Subtraction with borrowing (regrouping)
Why borrowing works An explanation of why (not how) borrowing/regrouping works when subtracting numbers
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- 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 one.
- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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