Scaling the Teaching Curve: A PCMI Teacher Program Residential Workshop Mathematics Session Summaries

Math 1 (Sunday evening)
In the initial problem, we counted regions determined by chords connecting points on a circle; the resulting sequence has a tempting pattern that is broken at the 6th number. Then we introduced the primary object of study for the week, "trains" consisting of Cuisenaire rod "cars." We determined a formula for the number of trains of a given length, developing the tools of adding and extending cars and the notions of cuts and joins.

Math 2 (Monday morning)
We started shaking hands to demonstrate the "handshake lemma." Returning to trains, we put them into a table, rows by length, columns by number of cars, and Pascal's triangle emerged. The rest of the session was devoted to combinatorial proofs of various identities, including Pascal's lemma about adding two numbers to find the number beneath them in the triangle, symmetry in rows, row sums, alternating row sums, and the "hockey stick" relation.

Math 3 (Monday afternoon)
An exploration of graph theory included the notion of vertex degree, connecting to the handshake lemma, and Euler's formula connecting the number of vertices, edges, and faces. We then combined these ideas with choose numbers to solve the initial circle problem, seeing that the number of regions is the sum of the first five terms in each row of Pascal's triangle.

Math 4 (Tuesday morning)
We considered three kinds of restricted trains: (a) trains with only length 1 and 2 cars, (b) all car lengths except 1, (c) only odd length cars. For each type, the count of all length n trains was a Fibonacci number. In each case, we showed that the restricted trains satisfied the Fibonacci recurrence. Then we worked on direct connections between the three types of restricted trains.

Math 5 (Tuesday afternoon)
Cuisenenarea! From one-dimensional trains, we consider two- and three-dimensional problems. Suggestions included counting tilings of 2 by n boxes with (i) red cars, (ii) white cars and three-square blocks in an L pattern, (iii) white and reds cars, (iv) any cars, and then filling a 2 by 2 by n box with reds. These led to different recurrence relations and connections to earlier work.

Math 6 (Wednesday morning)
Monday morning we partitioned trains by length and number of cars and found Pascal's Triangle. Now we created two other triangles: one by length and number of 1s, the other by length and number of odd cars. By Tuesday morning's work, we know the first columns of both triangles are the Fibonacci numbers. We worked on finding and justifying an analogue of Pascal's Lemma for each triangle in order to generate subsequent rows of the triangles.

Math 7 (Wednesday afternoon)
After telling the story of the childhood Gauss quickly summing the numbers from 1 to 100, we looked at a problem developed by Barry Cipra. A Barrycade (named by Richard Guy) is an ordering of 1 through n, corresponding to a train with 1 length 1 car, 1 length 2 car, ..., 1 length n car. A "breakfree Barrycade" is a collection of trains with at most 1 split per possible position; finding 4 trains with cars 1, ..., 6 making a breakfree Barrycade was a challenging puzzle. A "rainbow Barrycade" has distinct colors in each column; finding 3 trains with cars 1, ..., 5 making a rainbow Barrycade was an easier puzzle, but finding 3 trains with cars 1, ..., 6 making a breakfree rainbow Barrycade was very challenging.

Math 8 (Thursday morning)
As a way of seeing our progress, we came up with recurrence relations for trains made with: just 1s, 2s, and 3s (Tribonacci numbers); just 1s and 3s (Narayana's cow sequence); just 2s and 3s (Padovan numbers). And we dealt with trains having all even length cars. As a final activity, we observed that sums along diagonals in Pascal's triangle give the Fibonacci numbers. We verified the connection using trains: some showed that trains counted by the diagonals satisfy the Fibonacci recurrence relation (add a 1 to trains in the first diagonal, extend by 1 the trains in the second diagonal), others found an elegant connection to trains with no 1s (extend each car by 1).