Above the line, below the line

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).

Mutant sprouts

A while back I posted a recommendation for a popular pencil-and-paper game:

On the subject of topology, my game of choice is Sprouts, invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967 (as a general rule, you can't go wrong with a game if Conway had anything to do with it).

The rules are simple:

1. Start with some dots on the paper. The more dots you have the longer the game takes so you will probably just want to start with two or three.

2. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself.

3. The lines can be straight or curved but they can’t cross themselves or any other lines.

4. Each dot can have at most three lines connecting it

5. When you draw a line put a new dot in the middle.

6. The first player who can’t draw a line loses.

I was thinking about sprouts the other day and a few variations occurred to me. I don't know if they're particularly playable or if they add any interesting aspects to the game, but if you can't put a half-baked idea in a blog, what's a blog for?

Variant 1 -- Free sprouts

Played as above but with the following addition: for the first k moves of a game with n dots, the player, after drawing a line, adds a new dot.

Topologically the result is a game with n+2k dots (keep k small) but with the complication that lines are being drawn without knowing exactly how those lines partition the surface. This is still a game of perfect information but the variation should make it more difficult to think a few moves ahead.

Variant 2 and 3 -- Scored sprouts

Each player starts with a separate sheet of paper and proceeds to connect the dot according to the standard rules. After no more lines are possible, the players score their graphs based on the number of dots.

Score = 6


Score = 7

In variant 1 the player with the highest score wins. In variant 2, the win goes to the lowest.

Originally posted at Education and Statistics.

Topologists at play -- the game of Sprouts

It's important to have students think deeply about math in both structured and unstructured ways (I have a guilty feeling that I ought to say more about this, but that will have to wait for a future post). It's the unstructured part that tends to cause problems. That's one of the reasons I liked to make games part of my lessons when I was a teacher.

Games (at least the kind I recommend) require a great deal of focus -- you have to think about what you're doing or you won't do well -- and they encourage exploration and a playful attitude to the material. All of these things help build mathematical intuition.

On the subject of topology, my game of choice is Sprouts, invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967 (as a general rule, you can't go wrong with a game if Conway had anything to do with it).

The rules are simple:

1. Start with some dots on the paper. The more dots you have the longer the game takes so you will probably just want to start with two or three.
2. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself.
3. The lines can be straight or curved but they can’t cross themselves or any other lines.
4. Each dot can have at most three lines connecting it.
5. When you draw a line put a new dot in the middle.
6. The first player who can’t draw a line loses.

You can find a couple of sample games here.

Originally posted at Education and Statistics