K-12 Integer Sequences

The K-12 Integer Sequences Conference was organized by my good friend Dr. Gordon Hamilton (Dr. Pickle) of MathPickle.com and Dr. Neil J. A. Sloane, the founder of the On-line Encyclopedia of Integer Sequences (OEIS). I was fortunate enough to attend as a math educator among other educators, math professors, and curriculum developers.

The goal of the conference was to identify 13 integer sequences that could be promoted for use in classrooms from K-12. The sequences would help students practice or learn level appropriate curricular goals, but they would also introduce the students to the wonderful world of mystery found in integer sequences. You can watch our large group sessions in these videos. Three sessions were spent in smaller groups – mine sought to identify sequences for grades 10-12.

The event was hosted by the Banff International Research Station (BIRS) at the Banff Centre which means they provided the hi-tech and comfortable conference facility (pictured behind me) and the lodging for the attendees. BIRS is dedicated to math research and collaboration.

Naturally, if you are still reading this, you are keen to learn which sequences will be promoted as a result. As with many collaborative conferences, there are still some **loose ends to tie up. I will feature a few here though.

Kindergarten: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, …

A034326 This is the clock sequence. I think it’s brilliant. You can have kids do some skip counting on it where they will be exposed to factors of 12 (when you skip by 4s, you will always land on the same 4 numbers, when you skip by 5s, you don’t!). There is of course the excellent recursive nature of this sequence which mimics time.

Grade 3: 14, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 24, 22, 11, 9, 18, 16, 32, 30, 15, …

A254873 This sequence is a modified Recam├ín sequence developed by a 15-year old student. Starting at the seed number (14) the sequence continues by dividing, subtracting, adding or multiplying by the step number (2). Division gets precedence over subtraction which gets precedence over addition which gets precedence over multiplication. The new number must be a positive integer and not previously listed. The sequence terminates if this is impossible.

Grade 5: 4, 9, 7, 20, 6, 33, 13, 23, 16, …

Starting with 1, on the first step add 1/n, and on subsequent steps either add 1/n or take the reciprocal. What is the smallest number of steps needed to return to 1? This number of steps is the nth term of the sequence. (Note: the n=0 and n=1 terms are not defined, so the sequence actually starts with the 2nd term.)

eg. let n=2: 1, 3/2, 2, 1/2, 1 therefore the number generated is 4 because it took 4 steps
eg. let n=3: 1, 4/3, 5/3, 2, 7/3, 8/3, 3, 1/3, 2/3, 1 took 9 steps
eg. let n=4: 1, 5/4, 3/2, 7/4, 2, 1/2, 3/4, 1 took 7 steps

This one is a lot of work, but I think it works to help students see that just because you work with increasing fractions, it doesn’t necessarily mean it will require more steps. It really depends on the factors of n.

This time, I brought Amber along since I knew she would enjoy the tranquility, beauty and opportunity to relax in such surroundings.

My mom took the kids for most of the weekend, then Jasen for the last bit – we are very grateful.

Getting away to Banff now means being able to chow down and drink at the Banff Ave Brewing Company. Their beers are notable – especially their recent addition of an imperial IPA. Their food portions are ridiculously huge. I ate one burger for two meals.

The sequences have been determined in full! Here they are at the OEIS.org.

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