Learn Why Borrowing Works

Learn Why Borrowing Works Welcome to the presentation on why not how borrowing works and I think this is a very important a lot of people even know math fairly well or have a advance degrees still aren’t completely sure on why borrowing works and that’s the focus on this presentation. Let’s say I have the subtraction problem 1,000 that’s a zero 1,005 – 616 what I’m going to do is I’m going to write the same problem in a slightly different way we call this the expanded form. 1,500 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 zero hundreds plus zero tenths plus five. 1,005 is just a thousand plus zero plus zero plus five and then that’s minus 616 so that’s minus 600 minus ten minus six, six hundred six—six. 616 could be rewritten as 600 + 10 + 6 and I’ll put a minus a minus there because we are subtracting the whole thing. So, let’s do this problem, well we know if you’re familiar with how you barrow is this five is less than the six so we have top somehow make this five a bigger so that we could subtract the six from it. Well, we know from traditional borrowing that we have to barrow one from some place and make this into a 15. But what I want to see actually is understand where that one or actually where that ten comes from because your borrowing—string this five into a 15 you actually to have to add ten to it. Well, if we look at this top number the only place that a ten could come from is here, is from this thousand. But what we’re going to do, since this is a thousands place instead of borrowing ten from here which would make it kind of a very messy problem I’m going to borrow thousand from here. So I’m going to take I’m going to rid of this thousand ad I have a thousand that I took from this thousand. I have a thousand now that I can distribute into these three buckets, into the hundreds, tenths, and ones buckets. Well we need ten here so let put ten here, right so it’s ten plus five is equal to 15, we got our 15. So if we have ten if took ten from the thousand then we have 990 left, so we could put 900 here and 90 here, notice, we just said so we had a thousand and we just rewrote as 900 plus 90 plus ten and we add this ten to this five. And now we could do this subtraction just how you do in normal problem, 15 – 6 is nine, 90 – 10 it is 80, 900 – 600 is 300, so 300 + 80 + 9 is 389. And let’s see how we would done it in traditionally and see make sure that it would be kind of translated it 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 it say “okay, I need to get a, I need to turn this five into a 15. So I have to borrow one from something so we know from this side of the problem we actually borrow of ten that’s why it turn to 15. If we are going to borrow one I’d say where I can barrow the one from zero? No, can I barrow the one from this zero? No, I could barrow it from here but this is, I’m borrowing it from a hundred. So a 100 minus one is 99, so that’s the how I do it and I say 15 minus six is nine, nine minus one is eight and nine minus six is 300. So this way that I just did it is clearly faster and I guess you could say it’s easier but lot of people might say well Sal that’s looks like a little bit of a magic. You just took that five, put a one on it and then you barrowed a one from this hundred here. But really, what I did here is right here, I took a thousand from this one and I redistributed that thousand amongst the hundreds, tenths, and ones place. Let me do another example I think it might make a little bit more clear of why borrowing works. Let me do another a simple problem, actually I started at off with the problem that tends to confuse the most number of people. Let’s say I had a 732 minus—let me do a freely simple one minus 23, sometimes this three just come out weird. Well we just learn that it’s the same thing as 700 + 30 + 2 – 20 – 3. Well we see these two, two is less than three so we can’t subtract. Would be greater if we get a ten from some place? Or we get a ten from here, will make this into 20 and add the ten to the two and we get 12. And notice, 700 + 20 + 12 is still 732, so we really didn’t change the number of top of it all we just redistributed its quantity amongst in the different places. And now we’re ready to subtract, 12 – 3 is nine, 20 – 20 is zero and the you just bring down the 700, you get 700 plus zero plus nine which is the same thing as 709. And that’s the reason why this borrowing will work, well we say “oh let’s barrow one from the three makes it of two just becomes to 12 and then we subtract” 907. Let’s do another problem one last one and I think and once you don’t have to do it this way, you don’t have to, every time you do subtraction problem you do it this way although, if ever you get confuse you can’t do it this way and you wont make a mistake and you’ll actually understand what you’re doing. But if you are on a test and if you do 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 learned just the rules and they forget the rules and then they forget 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 had 512 – 38, well let’s keep doing that way I just showed you, that’s the same thing as 500 + 10 + 2 – 30 – 8, well two is less than eight, I needed ten from some place, well one option we could do is we could just get the ten from here so then that becomes to zero and this will become 12. You notice that 500 + 0 + 12 is the same thing as 512 too. So we could subtract 12 – 8 is four. But here we see this is zero is less than 30 so we can’t subtract but we can borrow from the 500, well, all we need is a 100, so let’s—if we tend this into a 100 then we took the 100 from the 500 this becomes 400. All I did I took a, I just rewrote 500 is 400 plus a 100. And I can subtract a 100 – 30 = 70 and bring down the 400. And this is the same thing as 474. And the way you learn how to do it in school is just say “well two is less than eight’ so let me barrow that one becomes 12 this becomes to zero, zero is less than three so let me barrow one from this five make this four and this becomes ten. So then you say 12 – 8 is four, 10 – 3 is seven and you bring down the four. Hopefully what I’ve done here will give intuition of why barrow works this is something that actually I didn’t quite understand until a while after I learn how to barrow. And if you learn this you’ll realize that what we are doing here isn’t really magic and hopefully you’ll never what you’re actually doing and you could always kind of think about what’s fundamentally happening to the numbers when you barrow. I hope you found that useful. Talk to you later. Bye.