Multiplication Graphs

Sam Vandervelde took the lead in developing this circle session on Multiplication Graphs for the 2011 ”Circle on the Road” workshop.

Strong middle school students would be best. Will also work with high school students with a decent but not extensive math background. (Some of each is also fine, but strong high school students will probably get bored with the pace.)

A multiplication graph is formed by choosing a positive integer n, numbering a collection of n-1 points from 1 to n-1, then drawing an edge between any two points whose product is a multiple of n. We will investigate the types of graphs that can (or cannot) be obtained in this manner and study other properties of these graphs. The circle will emphasize the research process, with plenty of opportunity for conjecture, proof, and/or refutation. We will also indicate towards the end of the circle how this topic is related to a current area of active research known as zero divisor graphs.

Apprentices: Katherine Cook, Ming Jack Po, Tingting Ma

A student hand-out (Multiplication_Table) and lesson plan are attached.

Jack – For the warm up activity, why do you care about divisibility by 13?

Sam – Because if the original number is divisible by 13 then the cute little “doubling” phenomenon will occur after multiplying only by 7 and 11, so won’t be as much of a surprise after the final multiplication by 13.

Secondly, I have a homework assignment for you: I’d like you to experiment with multiplication graphs yourself and come up with at least one conjecture of your own other than the ones mentioned in the lesson plan. (There I observed that when N=2pN=2p we get a star graph and when N=p2N=p2 we get a complete graph.) Bring your conjectures with you to our Friday afternoon planning session; we’ll see what we can come up with collectively at that point. You are also welcome to share observations before then via email or the wiki. But I’m leaving open the option of trying this out on the plane to Houston!