A few interesting math subjects — topics that we think about as fairly advanced, usually — came up in my Art of Problem Solving Prealgebra class tonight. I actually think they’re fairly normal things for students taking Prealgebra to think about, at least fleetingly, and that perhaps we don’t usually tell them how interesting these questions and thoughts are.
1) Minimum/maximum vs infimum/supremum. Many of the times that we touch inequalities, this is hiding in there. What’s the difference in an unfilled dot and a filled one, something inclusive and something exclusive, an open interval and a closed one, greater/less than vs greater/less than and equal to? These ideas are all quite close to (if not the same thing as) the idea of the difference between min/max and inf/sup. Today we were talking about rounding, and we introduced the convention of rounding 5 up. We asked the kids what values of x rounded to 0.6 when you rounded them to the tenths place. Because we were rounding 5 up, x can equal 0.55, but x can’t equal 0.65, so you need x greater than or equal to 0.55 but strictly less than 0.65. Some of the students knew x couldn’t equal 0.65 but proposed a less than or equal to cap like 0.64 (which is too low) or 0.6499999… (which is in fact equal to 0.65). These are both attempts to find a max when there simply isn’t one; the two students knew what the supremum was, though.
2) Density of the rationals in the reals. I think this started with the problem of converting 5.5555 to a fraction. When we arrived at a simplified but improper fraction, some of the students wanted to convert to a mixed number. We told them that was fine and would be correct, but we would generally just leave our answer as an improper fraction. One of the students, inspired by this, asked what the largest fraction that couldn’t be written as a mixed number was. I thought I knew what he meant, but I asked for clarification anyway, saying that we could write any fraction or integer as a mixed number if we let either the integer or fractional part be zero. The student clarified that he was interested in the largest fraction less than 1, and that’s where density of the rationals comes in. I told him that for any fraction I named, he would be able to find one that was larger than it but still less than 1. (I think I used 99/100 as an example of a fraction I could name, in which case 199/200 or 999/1000 are quick examples of even larger ones.) In fact, this meant that there were infinitely many fractions larger than the one I named but still less than 1. His response was, “Oh, right!” which I interpreted as meaning that this both made sense to him and that he’d thought about this before. (This is a student that I trust to ask questions when he’s unsure of something.)
3) The irrationality of sqrt(2). We were converting infinitely repeating decimals to fractions using the method of multiplying by a power of 10, subtracting, and then dividing. One of the students asked if we could use the same method to write sqrt(2) as a fraction. I told her that sqrt(2) is called irrational, which means that we can’t write it exactly as a fraction and that its decimal representation never ends or repeats. She seemed to think it was cool and surprising that there wasn’t a fractional representation and pushed a little more on that, so I tried to briefly sketch the direction of the proof. I told her to assume sqrt(2) = a/b where a/b was a fully simplified fraction, then to square both sides and try to reason her way to a contradiction. After a little bit, her response to this was, “Wow!” I don’t know if she went through the proof or not (she was also participating in the main classroom, so my guess is no), but I enjoyed seeing her excited about this and thinking that something I’d relegated to “simple contradiction proof examples” in my head was worthy of excitement.
These are the kinds of things that make me wish I were able to engage with AoPS students a little more. As an assistant, I do get to work with them one on one, but we also don’t want to talk with them about something too off-topic from the main class because we want to be sure they’re following along. Some of these rabbit holes are really worthwhile, though, and getting to engage in them just a little excited me as well as the students I was talking to (at least in the last two cases).
And like I said before, these arose fairly naturally from what we were discussing in class. The first is a connection between inequalities, something we teach to every student who takes algebra, with an idea that shows up in analysis and can feel new and weird. But it’s not new at all; it’s the same question the kids were wrestling with tonight when we asked about which values of x round to 0.6. The other two questions took a little more prompting from interested students, but they still weren’t very far afield from what we were doing. In the case of density of the rationals in the reals, that question that isn’t very far afield again points to real analysis…and the student had thought about it before. That’s kind of amazing, but it also says something about what our students can do when we let them wander mathematically.