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The Two-Box Addition Game [Uncertain Principles]
SteelyKid’s school does a “March Math Madness” thing, and this year all the kids in her class are being asked to practice “Math Facts” for ten minutes a night. This appears to be motivated by some requirement that students be able to rattle off basic addition problems at high speed. So there are flash cards and the like.
She’s good at this, but quickly gets bored, and does not hesitate for an instant in letting her boredom be known. It’ll be sort of interesting to see how this plays out if they actually expect her to answer 30 addition questions in a minute, or whatever the ridiculous requirement is.
Anyway, last night after the third or fourth “This is EASY,” I decided to shake it up a little, and try for something approximating actual, you know, math of the sort done by mathematicians.
“OK, I have a new game for you. How many ways can you add two numbers to get five?”
“What?”
“Here, let me show you. If you have box plus box equals 5, how many different ways can you fill in the boxes.”
“Hmm… I’ll have to write them down, OK?”
“Sure, that’s a good thing to do.”
“Do you want me to write down the turn-around facts, too?”
“Excuse me?”
“You know, where you turn them around? So, like, you have 2+3=5, then turn it around to get 3+2=5?”
“Well, we know those are there, right, so you don’t need to write them down if you don’t want to.”
“OK, I won’t… OK, there.”
“Great. So, there are three of these, and they each have a turnaround, so that’s…” (Note that their sets of addition facts always include zero, which gets you two extra ways of making five.)
“Six.”
“Right. Now let’s try it for box plus box equals six.”
“Hmmm… OK, I’m done.”
“Right, so how many are there?”
“Two, four, six, eight.”
“Is that right, though? What about this one, can you turn it around?”
“Oh, right, 3+3 is the same when you turn it around. So, 7.”
“Great. Now, how about seven?”
“OK, let me see…”
“Now, let’s think: there were six ways to fill the boxes to get five, right? And seven ways to fill the boxes to get six. So how many do you think there will be for seven?”
“Eight?”
“OK, let’s see.”
“Hmmm… Yeah! It’s eight!”
“Awesome! Nice work.”
At this point, it was time to head upstairs to bed, but we kept talking about it.
“So, what I wanted you to see is that this is another kind of thing you can do with math. You can look for these kinds of patterns, and see if they work.”
“So, like, if you had box plus box equals 100, you could figure out that there would be, like, 99 ways to do that?”
“Well, now, there were six ways to get five, and seven ways to get six, so…”
“Oh, it would be a hundred and one.”
“Right! A hundred is even, so you know there has to be an odd number of ways to fill the boxes–”
“Because there’s a double! It’s… fifty plus fifty, right?”
“Exactly. Nice work, honey.” She beamed, then went off to brush her teeth and pick out bedtime reading.
————
So, that’s my attempt at introducing SteelyKid to real math. If we’re feeling really ambitious tonight, we might try it with three boxes, but I need to work out the pattern for that myself, first…
Source: http://scienceblogs.com/principles/2015/03/05/the-two-box-addition-game/
