From what i understand though, it's the thinking and problems solving that the kids are stinking up the room in. At least at our school.

my sentiments precisely.

7's and 12's are hard because there is no pattern for them. the other numbers, most particularly 2, 4, 5, 9, 10, and 11 are quite easy when you recognize the pattern. and since those numbers constitute half of the other numbers, you can get half of them by reversing the digits. my son always had the most trouble with 7's and 12's. so did i, when i was his age.

that is what i call "creative reasoning".
another answer is each bus takes seven, and the last takes 8, and 6 kids can sleep on the bench seats.

it really is about thought: about levels of abstraction and problem solving. it is about making sense of the world- about how things relate to one another. i am not trying to be profound, but that really is how i think of math. it is an organizational method. but at a certain point, after you have the basic tools, learning shortcuts and developing strategies for math is the next level.

we had this discussion either on this board or another- but multiplication and division are either something you need to do on a calculator, or something you can mostly do in your head, depending on how you think. if you are trying to compute gas mileage (US units for now), you just got 11.8 gallons, and drove 276 miles. you can whip out the calculator, but it is really not terribly hard to do a 3 x 3 digit math problem in your head. but it takes strategies to do it.

the first part of the strategy is to know the goal. mileage is kinda meaningless if you can't get to at least two units, preferably three. so, if you approximate, it has to be on one of the two halves of the division, and only the last digit. in this case, if you say that the problem is 276/12, then you can "fudge" it later, by adding 2% (a good approximation) to the mileage.

2+7+6 is 15, so the numerator is divisible by 3, and that result is 92. the divisor then becomes 4. now divide both by 4, and you get 23. 2% of 23 is 0.46. but you need to round that down, because the "fudge" is actually less than 2%, not over. so the answer is 23.4

i don't know if that is right, or not. but doing that sort of thing in the head keeps you sharp. and it is all based on strategies, knowing what you have, what you know, and what you want.

or, to use the terms that they used in engineering school: accuracy -vs- precision.

11.8 ~ 12
12*12=144
2*144=288=24*12
288-276=12
23*12=276
Estimate ~23 miles per gallon
That's how my german brain gets the estimate in a few seconds.
(12*12=144 >known result since age 6 or 7
2*144=288 >simple doubling each digit
288-276 >comparing selected digits)

12 is easy. Just multiply by 10 and then add twice the number.

that's really good, jab.

i think they are all easy for that reason. you can break it into addition using the distributive property.

if you know your 5's and 11's, you know your 16's.
if you know your 9's and 10's, you know your 19's.

you can use subtraction too, for the harder ones......just have be clever in how you approach problems.

ie. You have to understand how numbers work.