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[LeoLeonardoIII]

I actually envision a sheet of paper. I "write" the solution for one part and then move on to the next part, and add the two together after.

Like, if I had 65 x 25 I would do 65 x 20 = 1300 and 65 x 5 = 325. Total 1625.

I've seen better ways to do mental multiplication but I haven't gone through the effort to really learn it and get good at it.

[Vector]

I see a big rectangle in my head (the multiplied number) that I break into smaller rectangles whose sizes I know.

For really hard problems, though, I also do the "piece of paper" method.

[Remuthra]

I see number relationships.

314. 3+1=4

[Urist McScoopbeard]

I just see numbers, and break them down in to some smaller ones. then multiply and add those.

[LordBucket]

Took me a while to figure out why the -10 and +10 were happening. How people keep all this stuffed into their heads and still remember what question they're trying to answer, I'll never know.

The +10 -10 is part of what makes it easier to remember. Don't think of it as "(990 +10) + (45 - 10)" That's unweildly. Think of it as "990 and 45." Now, move 10 over from one number to the other number.

You try. In your head, think "500 and 200." Now, move 100 over from one number to the other. Isn't that easier than thinking of it as a subtraction operation separate and distinct from an addition operation?

Now, consider the following problem: "372 minus 89." Try to do that in your head. If you think it in the manner you were probably taught to do math on paper, it will be needlessly difficult. Instead, think of it this way: "372 and 89. Take 1 away from 3. 272. Add 11. 283."

It's trivial to look at 89 and see that it's 11 less than 100. 100 Is a nice easy number to work with. Subtracting 100 from 372 is easy because all you're really doing is 3 - 1 = 2. Adding 11 to 272 is easy all your doing is 7+1 =8 and 2+1=3, and you can do both of those additions in a single operation.

372-89 can be performed effortlessly in your head by doing nothing other than adding or subtracting the number 1. Yes, doing math that way might be something that needs to be learned, but learning it is no different than learning to do math in other ways.

[Moogie]

LordBucket, that's one of the best ways I have ever seen mental arithmetic explained in a way that makes it seem like not such a daunting skill. Are you a teacher? If not, you totally should be!

Having said that, the example makes it seem easier than it probably is the majority of the time. It kinda falls apart a bit when you have something more like 322 minus 157. The first part doesn't change, getting rid of the hundred is still easy. But what now, you still have to start tracking multiple columns when subtracting larger digits from smaller ones (2 - 5 and 2 - 7). That's where I give up and start scribbling notes, otherwise I get lost in a sea of "carry the ones". What you say about how you think about the problem is so true. I don't remember ever being taught the tricks for mental problem solving, only the ones that require writing them down. So that's probably the biggest reason I find it so impossible.

To be fair, I'd probably find it easy if I put effort into it, so I probably sound a lot dumber than I am... but really, I've always struggled. I was always better at algebra than real math, anyway.

[GlyphGryph]

For the complicated ones remember that you can add the same amount to "both sides" of the subtraction and have it balance out. So you can do quite a bit and still only have a few numbers at a time, moving things from one to the other.

For example, add 3 to both sides~

322 - 157 +3-3 is the same as 325 - 160, which is a much easier bit of subtraction to do.

If you want to get really extreme, you can then do
322 - 157 = (322+3) - (157 + 3)    : Make the number on the right in the ones place into a "zero", which you know will be less than or equal to whatever is on the lft
= 325-160 = (325+40) - (160+40)     : Now do the same with the tens columns, making it into a "zero", which you know will be less than or equal to whatever is on the left
= 365 - 200    : And now just subtract the hundreds place
= 165

[LordBucket]

LordBucket, that's one of the best ways I have ever seen mental arithmetic explained in a way
that makes it seem like not such a daunting skill. Are you a teacher?

I have tutored math on occassion.

It kinda falls apart a bit when you have something more like 322 minus 157. The first part doesn't change, getting
rid of the hundred is still easy. But what now, you still have to start tracking multiple columns when subtracting
larger digits from smaller ones (2 - 5 and 2 - 7).

322 minus 157

300 minus 150 is 150

22 - 7 is 15

150 plus 15 is 165

In your head, easy.

the example makes it seem easier than it probably is the majority of the time.

That's a complaint I often hear from people who have difficulty with math. In the above linked thread I remember the discussion with a few people getting a bit heated. One can point out that a specific solution might be vastly faster and easier, but that it "only works" in specific cases. For example, in my last post the "go to 100" and "all you're doing is adding or subtracting 1" won't be true in all cases. But it was true tin that case, so that solution worked perfectly well. But if there'd been a different problem, a different solution would have presented itself. Math is all about patterns, and patterns manifest more often than they don't. Feynman vs the abacus comes to mind. Through no deliberate intent from the person choosing the problem, he "happened" to choose something easy to solve via a prticular method that wouldn't have worked on other problems. But if he'd chosen different numbers, odds are very good there still would have been a simple non-rote solution for them. It just would have been a different solution.

Attempting to apply one-size-fits-all cookie cutter solutions is valid for some purposes...but often results in things being massively more difficult than they need to be.

Like in that other thread, if you want to go somewhere, "consult google maps, print out directions to your destination, get in your car, follow directions you printed out" is an effective method of getting places that works in most cases. But it's simply not necessary to go to all that effort if your destination is your neighbor's house. You can see it from your front door, there's no need to print out driving directions or get in your car at all.

The more comfortable one is with math, the more often those simple solutions are apparent. Whereas if one trained to get into the habit of always using rote solutions, it's more likely those simple solutions will be missed.

[LeoLeonardoIII]

When I posted my thing I specifically tried to use numbers that started out simple and then later sets of numbers required more complex steps to solve. I think it's worthwhile to do that, otherwise you're trying to teach everything at once. So if you stop someone in the middle of an explanation and say, "That problem is too easy", you might be right - but it was intentional.

LB's 990-45 and 372-89 were good choices to illustrate what he was talking about at the time.

[Mongol13524]

The way they happen to teach it in the Occident isn't the only way to multiply. I'd look into Japanese Multiplication. (Or whatever you want to call it)

[Remuthra]

The way they happen to teach it in the Occident isn't the only way to multiply. I'd look into Japanese Multiplication. (Or whatever you want to call it)

Fixed.

[LordBucket]

The way they happen to teach it in the Oxident isn't the only way to multiply. I'd look into Japanese Multiplication.

It's a cute trick, but it's terribly slow drawing all those lines. Watching the video I was able to do every one of those problems in my head long before he got his answer and most of them before he even started drawing the lines. Brains are faster than fingers.

Take a look at this guy.

[Skyrunner]

I don't know why people always advocate breaking multiplication down :x It looks painfully slow, and there's the equally easy but faster way that involves knowing the 9x9 multiplication table and carrying numbers.

mind you, I haven't tried the multiplication breaking-down-of seriously so far.

[Remuthra]

I don't know why people always advocate breaking multiplication down :x It looks painfully slow, and there's the equally easy but faster way that involves knowing the 9x9 multiplication table and carrying numbers.

mind you, I haven't tried the multiplication breaking-down-of seriously so far.

More than anything, it's a way to explain the process .

[Tiruin]

I'd say that the ease of something depends on your memorization or learning style regarding it. Or how you do it.

Like

I don't know why people always advocate breaking multiplication down :x It looks painfully slow, and there's the equally easy but faster way that involves knowing the 9x9 multiplication table and carrying numbers.

mind you, I haven't tried the multiplication breaking-down-of seriously so far.

This is how we were taught. xD
Insomuch that it becomes like seeing the back of your hand in doing large-number multiplication. It's like multiply-addition.