Teacher Information: Pascal's Triangle

Few constructions arise in so many different contexts or are more fertile in terms of providing stimulating questions at many levels than Pascal's Triangle of Binomial Coefficients. This array of numbers was certainly studied by Pascal, but properties and portions of it were familiar to the Babylonians and the Chinese many hundreds of years before Pascal.

Traditionally, Pascal's Triangle appears in algebra and pre-calculus courses in connection with the expansion of binomials, and it is a key topic in combinatorics, number theory, and probability courses as well. There is no need, though, to wait for formal treatment. The patterns and fascination offer abundant enrichment opportunities for the curious individual or for class exploration.

Counting experiences (how many paths, how many different pizza combinations with a fixed number of toppings, etc.) lead naturally to the coefficients in the first few rows. Indeed, the "combination" notation displayed beside the color palette when an entry is clicked, is read: "4C2 is the number of combinations of 4 objects taken 2 at a time," the number of distinct pairs from a set of 4 objects. Students should actually make pairs from sets of four objects and see that there are six possible pairs. The entry 4C2 also represents the "address" of the entry 4C2 is the 2nd element in the Row 4, where both rows and entries start at 0. Thus in Row 2, in terms of combinations, from a set of 2 elements, there is only 1 way to select a set with no elements (2C0 = 1), 2 ways to select a set with 1 element (2C1 = 2), and 1 way to select a set with 2 elements (2C2 = 1), so that the 2nd row has entries: 1 2 1.

The key defining properties of Pascal's Triangle are observable: The outside entries on each row are always 1; every row is symmetric, reading the same right or left; and the sum of two adjacent entries gives the entry immediately below, in the next row (as for example, in Row 6, 6 + 15 = 21, in Row 7).

Students have an almost endless supply of questions to explore, just for challenging fun or, with more algebraic experience and sophistication, to prove. What kinds of patterns are discernable with the options at the bottom of the manipulative? Given the patterns of multiples of 2, 3, and 4, predict a pattern for multiples of 5? Check it out. The primes appear in interesting places. Will they always appear only near the margins? Why? Discounting the end 1s, are all the entries in Row 5 multiples of 5? How about Row 7? What does that suggest? What is the sum across each row? Knowing how one row is generated from the previous one, if the sum of the entries on one row is a power of 2 (what power?), what does that imply about the sum of the entries in the next row? Reading down the second column, the entries are 1, 2, 3, 4, ..., just the numbers in order. Is there a simple formula for the entries in the next column, 1, 3, 6, 10, ...? Count the sums of integers, 1, 1+2, 1+2+3, ... . Where do those sums appear? If you sum down any one column, where are the sums in the Triangle? How about the sums of squares, of cubes?