My seventh graders have been grappling with repeating decimals. Lots of cool stuff. One thing that’s been great has been the interplay of calculator and pencil-and-paper work. Today one of my students was doing some calculator experiments and found that 16/33=.48484848…
This surprised her, because in many of her other experiments there was a more direct relationship between fraction and the decimal. For example, 14/99=.14141414… After some nudging and some work on her part, she put together the revealing fact that 16/33=48/99. She was very proud of her discovery and shared it with her classmates at the board.
It’s really neat when more “sophisticated” mathematics provides occasions for reconsidering “easier” content. This student’s work was a concrete example of how “knowing the basics” isn’t required to do more sophisticated work. Rather, doing this harder work provided meaning and context for a skill that she wasn’t wholly confident with.
PS In my head, I call this kind of concrete, informifying, mountain-top experience of a concept a “place to hang your hat on”.