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[LordBucket]

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.

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]

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]

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

22 times 40

becomes

 22
x40

Spoiler: Basic multiplication of two 2-digit numbers (click to show/hide)

You start by multiplying the digit in the ones place of the lower number by the ones place of the upper number. The result is written below.

 22
x40
  0


Now you multiply the bottom ones place by the next digit of the upper number, which is the tens place, and write that result below.

 22
x40
 00


The first answer row is complete because we have multiplied the bottom number ones digit by the whole upper number.

Now we need to start a new answer row, below the first one, and every time you add a new answer row you lead it with a fresh 0 to shift it to the left. Here is our problem with the new zero.


 22
x40
 00
  0


We are now engaging the "4" which is the next number to the left of the bottom row of the problem, the tens place. We need to multiply that "4" by the "22" above. Again we do it in two steps, starting with multiplying the "4" with the rightmost "2" in the upper number's ones place.


 22
x40
 00
 80


Then we multiply the "4" with the "2" in the tens place on the top row.


 22
x40
 00
880


In all cases we're loading the new digit to the left in our answer row until we run out of digits in the top row to multiply.

Right now we would check to see if there are any more digits in the "40" row to continue the process. There aren't. We're finished multiplying.

Now we add everything in our answer row, vertically.


  22
 x40
  00
+880
   0




  22
 x40
  00
+880
  80




  22
 x40
  00
+880
 880


And there is our answer.

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

Spoiler: Multiplying numbers that require carrying a digit (click to show/hide)

Let's say we start with 23 x 85.


 23
x85


The first step would be multiplying the 1s digit of the lower number by the 1s digit of the upper number.


 23
x85
 15


But then the next space we should enter an answer digit is already taken up by that 1! Such an unscrupulous 1, sitting in someone else's seat. Instead of writing it in the answer row, we carry it up to above the next top-row digit - the tens place.


 1
 23
x85
  5


Now we do our next multiplication ...


 1
 23
x85
105


... and then add the carried 1 from above to the answer ...


 1
 23
x85
115


... and then dang, we're over one digit, so we have to carry again like so.


11
 23
x85
 15


This next part requires that you make a leap - of imagination! Any number actually has zeroes in front of it. So if I write "25" it's actually equal to "000025". This is because I'm really saying "this is a number with 5 in the ones place, 2 in the tens place, 0 in the hundreds place, 0 in the thousands place" etc. But normally it doesn't matter that there's nothing in all those higher digit places - nobody cares because they don't do any work. WELL THEY DO NOW BUDDY.

You're going to imagine there's a zero in front of our lovely "23" up there. You multiply like so (except without actually writing out the zero in real life, because we're too cool for that):


11
023
x85
015


Here we just said "5 times 0 equals zero", but we need to add the carried "1" to that result, which makes it just "1" below.


11
023
x85
115


Great! Excellent! Done with the first part. We've multiplied the 1s place of the lower number by the entire upper number. You may ask, "How did you know to stop adding zeroes and keep multiplying forever? I stopped because I knew any more zeroes would just result in an answer of 0 again, because 5 x 0 = 0 and there are no more carried leftovers above to add up to anything.

Now let's start on the lower row. LEAD WITH A ZERO because we're starting a new answer row. Also I'm crossing out all my carries up top so I don't get confused, and writing all my new carried numbers above it. In handwriting you'd use tiny numbers but I tried and the forum makes them look confusing.

Here's what it looks like after I struck out my old carried digits, which we won't use again, and add that leading zero to my second answer row.


11
 23
x85
115
  0


Now I multiply the top row using the tens place digit of the bottom row, starting with the top row's 1s place.


 2
11
 23
x85
115
 40


You'll notice for convenience I stick the carried digit above the next thing I'll be multiplying.


12
11
 23
x85
115
840


Here I said "8 x 2 = 16, and add the carried 2, =18. Of that 18, the 8 goes in the answer row and I carry the 1 to the row above."

Now we again have a carried digit and nothing to multiply. I'll add that "phantom zero" to show what's happening: 8 x 0 = 0, add that carried 1, the answer is 1, which we add to the left side of the answer row.


 12
 11
 023
 x85
 115
1840


Now that we're out of multiplication to do, we add.


  12
  11
   23
  x85
  115
+1840
    5


And Then


  12
  11
   23
  x85
  115
+1840
   55


But Then


  12
  11
   23
  x85
  115
+1840
  955
 

And So (RETURN OF THE PHANTOM ZERO)


  12
  11
   23
  x85
 0115
+1840
 1955

Spoiler: Why does this work? (click to show/hide)

In short, there is a Rule in multiplication that says you can break up and mix up stuff and still multiply it as long as you do it exactly right. Long multiplication like this is a way of making sure you get it right. Another way to explain it is that when you multiply 11 x 5, what you're actually doing, is multiplying 10 x 5 and also 1 x 5 and then adding the result together. If you wanted to multiply 111 x 5, you would do 100 x 5 plus 10 x 5 plus 1 x 5. As long as you keep the digits you're multiplying in exactly the right places (as in, ones place, tens place, etc.) it will work out. And the reason you would do it that way? Because we know that multiplying by zero equals zero! So I can easily skip past a bunch of steps and spot 500 + 50 + 5 right away.

When you want to multiply a two digit number by a two digit number, you're splitting it up into a whole lot of individual multiplications. You're taking, say, 11 x 11, and doing the following:

10 x 1 = 10
1 x 1 = 1
10 x 10 = 100
1 x 10 = 10

And adding the results together (121, which is what you get with a calculator or long multiplication doing 11 x 11). Again, long multiplication is a way to make sure you didn't miss any of those tasks - such as when multiplying a ten digit number by another ten digit number. It also saves space, believe it or not.

Spoiler: Stupid math tricks (click to show/hide)

Don't be surprised if stupid math tricks don't work for you until you go through algebra. It changes the way you think about some things and the processes you learn can really help your basic math.

Let's say you have two numbers you want to multiply, like 55 x 12. I can break that down in my head into two operations that are simpler: 55 x 10 and 55 x 2.

The 55 x 10 is super easy because it follows a kinda pseudo-rule that I don't think they bother to put into the textbooks until algebra: multiplying by 10, just add a zero to the right side of the other number. If you need to multiply by some larger one like 10,000 just strip the zeroes off one and stick em on the other: 55 x 10,000 means strip four zeroes off the right side and stick them on the left. So you end up with 550,000.

Anyway, that means our first part is 550. What about 55 x 2? I need to break that down into 50 x 2 and 5 x 2. Suddenly it's clear: 50 x 2 = 100, and 5 x 2 = 10, so you add them together and get 110. Now we go back up and add the two answer results: 550 + 110 = 660.

With certain numbers you can do this in your head pretty well. It's kinda tough to do unless you have a good handle on your higher-end multiplication table. Also it's easier for me to do when I imagine a sheet of paper with the numbers written on it - although your method may vary. 

Spoiler: Boxes (click to show/hide)

The boxes method is great but only if you have a handle on that multiplication trick I mentioned above with the 10s. This is because you'll need to be able to figure out in your head what 60 x 50 is for example. This requires an extra process: stripping away the two 0s, multiplying 6 x 5, and then slapping the two zeroes back on at the end. Like so:

60 x 50
6 x 5 (with 00 off to the side)
30 (with 00 off to the side)
3000

Which kinda requires knowing the various Rules of Multiplication that they don't teach until algebra.

I suspect Boxes may be fast if you're good at it, but so is long multiplication, and both get you to the same place.

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 >_<