Even in the tightest, most focused class, there will be some slack time. Maybe you don't have a quorum of students, or you're waiting to be called to an assembly, or you finished a lesson early or you just realized that the kids need a break.

Particularly for inexperienced teachers, slack time can present a real threat to classroom management. When not being challenged with mathematical concepts, those active young brains will quickly turn to the problem of finding clever and effective ways to test authority.

But slack time can also be an opportunity. In a relaxed setting, doing something that doesn't trigger their math-is-hard defenses, you can often sneak in some useful problem solving and pattern recognition practice. Even more importantly, you can reinforce the alertness essential to being good at math.

Which brings us to this family of puzzles.

Go up to the board (preferably without saying anything), draw a horizontal line, then start writing the sequence A, B, C... or 1, 2, 3..., putting some of the characters above the line and some below based on some property of the letters or digits. Somewhere in the middle of this you stop, turn to the class and ask where the next letter goes.

A B D
______________________________________________________________________________

C E F G

If you get the wrong answer, shake your head, write the character in the proper place then ask about the next. You may have to give a couple of hints but usually someone will let out an "Ooooh" of realization and will start eagerly shouting out the answers.

The property here is topological and you can, if you want, use this introduce topology or even to open a lesson on homotopy equivalence.

The important here though isn't introducing the topological terms; it's getting students to think topologically, to add these properties to the things they look for.

The properties in an above/below puzzle can be anything your students could reasonably be expected to spot, from prime vs. composite for numbers to the phonetic spelling of letter (there's a clever New Yorker cartoon that put the alphabet in alphabetical order but I can't find it on line).

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