### Using Isomorphism to Derive Pascal’s Identity

PurposesStudent collaboration; Student elaboration; Reasoning
DescriptionThis session begins with Jeff, Michael and Romina discussing the binomial expansion. Michael remembers that finding the coefficients for binomial expansion relates to n choose r. This prompts Romina to remind Michael that n choose r was what the students had learned with the towers problem. Michael concludes this connection explaining that each term of the binomial relates to the colors that are being chosen from in the towers problem. Thus, finding the coefficient of the third term of (a+b)^10 is similar to finding the number of towers 10 tall choosing 2 blocks of one specific color.
In the second event, the students recall that choose relates to Pascal’s Triangle and draw the triangle on the board. The students explain why addition of adjacent elements is used to find additional rows. Michael explains that moving from the third row to the fourth row is similar to considering when going from three pizza toppings to four pizza toppings. This means that each option in row three may either have one topping added to it or no toppings added to it. So, if all the one topping combinations from three topping choices have no toppings added they will still have one topping. The no topping combination from three topping choices can have one topping added, so in total there are four one topping combinations from four topping choices.
The students in event 3 rewrite Pascal’s Triangle in choose notation. The students re-explore the relationship between rows using this notation. Michael explains that from three pizza toppings choosing one, an additional topping added to these pizzas (so two toppings) would make this element now choose two (from four). He also explains that from three pizza toppings choose two, no additional toppings would also make this element choose two toppings (from four). So three pizza toppings choose one plus three pizza toppings choose two make four pizza toppings choose two. This reasoning is used to support the conclusion that four choose two plus four choose three equals five choose three.
In the final stages of this session occurring in event 4, the students generalize the nth row of Pascal’s Triangle. The students collectively explain that n choose x will combine with n choose x+1 to make n+1 choose x+1. Ankur and Michael explain the n choose x will gain an additional x (or topping), n choose x+1 will not gain an additional x (or topping) and that this happens because they are adding from the total possible topping choices (n+1).
Jeff explains in terms of the pizza toppings the general rule of adding adjacent elements to produce new rows of Pascal’s triangle to Brian. He explains that n increases by one since the number of possible toppings increases. He also explains that x increases by one since all the choices of n choose x gain a topping while the choices of n choose x +1 do not gain a topping so in total n choose x combines with n choose x+1 to produce n+1 choose x+1.
In this night session, Jeff, Michael, and Romina (and joining later Ankur and Brian) explain Pascal’s Identity using recognition of isomorphic tasks to support their conclusion. The students’ rich understanding of combinatorics problems including the tower problem and the pizza topping problem piece together to help the students arrive at their own derivation of Pascal’s Identity.
In this analytic reference is made to the tower problem. In the tower problem, students are asked to find all possible combinations of n-tall towers that can be made when selecting from two colors. Also referenced in this analytic is the pizza topping problem. In the pizza topping problem students find the number of different choices for pizza that customers have when from ‘n’ total toppings. 