Unringable Numbers

To succeed in doing division, especially with large numbers, there are two prerequisites. The first is that you have to know a lot of multiplication facts like THAT . Our 3-4 students--fourth graders in particular--have done a great job with this. The children, as a rule, seem to recognize the progress they've made in committing these facts to memory, and also in understanding the broader relationships between numbers (if you know that 2x8=16, how does that help you find 4x8?). It's been gratifying to get to the point in the year where I say "7x5" to a group and hear a grand chorus of "35" shooting back at me.

The second prerequisite is to think flexibly. Now that children have gotten good at telling me the product for a given expression, we need to switch the task (isn't that always the way??). So these days I give them the product and have them tell me the factors (the numbers to be multiplied to make my number). When I say "35," then, the students need to respond "7x5" (or 5x7); when I say "36," they should tell me "6x6" or "9x4." The language I use with children is that numbers such as 35 should "ring a bell"--that is, they should spark an immediate connection in children's minds with 5x7.

This ability is critical for doing division. One thing that makes it hard, though, is that not all numbers ring bells. An example is 17--the only multiplication expression that goes with 17 is the dull-and-boring 17x1/1x17. 43 won't ring any bells, either. Children not only have to be able to shoot back "3x9" when I say 27; they have to be able to recognize which numbers don't go with any expressions. That's new and different, and it takes children a while to figure out what's going on. Fortunately, young brains are quite malleable, and when they've gotten the idea, they're usually quite good at it.

So we were practicing some of these relationships this afternoon in one of the classes. 49, I said to the fourth graders I was working with, and they quickly responded with 7x7. How about 30? I asked, and got, variously, 5x6, 2x15, and 3x10. Twenty-five? That was easy, they scoffed--5x5. Then up came 37. Ring any bells? I asked.

There was silence for a moment as the students considered. At last, one child raised her hand. "It's unringable," she said confidently.

The usual mathematical term for this concept is prime, a term that this child knew but had temporarily forgotten. I must admit, though, that I have a certain fondness for unringable. Perhaps we can get the mathematicians' union to recognize it as an alternative. In a discipline that features the Pigeonhole Principle and the Generalized Ham Sandwich Theorem, I'd say it isn't entirely out of the question...