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Author Topic: multiplication easy...WHAT?!?  (Read 3935 times)

LordBucket

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Re: multiplication easy...WHAT?!?
« Reply #30 on: March 23, 2014, 06:26:58 am »

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.

Breaking down multiplication is just one tool.

Let's be sure we're comparing the same things though. The methods I've been talking about are for doing math in your head. Your 9x9 and carry method, can you do it in your head or do you need paper? Let's try this. Let's go to random.org and generate two random two digit numbers. Clicking twice I get 12 and 87. Do 12 * 87 in your head. 1044. Should take no more than a couple seconds.

Can you do (12 * 87) in your head in, say...3 seconds? How about 5? Or would you still be writing the 12 and the 87 on a piece of paper without even having started the multiplication yet by the time someone using a mental method had already given you the answer?

Because I can. Break down into easy factors, multiply then add, going left to right instead of right to left.

Here's roughly the thought process for doing 12 * 87 in your head:
870, 160, 970, 1030, 1044.

Done.

Here's the long formal explanation for the above

Recognize that (12 * 87) is the same as (10 * 87) + (2 * 87). You don't actually think this. You look at the 12 and realize that multiplying by 10 and multiplying by 2 are way easier than multiplying by 12. So do that instead. This is the "breaking down multiplication" part that you mentioned.

(87 * 10) is really easy. You don't actually need to do any multiplication. You just think "87" and than you think "0" and the result is "870." This is of course vastly simpler than the more traditional method of writing it out in reverse order for no good reason, multiplying (1 * 0) then (1 * 7) then (1 * 8 ) backwards. Think about that. If you're doing 870 * 10 on paper via the conventional method, you would write from right to left, zero, then 7, then 8, even though the number you're writing is 8, then 7, then zero. Why is the paper method backwards? There's no reason for that, and it slows you down on simple "add a zero at the end" problems like this one. Because that's what this is. In your head, "multiply by ten" is exactly the same as "think 0." "Think 0" is much faster than a three step, reverse order multiplication process.

Multiply (2 * 80). The thought process is, again, left to right: (2 * 8 ) = 16, so you think "16" then you think "0" and, like above, thinking "16" "0" results in the thought "160" with no actual multiplication by ten required.

The next step is way easier to think than it is to explain. We're doing (870 + 160). The way that we do this, written on paper, is (870+160) = (870+100) + (160-100) = (970+60) = (970+30) + (60-30) = (1000+30) = 1030. You don't think this. But that is the process. What you actually think is roughly: "870, 160, 970, 1030." You're not really "adding" here so much as you're moving portions of one number over to the other number. If I give you a shovel and ask you to dig a hole in the ground and make a pile of dirt somewhere, do you think of that as being a multiple step process of "subtracting" dirt from over there and "adding" dirt to over there? No. You move the dirt as a single process. That's what we're doing here. There's no need to "subtract" 100 from 160 to add it to 870. All we do is see that we're adding 870 + 160, and 8+1 = 9. Look at that again: we're literally adding one to 8. This is incredibly simple. It takes a tiny fraction of a second to add 8+1 in your head. And by virtue of adding 8+1 in your head, the 1 you added to the 8 is now gone, and all you need to do is add the 60. 970 + 60 involves "carrying" a digit. Carrying is slow. So we don't do it. Instead we see that 1000 is a very easy number to work with, and that we only need to move 30 over to turn that 970 into 1000. So we move 30 over from the 60 to get 1000...and then no further operations are required, because we're adding 1000 plus 30, and the process for adding (1000+30) in your head is to think, from left to right "1000" "30" which is "1030." Try that in your head. Add 1000 plus 30. Just don't think "plus" and don't think "add." Think in your head "1000" and think in your head "30." And think those two thoughts right next to each other and the addition process is done for you automatically simply by thinking those two numbers. (Incidentally, with practice this entire paragraph isn't even necessary. I'm explaining the long process, but if you get into the habit of doing math in your head, you can get so you can simply think 870, 160, 1030 without even 970 in the middle.)

Finally, we multiply the remaining (2*7), get 14, and add (1030+14). Again, no carrying is required, because we can add these numbers left to right. Think (1) then think (0) then think (3+1=4) then think (4). Answer: 1044.


Quote
It looks painfully slow, and there's the equally easy but faster way

Is it?



Related, true story: earlier this evening I was helping a friend install a tv mounting bracket on a wall. We were trying to approximately center the TV over a fireplace mantle, but needed the bracket screws to line up with studs. The tv was 55 inches wide, and studs are supposed to be 16 inches apart. We'd located one stud at almost exactly the middle of where we wanted the tv centered, which introduced the question of whether we could center the mounting bracket on that stud, or whether we'd need to put it off-center because it might not be wide enough for the edges to reach any other studs if we put the middle on that stud. We ended up needing to divide 55 by 16. Immediately we both said (at the same time. He's an accountant. I like math. The timing was funny.) "3, and change." This is part of the benefit of doing math left to right rather than right to left. If you were to do (55/16) on paper using the conventional method, you would end up wasting a lot of time on the least important digits, for no good reason. We, doing it in our head, did no division at all, and instantly generated useful information by multiplying (16 * 3) = 48 to get the leftmost part of the answer, the 3 we multiplied by, which in this case was the number of studs that would be available to bolt the bracket to within the 55 inches of horizontal space that the tv took up. Immediately we had useful information without taking the time to generate the full, precise answer. Then, rather than doing any division to get the precise answer, it was a simple matter of subtracting (55 - 48), and getting 7 (which, incidentally was done by the mental process of adding 5 plus 2 rather than subtracting anything.) And at that point, the problem is done because we can work with fractions. All we need to do is put that remainder over the divisor: 7/16ths. 55/16 = 3 and 7/16ths.


So there you go. Real life application of math story. Double digit division done in our heads at conversation speed. No paper involved.

LordBucket

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Re: multiplication easy...WHAT?!?
« Reply #31 on: March 23, 2014, 07:12:08 am »

Just as a "for fun" followup, let's say you wanted to convert that 7/16ths into decimal form. I stopped at 7/16ths while mounting the tv, but...what if? Can you do 7/16 in your head? Sure. It's not difficult.

First, immediately recognize that 7/16 is "a bit less than half." Then, take that 1/2 and change it to 8/16.

Recognize that the amount "bit less than half" that 7/16ths is, is 1/16. 7/16 is 1/16 less than 1/2.

What is 1/16? Well, we probably don't have that memorized. Personally, I don't. But we probably do have memorized that 1/8 is .125. So, 1/16 is half of .125. That's easy: .0625.

Therefore, 7/16 is .0625 less than 1/2.

.5 - .0625 = .4375

7/16 = .4375

Took me under 30 seconds, in my head. No paper. Might have been under 20. Wasn't watching a clock when I did it.

quinnr

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Re: multiplication easy...WHAT?!?
« Reply #32 on: April 01, 2014, 10:42:35 pm »

Sooo I'm not seeing it described how I would describe it.

22 times 40

becomes

 22
x40



There's an extra step if we're multiplying with digits large enough to make an individual multiplication give an answer over 9.


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Do this. It took me a really long time to get okay at my current math class, as our teacher disallows calculators for pretty much all the tests. I'm good with letters and stuff, but basic multiplication is hard, so I do it all out this way on every test. I think I have the messiest papers in the room >_<
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