in Clermont, France on June 19th, the son of a judge. . Moved to Paris after mothers death. Educated by father and looked after by sisters Gilberte and Jacqueline. . Admitted to the circle of French geometers which became the French Adademy. . Constructed an 'arithmetic machine.' . Patented a calculating machine. . Turned to religion. . Corresponded with Pierre de Fermat and established early probability theory. Had a visionary revelation which caused him to retreat from the world and join his sister in retreat. . Gave up home as act of self-denial and went to live with Gilberte. Died in Paris on August 19th. The basics of probability theory (with Pierre de Fermat The geometry of conics, including 'Pascals theorem' on hexagon inscribed in a conic. Pascals triangle which give coefficients of the expansion of a binomial. The geometry of the cycloid.
A Biography On Blaise Pascal On The 19th Of June, 1623, Blaise The following years Pascal studied geometry, and wrote a lots of texts concerning wasthe one that made the biggest breakthroughs working on pascals triangle. http://home.c2i.net/greaker/comenius/greaker9900/pascal/who.htm
Extractions: On the 19th of June, 1623, Blaise Pascal was born in Clermont in France. His mother died when Blaise was only at the age of 3, and it was his father, Etienne Pascal, who took care of Blaise and his three sisters. In 1631 the family moved to Paris. The reasons for moving were that the father could prosecute his scientific studies, and he could also give Blaise a good education in Paris.
Biopasca In 1632, the pascals left Clermont for Paris, where Blaise's father the sum of theangles of a triangle is two gave him a copy of a Euclidian geometry textbook http://www.andrews.edu/~calkins/math/biograph/199899/biopasca.htm
Extractions: Summary: Important Points Background Blaise Pascal, the only son of Etienne Pascal, was born on June 19, 1623 in what was Clermont (now Clermont-Ferrand), Auvergne, France. In 1632, the Pascals left Clermont for Paris, where Blaise's father took it upon himself to educate the family. Thus, Pascal was not allowed to study mathematics until the age of 15, and all math texts were removed from the house. Despite all this, Blaise's curiosity grew and he began to work on geometry himself at the age of 12. After discovering that the sum of the angles of a triangle is two right angles, his father relented and gave him a copy of a Euclidian geometry textbook. An Early Achiever Blaise Pascal made many discoveries between the ages of fourteen and twenty-four. At fourteen, he attended his father's geometry meetings, and at 16, he composed an essay on conic sections, which was published in 1640. Between the ages of 18 and 22, he invented a digital calculator, called a Pascaline, to assist his father in collecting taxes.
Patterns - Secondary Level In this book, students study history and geometry as they arithmetic, symmetry,design transformation, Latin squares, pascals triangle, repeating circles http://www.aw.com/canada/school/math/mr/pat/pattxt.html
- Numberful.net: Homework, Private Lessons And Lectures On Math Basic and Vector geometry, 2, May 18 1204 pm. Binomial theorem, pascals triangle,simple equations, solving equations and term shaping, inequalities http://www.numberful.net/homework/messages/board-topics.html
Extractions: Moderators: Main Category Posts Latest Post Basic Calculations and Sets October 01 - 11:10 am Numbers October 20 - 06:13 pm Usual sets of numbers, complex numbers, systems of numbers, period, greatest common divisor, smallest common multiple, prime numbers, fundamental theorem of arithmetics, ... Basic and Vector Geometry May 18 - 12:04 pm Triangles, squares, circles, areas and volumes, geometrical constructions, trigonometric formulas, vectors, lines, planes, dot and cross product, shortest distance calculations, intersections, linear dependencies, ... Algebra, Equations and Matrices October 01 - 11:12 am Binomial theorem, pascals triangle, simple equations, solving equations and term shaping, inequalities, polynomials and polynomial division,liear equations, matrices, determinants, problems, ... Sets, Groups and Vector Spaces October 26 - 03:59 pm Sets and operations on sets, groups, vector spaces, mappings, linear combinations, basis of a vector space, orthonormalization, cross product, dot product, ... Functions October 26 - 04:11 pm Linear functions, quadratic functions, polynomials, logarithms, roots, exponential functions, circular functions, graphs and concepts of functions, symmetry, maximum and minimum problems, ...
MathsNet: ICME 9 calculus visually and links between dynamic geometry and the snooker table Pythagoras'theorem conic sections pascals triangle, probability distributions http://www.mathsnet.net/courses/icme9acc/
Extractions: The conference took place in Makuhari, which is to the east of Toyko, in a new conference centre. At the Opening ceremony in Makuhari Event Hall a sequence of welcoming messages were delivered (with simultaneous Japanese/English translation) from key figures in Japanese education and organisers of ICME. Messages were quoted from the Japanese Prime Minister and the US President. At the International Round Table , video conferencing was used to link Japan with Singapore and USA. Mr Wee Heng Tin, Director General of Education, Singapore, talked about the possibilities of the Internet, with users customising content. 40% on Singapore households are on-line. But what about the "digital divide" - those with on-line access and those without? The US speaker Bruce Alberts talked about hands-on learning and learning how to learn. Akito Akihira discussed issues caused by students downloading without understanding. Gilah Leder from Australia was ambivalent about technology, mentioning again the digital divide in less developed countries compared to affluent homes. Teachers have inadequate training in new technologies; in fact their students are more proficient.
Class Reviews, 1998 Driscoll, 1995, Farther Out Keywords geometry, Technology; Finzer, Bennett, 1995 Green Hamberg, 1986, pascals triangle Keywords Connections, Issues; Kennedy http://www.stolaf.edu/people/wallace/Courses/MathEd/Reviews/Reviews98/reviewinde
Extractions: Compilation, 1995, September Calendar Hallowell, 1995, Case of Blue Schielack, 1995, Football Coach's Grassl, Mingus, 1998, Keep Counting ... Hansbarger, Stewart, 1996, Merging Mathematics BSwiggum1 Janareviews Joelreviews Krisreviews Loanarticles Loanreviews Loansreviews Maritreviews Maritsreviews Maritsreviews.asc Markreviews Reginareviews reviewindex98.html temp Return to Keywords Gonzales, Fernandez, Knecht, 1996, Active Participation Compilation, 1995, September Calendar Hallowell, 1995, Case of Blue Schielack, 1995, Football Coach's ... Return to Keywords Nicol, 1997, Physics Teacher Stacey, MacGregor, 1997, Ideas about Symbolism Dougherty, 1996, The Write Way Bethell, Miller, 1998, From an E to an A ... Return to Keywords Return to Keywords Vincent, 1996, Informal Assessment Driscoll, 1995, Farther Out Return to Keywords Dubinsky, 1995, Is Calculus Return to Keywords Return to Keywords Paul, 1995, Return of Matt
Java Resources - Graphing The applet graphically presents pascals triangle moduloan 14October-1999 State 503 - Service Unavailable - geometry and GIS triangle Applet http://www.infosys.tuwien.ac.at/melange/html/educational.math.graphing.10.html
Extractions: Description: This applet graphs a user-supplied function and computes the area under the curve in the specified interval. One may choose from rectangle rules, trapezoids, or adaptive quadrature (using Simpsons rule). P I find this useful to demonstrate the definite integral, numerical integration, and the definition of ln(x) as the area under 1/t from 1 to x.
Ask Jeeves: Search Results For "Pascal's Triangle Binomial Expansion" and math homework help from basic math to algebra, geometry and beyond. still underconstruction) I plan to discuss the pascals' triangle, Sierpinski triangles http://webster.directhit.com/webster/search.aspx?qry=Pascal's Triangle Binomial
7th Grade Pre-Algebra And 8th Grade And High School Algebra pascals father had originally banned any mathematic text or gave up playtime to secretlystudy geometry and amazingly He found that any triangle is made up of http://mcnutty-professor.tripod.com/
Extractions: "The direction in which education starts a man will determine his future life" - Plato This week in Algebra we are learning about monomials. Middle School Algebra will test on systems of equations on Wednesday. Pre-Algebra is continuing to work with probability and will test on this chapter on Wednesday also. The History of Mathematics Blaise Pascal Born June 9, 1623 in Clermont Auvergne France Blaise Pascal's work in the fields of physics, geometry, probability and calculus is outstanding. Not only is his work recognized in its own right, but his contributions lay the foundation for many area of mathematical discovery in the years that followed. Pascals father had originally banned any mathematic text or instruction from the household until young Pascal reached the age of 15. He asked that tutors concentrate first on language studies.
MAA Florida Section Newsletter--February 2001 One of the Great Theorems in algebraic geometry is Bezouts Theorem, which Room 244,A Variant of pascals triangle Dennis Van Hise Stetson University (student http://www.spcollege.edu/central/maa/archives/Feb01.htm
Extractions: Volume 22, Issue 2 A workshop on using TI Calculators in the Mathematics Curriculum by Doug Child - Rollins College A workshop on Successful Grant Writing by Bill Bauldry - Appalachian State University Presidential Welcomes Pleanary Address from Barry Cipra, Freelance Writer, Northfield, MN Room 101 Adventures in Number Theory via Mathematical Data Scott Hochwald - UNF Abstract: The harmonic series is usually thought of as a creature from Analysis. However, when we look at partial sums of the harmonic series, we enter a world full of number theoretic possibilities. I will present some partial sums and let you look for patterns. We will discuss the patterns. Room 265 My Erdos Number is Sqrt [-1] Li Zhou - Polk CC Abstract: Ill discuss some useful problem solving strategies, such as collecting data, using calculators and computers, working backwards, and so on. In particular, Ill illustrate these concepts using my solutions to Problem 667 (College Mathematics Journal, Jan. 2000), Problem 1597 (Mathematics Magazine, Apr 2000), and Problems 10771, 10798, and 10814 (American Mathematical Monthly, Dec. 1999, Apr. and Jun. 2000). Room 269 The Game of Cubic is NP-complete Erich Friedman - Stetson University Abstract: Cubic is a puzzle/video game/applet that can be played on the web. There are variously colored blocks that the player can drag around. The blocks are affected by gravity. When two or more blocks of the same color meet, they all disappear. The goal is to make all the blocks disappear. I will show that the question of whether or not a given Cubic position can be solved is NP-complete. That is, Cubic is easy enough so that a solution can be checked in polynomial time, but hard enough so that any polynomial time algorithm to solve Cubic positions would also yield a polynomial time algorithm to solve other hard math problems.
Untitled Pascal developed the mystic hexagon method as well as projective geometry theorems. inrow 4, place 2. This is shown in the diagram below of pascals triangle. http://www.mps.k12.nf.ca/mathematics/Grassroots/Pascal's Triangle/Pascal2.htm
Extractions: Blaise Pascal was born in Clermont, Auvergne, France on June 19 th , 1623. He died August 19th, 1662 in Paris, France. Pascal was one of four children and he was the only son of Etienne Pascal. Against his fathers wishes he refused to listen to his father and pursued his interest in Mathematics before the ago of 15. At the early age of sixteen Pascal developed the mystic hexagon method as well as projective geometry theorems. In later years Pascal was the second to invent the mechanical calculator, this was to assist his father's work as a tax collector. You start out with the top two rows: 1, and 1 1. Then to construct each entry in the next row, you look at the two entries above it (For example; the one above it and to the right, and the one above it and to the left). At the very beginning and the end of each row, when there's only one number right above, you put a 1. You might even think of this rule (for placing the 1's) as included in the first rule: for instance, to get the first 1 in any line, you add up the numbeabove and to the left, since there is no number there, you pretend it's zero, and the number above anto the right (1), and get a sum of 1. When referring to an entry in Pascal's Triangle, people usually give a row number and a place in that row, beginning with row zero and place zero. For instance, the number 6 appears in row 4, place 2. This is shown in the diagram below of Pascals Triangle.
Extractions: Blaise Pascal Blaise Pascal was the third of Etienne Pascals four children and was born on June 19, 1623 in Clermont, Auvergne, France. He died on August 19,1662. Blaise was a child with a marvelous gift, and was extremely fascinated with mathematics. When Blaise was 19 he invented the first working calculator and later the barometer, the ydraulic press, and the syringe. Pascal found probability a very interesting topic, in which was introduced to him by a gambler. He developed a triangular pattern, he then named after himself, which helped him make a better educated guess. The Chinese first discovered this pattern hundreds of years before, but it was Pascal who discovered all of the patterns it contained. This pattern was named Pascals Triangle. The Presentation of Pascals Triangle: How Pascals Triangle is Formed: This pattern keeps on going throughout the whole triangle. Where do we use Pascals Triangle? When you first look at Pascals Triangle, you just see a triangle made up of a bunch of numbers. When you actually explore the meaning of this triangle, you will learn that this triangle is much more than that. This triangle is used in two major areas. One is Algebra and the other is Probability/Combinatorics. Some Different Types of Patterns Found Throughout This Triangle: The Hockey Stick Pattern 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. After you have selected your diagonal, you must then choose a number that is adjacent to the diagonal below it. It should then look like a hockey stick, hence the name for the pattern. What is interesting about this pattern is that when you add up the numbers in the diagonal the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself. 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.
Algebra 2 the product of a binomial times a polynomial, and the use of pascals triangle tosolve This chapter begins with right triangle trigonometry 4. Fractal geometry. http://www.asfg.mx/highschool/Math/10th-grade.htm
Extractions: Up High School Math Department Student´s Projects Math Teachers ... Up ALGEBRA II 10TH GRADE MA. LETICIA GARCIA DE ESPINOSA leticia.garcia@warrior.asfg.mx In this Algebra II course students explore the language of algebra in verbal, tabular, graphical, and symbolic forms. Problem solving activities and applications encourage students to model patterns and relationships with variables and functions. It incorporates the use of the graphing -calculator for discovery, problem solving, and modeling. Students are required to use algebra skills and concept-analysis to solve real world situations. ALGEBRA II OUTLINE 2002-2003 1. Analyzing equations and inequalities: This chapter provides a review of essential skills and concepts in algebraic settings. Students survey evaluating expressions and the properties of real numbers. Then the procedures for solving linear equations are presented and extended to solving inequalities. The number line is used as a mathematical model to review absolute value, and equations involving absolute values are solved. This chapter ends with solving compound sentences and inequalities involving absolute value. (August-September) 2. Graphing linear relations and functions: This chapter begins with students graphing relations and identifying those that are functions. Next, they graph linear equations from a table of ordered pairs, identify the slope and intercepts, and use these to group other linear equations. Graphing technology is applied to graph linear equations and to approximate solutions of equations in one variable. Then students determine if the lines are parallel, perpendicular, or neither. Next, students draw scatter plots and find prediction equations to solve problems. They conclude their study by graphing special functions and linear inequalities. (October)
History Of Mathematics pascals triangle. Known to Chinese in 11 th Century. (a+b) 2 = 1a 2 + 2ab + 1b 2.1 2 1 is the 3 rd row in the Pascal triangle. Algebraic equations for geometry. http://www.math.sfu.ca/histmath/math380notes/math380.html
Organic Mathematics: Comments And Feedback similar to the one used to create the pictures of pascals triangle for a He has coauthoredMath A Rich Heritage; translated Sona geometry Reflections on the http://www.cecm.sfu.ca/organics/comments/comments.html
Extractions: If, three centuries ago, Machin would have chosen the input vector: "liste:=[Pi,arctan(1/3),arctan(1/5),arctan(1/8),arctan(1/239)];" he would have obtained: "vector_found=[0,1,-1,1,0]" thus missing his historic formula. You are certainly aware of the problem because it occurs whenever we are looking for a relation between Pi and other constants among which an integer relation (with smaller coefficients) already exists. Is it possible to modify the Maple program (or the algorithm) so as to introduce the constraint that the integer coefficient of Pi be different from zero?
