Recently I’ve been volunteering with the middle school math club at Penn Alexander, a PreK-8 school in my neighborhood. Today we did (among other things) a fun activity I’d never seen before, called “number bracelets”. The students seemed to enjoy it; it worked especially well with a bunch of students all working on it at the same time since they were able to compare notes.
Here’s the idea: start with any two one-digit numbers you like. For example, let’s choose 4 and 6. Next, add the two numbers: 4 + 6 = 10. Throw away the tens digit of your answer (if any); this is the next number in the sequence. In our case we get 0. Now we have 4, 6, 0 and we do the same thing with the last two numbers, 6 and 0: 6 + 0 = 6. So now we have 4, 6, 0, 6. Continuing, we get
4, 6, 0, 6, 6, 2, 8, 0, …
Try some different starting numbers. Do the sequences ever repeat? How many different sequences can you make? How long can they be? Can you generalize this to other sorts of rules for generating sequences?
A much better description (with pretty pictures, more questions for exploration, and spoilers) can be found here.