Stumbled across this post from the Shah/Kinnell 180 blog this morning, and spent some time exploring the question he poses:
Found it really fascinating, and went on to explore a related question: what the function looks like that models this:
across whole numbers. You should explore it; it has a really beautiful structure and the graph is fascinating. The general case for bases other than 2 is worth looking into as well.
The approach I take to questions like this is to pick some small numbers, see what pops out, and keep going until I see a pattern. Then, try to figure out where that pattern is coming from, test it a few more times, consider the extreme cases, and generalize. It’s exactly the kind of thinking I want my students coming out of class able to apply to a number of different situations. I’m curious how that skill could be broken down into smaller pieces that could be taught — and in particular in a low-risk environment where students can feel successful. Patterns are definitely one piece, but there’s something about picking some simple numbers and making a guess that is incredibly powerful but also a huge hurdle to students who struggle with math. This is a bit of what I tried to get at with my post on low-floor problems, but I’m still not sure what is most powerful at teaching that habit of mind.
Desmos graph of the function here.