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Exploring Pascal || Student Center || Teachers' Place
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= good places to begin

Triangle grid paper (to print out)

Ask Dr. Math: Archive - Search for Pascal
Practical Uses for Pascal's Triangle
One Person of Seven Born on Monday
Pascal's Triangle and the Ice Cream Cones
Binomial Expansions and Pascal's Triangle
Binomial Theorem
Proof by Induction
Or search the Dr. Math archives for "Pascal's Triangle" (just the words, not the quotes).

Ask Dr. Math FAQ: Pascal's Triangle
What is Pascal's Triangle? How do you construct it? What is it used for? Answers to questions frequently sent to the Math Forum's service Ask Dr. Math. See Pascal's Triangle to 19 Rows and Interactive Pascal's Triangle by Ken Williams of the Math Forum.

Biographies from the MacTutor Math History Archive (St. Andrews)
Blaise Pascal
Omar Khayyam
Yang Hui
Eugène Charles Catalan

The Binomial Expansion and Infinite Series - The MathMan
Triangular, tetrahedral, even the Fibonacci numbers are in Pascal's triangle. Sample problems with some answers from Chapter 9 of Don Cohen's worksheet book for children as young as 7.

The Catalan Numbers in Pascal's Triangle - Seth Johnson
The Catalan Numbers appear in at least two places in Pascal's Triangle, first in the middle column going directly down the center (see illustration), subtracting the element immediately adjacent to it. If one does this on the 2N row, the result is the Nth Catalan Number. Second, they appear one row above: take the Nth term over and subtract the term immediately to the right...

Combinatorics Topics for K-8 Teachers - Roger Day
Pascal's Formula and Pascal's Triangle; Exchanging Conjectures and Justifications; Finding Fibonacci's Sequence in Pascal's Triangle. Notes from sessions of a 1996 course for professional development in Mathematics Education at Illinois State University in Normal, Illinois.

Connections with Combinations and The Binomial Theorem - Hartig
An exploration of the properties of Pascal's triangle, this site illustrates how Pascal's Triangle is an infinite triangular array of integers, listing some properties and showing evidence that suggests there is a close connection between the numbers in Pascal's Triangle, the numbers generated when counting combinations, and the Binomial Theorem, which identifies the coefficients appearing in expansions of powers of a + b. From the PSW Publishing Company.

Mathworld (formerly Eric's Treasure Trove of Mathematics) - Pascal's Triangle
Mathematical definitions and descriptions of numbers you can find in Pascal's triangle.

Generating Pascal's Triangle - Karl F. Kuhn
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 adding together the two numbers above it. The numbers may also be generated by using the idea of combinations found in probability theory. To do this assign a column and row number to each value. Then use the combinations formula to produce the value in question...

Math on Wheels - Pascal's Triangle - David Fayegh
Studying the pattern of even and odd numbers in the triangle provides a basis and motivation for visualization. Display even numbers in the triangle using yellow dots and odd numbers using black dots. Noting that each element of the rows of the triangle is just the binomial coefficients n choose k as k runs from 0 to n, we can write Maple code that computes the elements of Pascal's triangle...

Number Patterns in Pascal's Triangle - Ulysses Harrison
A lesson plan designed to enable students at grade 5 or higher to recognize the integers, rows and columns that comprise Pascal's Triangle. The main objective of the lesson is to enable students to reproduce the first eleven rows of Pascal's Triangle by recalling number patterns given in the lesson without having to look again at the original triangle.

Pascal's Fractals - MAA Online, Ivars Peterson (MathLand)
It's possible to convert Pascal's triangle into eye-catching geometric forms. For example, one can replace the odd coefficients with 1 and even coefficients with 0. Continuing the pattern for many rows reveals an ever-enlarging host of triangles, of varying size, within the initial triangle. In fact, the pattern qualifies as a fractal. The even coefficients occupy triangles much like the holes in a fractal known as the Sierpinski gasket. An article written by the mathematics and physics writer and online editor at Science News.

Pascal's Pyramid Or Pascal's Tetrahedron? - Jim Nugent
An advanced exploration. The three-dimensional analog of Pascal's triangle is of interest for the same reasons that Pascal's triangle is of interest. It touches on number theory, the distribution of primes and twin primes, divisors, factors, combinatorics, and geometry. Article abstract: A lattice of octahedra and tetrahedra (oct-tet lattice) is a useful paradigm for understanding the structure of Pascal's pyramid, the 3-D analog of Pascal's triangle. Notation for levels and coordinates of elements, a standard algorithm for generating the values of various elements, and a ratio method that is not dependent on the calculation of previous levels are discussed. Figures show a bell curve in 3 dimensions, the association of elements to primes and twin primes, and the values of elements mod(x) through patterns arranged in triangular plots. It is conjectured that the largest factor of any element is less than the level index. With illustrations (visualizing Pascal's tetrahedron as a stack of marbles...)

Pascal's Triangle Applet - developer.com
An applet that computes the binomial coefficients, graphically presenting Pascal's triangle modulo an integer number p. It can draw triangles of up to 650 rows with a p value between 2 and 15,000. The resulting pictures may be good graphical illustrations of the binomial theorem and the laws of division of natural numbers.

Pascal's Triangle From Top to Bottom - Matthew Hubbard and Tom Roby
The section "Applications" addresses questions such as "What types of questions are answered by the binomial coefficients?" and draws on Java applets to address "What does Pascal's Triangle look like mod n?" The section "Identities" organizes identities and proofs into the categories sums, products, sums of products, products of sums, factorization identities, identities by name, and identities involving famous numbers (e.g., Euler numbers, Fibonacci numbers, and Stirling numbers).

Pascal's Triangle Interface - Andrew Granville
A form that lets you visualize the entries of Pascal's triangle with respect to a modulus between 2 and 16. Select values for the number of rows, modulus, and the size of the image, and submit. Also see Granville's The Arithmetic Properties of Binomial Coefficients I.

Pascal's Triangle mod 2 - Riddle
From the Iterated Functions Systems site by Larry Riddle of Agnes Scott College. An illustrated explanation: the figure represents the first 128 rows of Pascal's triangle. The coefficients in the triangle that are odd are displayed as red boxes. The coefficients in the triangle that are even are displayed as black boxes.

Pascal's Triangle and Programming - Brian Ward
A common programming project for new students, Pascal's Triangle is useful in number theory, probability, and having fun (among other things). This version came into being after I saw someone in the process of constructing one by hand. I thought it was silly to do that when we have computers, which can save you a lot of time and pain (especially if you add something incorrectly). I remembered that the only triangle programs I'd ever seen were those silly beginner's course programs... so I wrote mine.

Pascal's Triangle and Related Triangles - Helena Verrill
Links, Puzzles, and Related Triangles: Circle regions problem, Fibonacci sequence and other diagonals of Pascal's triangle, Clown Problem, Tchebychev Polynomials, Bessel Polynomials, and Stirling numbers.

Pascal's Triangle using Clock Arithmetic - Part I - Jay's Corner
Exploring Pascal's triangle when the modulus is a prime; when the modulus is a power of a prime; and when the modulus has at least two different prime divisors.

Patterns in Pascal's Triangle - Jeremy Baer
An applet for exploring patterns in the numbers contained in Pascal's triangle, which colors the cells in the first 128 rows of the triangle depending on whether or not they are divisible by some number x, where x is entered by the user. Many of the patterns created tend to be self-similar; for example, for x=2, the resulting pattern looks like Sierpinkski's triangle. Some information about Pascal's triangle and a number of related links are also included.

Pinball and Pascal's Triangle - Eric Hiob
The picture shows a type of pinball machine that you can build yourself using 10 finishing nails, 5 small cups, a wooden board and a pinball (marble). Nail the nails part way into the board in the triangular pattern shown, with one nail in the top row, two in the second, three in the third and so on, and with enough space for the pinball to fit between the nails. How many different paths are there through the pinball machine and what are they? If we superimpose Pascal's triangle on top of the pinball machine then we see the connection between the two: Each number of Pascal's triangle represents the number of distinct paths that a pinball can take to arrive at that point in the pinball machine... The numbers in Pascal's triangle can also be gotten using the combination function... Pascal's triangle is essentially a listing of all the possible values of the combination function.

Pizza and Pascal's Triangle - Morganna Letsch
The Pizza Place offers pepperoni, mushrooms, sausages, onions, anchovies and peppers as toppings for their regular plain pizza. How many different pizzas can be made? The first possible method of solving this problem is to use combinatorics and the following formula: Pt = nC0+nC1+nC2+nC3+...+nCn . Where Pt=total number of pizzas that can be made and n=the number of possible toppings This means: Pt = 6C0+6C1+6C2+6C3+6C4+6C5+6C6 . These terms correspond to the rows in Pascal's triangle...

Polynomials from Pascal's Triangle
There are many interesting things about polynomials whose coefficients are taken from slices of Pascal's triangle. (These are a form of what's called Chebyshev polynomials.) For example, the numbers 1,9,28,35,15,1 are taken from the 11th diagonal of Pascal's triangle...

A Short Account of the History of Mathematics, W. W. Rouse Ball (4th ed., 1908)
From the School of Mathematics, Trinity College, Dublin.

Sierpinski meets Pascal - Cynthia Lanius
Have you ever seen the triangular pattern of numbers (shown) named after the famous French mathematician Blaise Pascal? Do you see the pattern of how the numbers are placed in the triangles? Print out or copy the triangle and fill in the missing numbers; then check your answer here and see what Pascal's Triangle has to do with Sierpinski's Triangle.

Spreadsheets, Pascal's Triangle and Sierpinski Gaskets - Aarnout Brombacher
Download an Excel spreadsheet to generate diagrams like the one illustrated.

The Twelve Days of Christmas and Pascal's Triangle - Judy Brown
Using Pascal's triangle, find the number of items given each day in the song, "The 12 Days of Christmas." See also Pascal's Triangle Activities.

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