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?"
"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?"
"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.)
"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?"
"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...
Trying a few numbers by hand: 0 has a unique solution (0+0+0). 1 has three possibilities (1+0+0 and permuatations thereof). 2 has six possibilities (1+1+0, 2+0+0 and permutations thereof). 3 has ten possibilities (1+1+1 is a singleton, 2+1+0 has 6 permutations, 3+0+0 has 3 permutations). I think I see the pattern here.
Conjecture: if you have K boxes totaling N, the number of ways to construct the sum will be choose(N+K-1,K-1). I'm not sure how to prove that, but it's consistent with the results for one and two boxes, as well as what I have tried for three boxes. Possibly a proof by induction will work, i.e., if it works for a particular choice of N and K, then it must work for N+1 and K, and N and K+1.
And that sounds like a great game to play with SteelyKid. Encourage her inner mathematician while you can.
There was an article in Scientific American last year touching on this. I don't think Eric's formula works, but I don't remember. I do know that if you look at the ways to sum to a number, i.e. using 2 numbers - three numbers - four numbers ..., that the partitioning of every fifth number can be divided by 5, every 7th partitioning is divisible by 7 and every 11th partitioning is divisible by 11. Alas no other obvious ones (but other patterns occur).
The three-box answer summing to N is the sum of the two-box answers for numbers up to N. This makes some sense if you write the possibilities out sorting them by the number in the first box; at each step, the number of configurations with a zero in the first box is the same as the number of ways to sum two boxes and get N, and the number of configurations with 1 in the first box is the same as the number of ways to get N-1 with two boxes, etc.
Since there's only one way to sum two numbers and get zero, this works out so that the total number of configurations is just the sum of the integers from 1 to N+1. Which is famously (N+1)(N+2)/2.
This reminds of when my little sister realized that adding a fraction with numerator 1 and denominator being any real number with another fraction of the same conditions, the answer would be (addition of the denomiantors/multiplication of the denominators). It's amazing how kids can make you appreciate maths and its tricks.
Good job taking an exercise she had beaten to a meaningful level. I won't criticize the original one, having dealt with my fair share of students that can't sum 5+7 without a calculator, losing track of why they are doing it due to the attention shift, and still getting the wrong result, ending up with the velocity of a bicycle being superluminal, but it is nice to see a logical followup.
Try this with beginning fractions: 16/64 =1/4. 19/95=1/5. Why doesn't 14/43=1/3? It all leads to the understanding that arithmetic is really polynomial algebra....
If your student loses track of 'why' they are doing something when they use a calculator, there is a much bigger problem going on than the inability to do mental calculation.
How come it used to be the 'smart kids who end up studying physics and engineering' were the ones who carried a slide rule around, but now people like you keep complaining about calculators?
If you read back in Chad's post on "thanks CC", and in my and others' experience, reflexive calculation and algorithmic equation-solving do not predict later success. Sometimes I even used to think the opposite, but that might have been frustration at the difficulty of 'deprogramming' misguided egos.