Exploring Discrete Mathematics Summary

Monday - Friday, July 2 - 6, 2007

Monday - KEY IDEA: Lattice Paths

We started by examining the number of paths on a grid from the origin to (2, 2) only allowing steps that go up or to the right. We looked at the number of similarly defined paths from the origin to other points. Our discussion included permutations and combinations and Pascal's Triangle. We found a general formula for the number of paths from the origin to a point (m, n) and we discussed the sequence of sums of rows in Pascal's Triangle.

We looked at paths on a grid again, but this time we allowed one more kind of step: a diagonal up and to the right. This led to a discussion of the Delannoy Triangle which made sense because we counted the number of paths from each of the penultimate points (the one to the left, the one below, and the one diagonally to the left and below). Homework: Can you explain the sequence of sums of rows in the Delannoy Triangle?

Tuesday - KEY IDEA: The determinant of a matrix counts the number of non-intersecting paths

Today we investigated pairs, triples, or higher orders of paths from n points to n points on a grid only allowing steps that go up or to the right. We related the numbers of combinations of paths (total and non-intersecting) to the permanent and determinant of a matrix. For a couple examples, we found a correspondence of intersecting paths (called a killing involution) which helped us count the number of non-intersecting paths (the survivors) then relate this idea to the determinant of a matrix.

Our task is to look at what we learned yesterday and today and maybe some other things then come up with a project that we and other teachers could use in our classrooms.

Thursday - KEY IDEA: Our project will center around Pascal's Triangle

We divided up into three subgroups that will focus on different areas of our group project: Plinko (Mark, Dawn, and Lars), Kids Walking Paths (Kym and Alice), Pascal Identities via paths (Tanya, Jocelyn, and Malvia).

Friday - KEY IDEA: Subgroup Work Begins

We began working in our three subgroups. The Plinko group went shopping to get supplies to build the board. The other two groups began working on materials that teachers could use for classroom activities.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.