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Welcome to the presentation on why, not how, borrowing works.

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And I think this is very important because a lot of

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people who even know math fairly well or have an advanced

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degree still aren't completely sure on why borrowing works.

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That's the focus of this presentation.

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Let's say I have the subtraction problem

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1,000-- that's a 0.

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1,005 minus 616.

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What I'm going to do is I'm going to write the same problem

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in a slightly different way.

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We could call this the expanded form.

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1,005-- what I'm going to do is I'm going to separate

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the digits out into their respective places.

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So that is equal to 1,000 plus let's say zero 100's

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plus zero 10's plus 5.

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1,005 is just 1,000 plus 0 plus 0 plus 5.

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And then that's minus 616.

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So that's minus 600 minus 10 minus 6.

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616 could be rewritten as 600 plus 10 plus 6.

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And I put a minus there because we're subtracting

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the whole thing.

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So let's do this problem.

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Well, if you're familiar with how you borrow is, this 5 is

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less than this 6, so we have to somehow make this 5 a bigger

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number so that we could subtract the 6 from it.

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Well, we know from traditional borrowing that we have to

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borrow 1 from someplace and make this it into a 15.

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But what I want to see actually, is understand where

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that 1 or actually where that 10 comes from.

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Because if you're turning this 5 into a 15 you actually

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have to add 10 to it.

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Well, if we look at this top number, the only place that

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a 10 could come from is here, is from this 1,000.

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But what we're going to do since this is the 1,000's

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place, instead of borrowing 10 from here, which would make it

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kind of a very messy problem, I'm going to borrow

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1,000 from here.

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I'm going to get rid of this 1,000.

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And I have a 1,000 that I took from this 1,000.

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I have 1,000 now that I can distribute into

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these 3 buckets.

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Into the 100's, 10's and 1's buckets.

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Well, we need 10 here, so let's put 10 here.

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So it's 10 plus 5 is equal to 15.

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We got our 15.

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If we took 10 from the 1,000 then we have 990 left.

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So we could put 900 here and 90 here.

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Notice, we just said-- so we had 1,000 and we just rewrote

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it as 900 plus 90 plus 10.

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And we added this 10 to this 5.

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And now we could do this subtraction just how we

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would do a normal problem.

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15 minus 6 is 9.

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90 minus 10 is 80.

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900 minus 600 is 300.

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So 300 plus 80 plus 9 is 389.

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And let's see how we would have done it traditionally and make

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sure that it would have kind of translated into the same way.

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Well, the way I teach it and I don't know if this is actually

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the traditional way of teaching borrowing, is I say, OK, I need

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to turn this 5 into a 15.

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So I have to borrow a 1 from someplace.

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Well, we know from this side of the problem that we actually

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borrowed a 10 because that's why it turned to 15.

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If we're going to borrow 1, I'd say, well, can I

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borrow the 1 from the 0?

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No.

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Can I borrow the 1 from this 0?

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No.

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I could borrow it from here, but I'm borrowing

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it from 100, right?

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So 100 minus 1 is 99.

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So that's the how I do it.

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And I say 15 minus 6 is 9.

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9 minus 1 is 8.

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And 9 minus 6 is 300.

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So this way that I just did it is clearly faster and, I guess

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you could say it's easier, but a lot of people might say, well

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Sal, that looks like a little bit of magic.

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You just took that 5, put a 1 on it, and then you borrowed

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a 1 from this 100 here.

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But really, what I did is right here.

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I took 1,000 from this 1 and I redistributed that

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1,000 amongst the 100's, 10's, and 1's place.

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Let me do another example.

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I think it might make it a little bit more clearer

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of why borrowing works.

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Let me do a simpler problem.

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I actually started off with a problem that tends to confuse

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the most number of people.

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Let's say I had 732 minus-- Let me do a fairly simple one.

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Minus 23.

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Sometimes those 3's just come out weird.

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Well, we just learned that's the same thing as 700 plus

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30 plus 2 minus 20 minus 3.

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Well, we see this 2, 2 is less than 3, so we can't subtract.

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Wouldn't it be great if we could get a 10 from someplace?

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We could get a 10 from here.

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We make this into 20 and add the 10 to the 2 and we get 12.

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And notice, 700 plus 20 plus 12 is still 732.

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So we really didn't change the number up top at all.

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We just redistributed its quantity amongst the

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different places.

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And now we're ready to subtract.

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12 minus 3 is 9.

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20 minus 20 is 0 and then you just bring down the 700.

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You get 700 plus 0 plus 9, which is the same thing as 709.

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And that's the reason why this borrowing will work.

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Well, we say, oh, let's borrow 1 from the 3.

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Makes it a 2.

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This becomes a 12.

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And then we subtract.

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9 0 7.

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Let's do another problem, one last one.

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And once again, you don't have to do it this way.

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You don't have to every time you do a subtraction

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problem do it this way.

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Although if you ever get confused, you can do it this

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way and you won't make a mistake, and you'll actually

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understand what you're doing.

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But if you're on a test and you have to do things really fast

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you should do it the conventional way.

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But it takes a lot of practice to make sure you never are

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doing something improper.

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And that's the problem.

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People learn just the rules, and then they forget the

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rules, and then they forgot how to do it.

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If you learn what you're doing, you'll never really forget it

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because it should make some sense to you.

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Let's do another problem.

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If I had 512 minus 38.

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Well, let's keep doing it that way I just showed you.

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That's the same thing as 500 plus 10 plus

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2 minus 30 minus 8.

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Well, 2 is less than 8.

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I need a 10 from someplace.

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Well, one option we can do is we can just get

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the 10 from here.

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So then that becomes 0.

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And then this will become a 12.

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Notice that 500 plus 0 plus 12, same thing as 512 still.

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So we could subtract.

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12 minus 8 is 4.

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But here we see this 0 is less than 30, so we can't subtract.

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But we can borrow from the 500.

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Well, all we need is 100, so if we turn this into 100 then we

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took the 100 from the 500.

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This becomes 400.

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I just rewrote 500 as 400 plus 100.

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Now I can subtract.

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100 minus 30 is 70.

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Bring down the 400.

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And this is the same thing as 474.

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And the way you learn how to do it in school is you say, oh,

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well, 2 is less than 8, so let me borrow the 1.

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It becomes 12.

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This becomes a 0.

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0 is less than 3, so let me borrow 1 from this 5.

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Make this 4.

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This becomes 10.

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So then you say 12 minus 8 is 4.

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10 minus 3 is 7 and you bring down the 4.

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Hopefully what I've done here will give you an intuition

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of why borrowing works.

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And this is something that actually I didn't quite

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understand until a while after I learned how to borrow.

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And if you learned this, you'll realize that what you're doing

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here isn't really magic.

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And hopefully, you'll never forget what you're actually

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doing and you can always kind of think about what's

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fundamentally happening to the numbers when you borrow.

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I hope you found that useful.

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Talk to later.

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Bye.

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