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Pascal's Triangle for High School Teachers Tips for presenting the ideas of the binomial coefficients to young students

        Many are the challenges for teachers of math to high school students; two of the most difficult are keeping those who have an aptitude for math challenged by new and interesting material, and reaching those who have "tuned out" math and, with some luck, returning them to the fold.  Pascal's Triangle can help on both these fronts, as it is both a treasure trove of unexpected relations between numbers and an object easily created by the application of simple rules.

        While there are interesting pieces of information throughout the website, we have compiled these links as a good place for high school teachers to start looking for ways to present the binomial coefficients to their students, and even bringing the topic into play in different parts of the curriculum.

        History:  It may seem an impertinent question, but the common student plea of "When are we ever going to use this in real life?" should be viewed as the one of the most central questions in math.  With the history of the binomial coefficients, you can begin to answer these questions by showing how others studied these ideas to answer practical questions around the world.  While polynomials may seem to students like arbitrary objects invented solely to torture them, they can find out about recipes and music in India, the Precious Mirror in China, poets in the Middle East, arguments between people in the 1600's that are still important today, and a guy getting stabbed in the face.  (The last guy mentioned is Nicolo Tartaglia; you will be sure to increase your student's level of attention telling his story.)

        Applications:  In the applications section of the website, there are many questions related to probability and statistics; several have Java applets that can let the student explore a problem interactively.  We also have an applet that explores the famous fractal design known as Sierpinski's Gasket, in which the numbers in Pascal's Triangle are replaced with their remainders when divided by some number n between 2 and 22 and represented with different colored dots.  (Note: while there is actual math content in Sierpinski's Gasket, we should not denigrate the educational value of fooling around and making pretty pictures.)

        Proofs:  Many of the proofs of the identities involving Pascal's Triangle involve concepts no more difficult than the adding and multiplying of whole numbers; even simpler, some can be presented with picture proofs, many of those on this website taken from Roger B Nelsen's Proofs Without Words.  Here are the links to those proofs.

Sums of powers of nine are triangular numbers	The Christmas Stocking Theorem
The sums of odd cubes are triangular numbers	The Hockey Stick Theorem
The sum of reciprocals of triangular numbers = 2	Pascal's Triangle to the Fibonacci numbers
Two consecutive triangular numbers add up to a square number	Two ways to sum the entries of an n×n multiplication table
The basic additive identity of Pascal's Triangle (combinatorial proof)	The symmetry of Pascal's Triangle (combinatorial proof)
The sum of a row of Pascal's Triangle is a power of 2 (double the sum of the previous row)	Another proof involving triangular numbers
Yet another proof involving triangular numbers	Again with the proofs involving triangular numbers?!? (Oy.)