For this week’s work, I decided to dive even deeper into the rascal triangle. I was just so impressed by the work that these three students put forth and the fact that they weren’t going to let their teacher’s discouraging words bring them down. This just shows that students should still pursue the “wrong” answers, because they never know what discoveries they may make. I am going to include some of the work that I wrote out for my daily work and then elaborate on it via this blog post.
From my research, the values in the first few rows of the rascal triangle support the claim that each element of the nth diagonal is congruent to 1 modulo (n-1). To explain this, in the fourth diagonal, each of 1, 4, 7, 10… is congruent to 1 modulo (4-1)=1 modulo 3. So, in these cases, n=1, 2, 3,… Moreover, we can see that the nth number on the rth row a∨(nr) is the nth number on the (r-n)th diagonal and n, r=1, 2, 3,… and the 0th row is excluded. This observation is fundamental in proving that the rascal triangle consists strictly of integers. Also, the sums of the rows of the rascal triangle are “cake numbers” (I had never heard of these before…). A cake number is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly n planes.
Another way to prove that all of the entries in the rascal triangle are integers, is by doing what the boys did…
They realized that the NE-SW diagonals exhibit a simple pattern: the mth diagonal (starting the count with m = 0) is an arithmetic progression with m as a difference and 1 as the first term. The entry #n in the diagonal #m is simply nm + 1. The formula to generate the triangle is then
(m + 1)(n + 1) + 1 = [((m + 1)n + 1)(m(n + 1) + 1) + 1] / (mn + 1)
I show this (or at least attempt to) in the picture I included, so that people could put a visual with the written work. I am so inspired by these students, and it makes me wonder how often we as math teachers shut down valid ideas. Not intentionally, of course, but without even realizing that our students could really be onto something. It just makes me think about how I’ll go about teaching in the future.