Welcome to the presentation on why, not how, borrowing works.
And I think this is very important because a lot of
people who even know math fairly well or have an advanced
degree still aren't completely sure on why borrowing works.
That's the focus of this presentation.
Let's say I have the subtraction problem
1,000-- that's a 0.
1,005 minus 616.
What I'm going to do is I'm going to write the same problem
in a slightly different way.
We could call this the expanded form.
1,005-- what I'm going to do is I'm going to separate
the digits out into their respective places.
So that is equal to 1,000 plus let's say zero 100's
plus zero 10's plus 5.
1,005 is just 1,000 plus 0 plus 0 plus 5.
And then that's minus 616.
So that's minus 600 minus 10 minus 6.
616 could be rewritten as 600 plus 10 plus 6.
And I put a minus there because we're subtracting
the whole thing.
So let's do this problem.
Well, if you're familiar with how you borrow is, this 5 is
less than this 6, so we have to somehow make this 5 a bigger
number so that we could subtract the 6 from it.
Well, we know from traditional borrowing that we have to
borrow 1 from someplace and make this it into a 15.
But what I want to see actually, is understand where
that 1 or actually where that 10 comes from.
Because if you're turning this 5 into a 15 you actually
have to add 10 to it.
Well, if we look at this top number, the only place that
a 10 could come from is here, is from this 1,000.
But what we're going to do since this is the 1,000's
place, instead of borrowing 10 from here, which would make it
kind of a very messy problem, I'm going to borrow
1,000 from here.
I'm going to get rid of this 1,000.
And I have a 1,000 that I took from this 1,000.
I have 1,000 now that I can distribute into
these 3 buckets.
Into the 100's, 10's and 1's buckets.
Well, we need 10 here, so let's put 10 here.
So it's 10 plus 5 is equal to 15.
We got our 15.
If we took 10 from the 1,000 then we have 990 left.
So we could put 900 here and 90 here.
Notice, we just said-- so we had 1,000 and we just rewrote
it as 900 plus 90 plus 10.
And we added this 10 to this 5.
And now we could do this subtraction just how we
would do a normal problem.
15 minus 6 is 9.
90 minus 10 is 80.
900 minus 600 is 300.
So 300 plus 80 plus 9 is 389.
And let's see how we would have done it traditionally and make
sure that it would have kind of translated into the same way.
Well, the way I teach it and I don't know if this is actually
the traditional way of teaching borrowing, is I say, OK, I need
to turn this 5 into a 15.
So I have to borrow a 1 from someplace.
Well, we know from this side of the problem that we actually
borrowed a 10 because that's why it turned to 15.
If we're going to borrow 1, I'd say, well, can I
borrow the 1 from the 0?
No.
Can I borrow the 1 from this 0?
No.
I could borrow it from here, but I'm borrowing
it from 100, right?
So 100 minus 1 is 99.
So that's the how I do it.
And I say 15 minus 6 is 9.
9 minus 1 is 8.
And 9 minus 6 is 300.
So this way that I just did it is clearly faster and, I guess
you could say it's easier, but a lot of people might say, well
Sal, that looks like a little bit of magic.
You just took that 5, put a 1 on it, and then you borrowed
a 1 from this 100 here.
But really, what I did is right here.
I took 1,000 from this 1 and I redistributed that
1,000 amongst the 100's, 10's, and 1's place.
Let me do another example.
I think it might make it a little bit more clearer
of why borrowing works.
Let me do a simpler problem.
I actually started off with a problem that tends to confuse
the most number of people.
Let's say I had 732 minus-- Let me do a fairly simple one.
Minus 23.
Sometimes those 3's just come out weird.
Well, we just learned that's the same thing as 700 plus
30 plus 2 minus 20 minus 3.
Well, we see this 2, 2 is less than 3, so we can't subtract.
Wouldn't it be great if we could get a 10 from someplace?
We could get a 10 from here.
We make this into 20 and add the 10 to the 2 and we get 12.
And notice, 700 plus 20 plus 12 is still 732.
So we really didn't change the number up top at all.
We just redistributed its quantity amongst the
different places.
And now we're ready to subtract.
12 minus 3 is 9.
20 minus 20 is 0 and then you just bring down the 700.
You get 700 plus 0 plus 9, which is the same thing as 709.
And that's the reason why this borrowing will work.
Well, we say, oh, let's borrow 1 from the 3.
Makes it a 2.
This becomes a 12.
And then we subtract.
9 0 7.
Let's do another problem, one last one.
And once again, you don't have to do it this way.
You don't have to every time you do a subtraction
problem do it this way.
Although if you ever get confused, you can do it this
way and you won't make a mistake, and you'll actually
understand what you're doing.
But if you're on a test and you have to do things really fast
you should do it the conventional way.
But it takes a lot of practice to make sure you never are
doing something improper.
And that's the problem.
People learn just the rules, and then they forget the
rules, and then they forgot how to do it.
If you learn what you're doing, you'll never really forget it
because it should make some sense to you.
Let's do another problem.
If I had 512 minus 38.
Well, let's keep doing it that way I just showed you.
That's the same thing as 500 plus 10 plus
2 minus 30 minus 8.
Well, 2 is less than 8.
I need a 10 from someplace.
Well, one option we can do is we can just get
the 10 from here.
So then that becomes 0.
And then this will become a 12.
Notice that 500 plus 0 plus 12, same thing as 512 still.
So we could subtract.
12 minus 8 is 4.
But here we see this 0 is less than 30, so we can't subtract.
But we can borrow from the 500.
Well, all we need is 100, so if we turn this into 100 then we
took the 100 from the 500.
This becomes 400.
I just rewrote 500 as 400 plus 100.
Now I can subtract.
100 minus 30 is 70.
Bring down the 400.
And this is the same thing as 474.
And the way you learn how to do it in school is you say, oh,
well, 2 is less than 8, so let me borrow the 1.
It becomes 12.
This becomes a 0.
0 is less than 3, so let me borrow 1 from this 5.
Make this 4.
This becomes 10.
So then you say 12 minus 8 is 4.
10 minus 3 is 7 and you bring down the 4.
Hopefully what I've done here will give you an intuition
of why borrowing works.
And this is something that actually I didn't quite
understand until a while after I learned how to borrow.
And if you learned this, you'll realize that what you're doing
here isn't really magic.
And hopefully, you'll never forget what you're actually
doing and you can always kind of think about what's
fundamentally happening to the numbers when you borrow.
I hope you found that useful.
Talk to later.
Bye.