For instance, we found one 6-number sequence that *looked* like it only had one repetition in the final pattern, instead of two as predicted by the proof. That was because the second iteration went back over all the same lines as the first, but did them in a different order. It was 1-2-1-4-1-2.

And we found that an eight-number sequence (or other, longer multiples of four) can be chosen so that it comes back to the starting point at the end of the first sequence pattern. Then, because it starts the next sequence in the same direction, it will keep going around the circle rather than walk off into infinity.

As a homeschooler, I haven’t had to study the CCSS, so I can’t answer that question. For my dd, there was plenty of mathematical reasoning that involved modeling, vectors, cyclical functions, modular math, multiple representations, and justifying her statements. YMMV.

So as a separate issue, what CCSS math standards does this exploration support? I’m thinking of sharing this with teachers and it seems strong in art and technology (computer programs the model the sequences) but not so strong in the math standards.

The 5-4-3-2 pattern has four numbers, so it’s a 0 mod 4 pattern. As you repeat the pattern, each new iteration starts pointed in the same direction as the first one, so it never comes back around. It diverges.

But the 5-4-3 pattern has three numbers. Each time you repeat, the first line is pointing in a different direction, and after drawing the pattern four times, you have come full-circle back to your beginning.

My daughter’s proof is based on seeing the sequence as a roundabout way to describe a vector from point A (start of the sequence) to point B (end of the sequence). With 4-number sequence (or multiple of 4), the vectors all point the same direction, and the patterns walk off into the distance.

But with the other sequences (not a multiple-of-4 numbers long), as you repeat the patterns, the vectors take a “walk around the block” and come back to where they started. whether the vectors “walk” clockwise or counterclockwise and how many iterations it takes to come back to the beginning can be predicted by looking at how many numbers are in the sequence — and particularly, at the remainder of that number when divided by four.

Does that help? It’s hard to explain in words, without diagrams…

I know a 5, 4, 3, 2 sequence will diverge but a 5, 4, 3 sequence will converge.

I would like to understand the proof of being able to predict the behavior of sequences. Explain it as simply as you can, please. Thanks!