Genre 2: Pascal's Triangle

Blaise Pascal is known by many from learning about Pascal's triangle. It was originally not a triangle that Pascal created, but rather a square table of binomial coefficients. If were to turn his table so that the top left corner is facing up we would get Pascal's triangle (shown below). Pascals triangle has many numerical patterns. An engaging activity for students would be to see how many of these patterns they can find. (http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html)

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 One pattern can be seen in the way the triangle is created. Each entry of the triangle is the sum of the two entries above it (the outermost diagonals of the triangle are always 1's).

Another pattern that is easy to see is the diagonals that are the natural numbers.

If we look at the powers of 11 we can see that they match up with the rows of the triangle. (At the fifth power you have to manipulate the numbers for the pattern to continue.)

When we add each row we get the powers of 2.

Other famous numbers patterns, such as the triangular numbers can also be seen in the triangle. One of the most useful features of the triangle is finding combinations in counting theory, this will be further explained in the third genre.