Partial Quotient Division An alternate to traditional long division
Partial Quotient Division
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- Let's say we had to figure out
- how many times 16 goes into 1,388.
- And what I want to do is first think about how we
- traditionally solve a problem like this, and then introduce another method that allows for a little bit more approximation.
- So traditionally you would say, well sixteen does not go into one any times,
- so then you move over one spot, and then how many times does it go into 13?
- Well it still does not go into 13, and then you go all the way into 138.
- And then you say, well, sixteen does go into 138, but how many times does
- it go into 138? And you might try nine first, and I'll do all my arithmetic on the
- right side so you ight say 16 times 9, 6 times 9 is 54, 1 times 9 is nine,
- plus five is fourteen, so that's a hundred and forty-four times, but still, that's larger than a 138.
- So it's going to go into it eight times, eight times is going to be less than 138,
- so we stick the eight here. And notice to do this little trial and error here
- I had to make sure I got the right exact answer with that eight right over here. Then when we say eight
- times six is 48, and then eight times one is eight, plus four, is twelve.
- So eight times sixteen is 128, so when I subtract, I get the remainder from 138, so I get a remainder
- of eight minus eight is zero, 3-2 is 1, and these cancel out.
- So I have a remainder of ten, but I still have a remainder of eight so I bring
- that down and I have a hundred and eight. And then I do the same thing again.
- Let me get rid of this so everyone doesn't get distracted, we say how many times does
- sixteen go into 108? And you can approximate and say well, it's definitely not eight times, eight times
- is 128, is it, maybe, seven times?, and you might do a little math on the side,
- so its 16 times 7, 6 times 7 is 42, one times 7 is seven plus four is eleven, so you get 112. So that's
- still to large, so that's going to be six, but notice, we had to do this little sidework on the side
- right over here to come up--to realize it wasn't seven, we know six is the largest how many times it
- can go into 108 without going over it. So 6 times 6 is 36,carry the three and regroup the 3 depending
- on how you think about it. 6 times 1 is 6 plus 3 is 9, plus six is -- or plus six is nine/ Then you subtract
- again, eight minus six is two, and then you could just say ten minus nine is one, or you can even borrow,
- you could make this ten and then this goes away, 10-9 is 1, so then you have 12. And for not going
- into decimals, you're kind of done, because 16 does not go into 12,
- so 16 goes into 1288 eighty-six times with a remainder of 12. And that's all a decent way of doing it.
- And that's the way you traditionally know how to do it, but what I wanna do is to
- introduce another, maybe a little more interesting way to solve a long division problem.
- So once again, 16 goes into 1388. What we're going to do is a much more ??? way
- for approximation or for essentially guessing. And what we want to do is just guess
- we are going to make guesses for how many times 16 goes into the numbers without overestimating,
- without jumping too high. And now we're going to talk, we're not just going to think about 1 or 13 or 139.
- We are going to think about the whole number as a whole, and before we do that I am going to get
- two things out of the way, just because it will help us. I am just going to remind ourselves
- what 16 times 2 and 16 times 5 are. I am just picking these as random numbers that we can use
- to approximate. You do not have to use 2 and 5, you can use any numbers.
- Maybe I'll show other examples there. So 16 times 2, we know, is 32.
- 16 times 5 is 50+30 which is 80. Let's just keep these two results in mind while we try
- to tackle this right over here. So the first thing to think about is the best guess
- for how many times does 16 go into 1388. Or another way to think about, how many times
- does 16 go into a thousand, let's just do something as a very rough approximation.
- Well, we know it is not going to be 100, because 100 times 16 would be 1600.
- You just roll those two zeros at the end of it. And you say how many times does it go into a thousand.
- We know 16 times 5 is 80. We know that 16 times 50 would be 800. Let's use that.
- I'm using 5 instead of 2, I'm multiplying it by another 10 to get to 50,
- because 800 is a lot closer than 320 to 1000 that we care about.
- We could say, well, 16 times 50 will get us to 800. Once again, how do I know that?
- Well, 16 times 5, I know, is 80, so 16 times 50, I have multiplied by 10, is 800.
- And then I just subtract. 8-0 is 8, and then you can say 13-5 is 588.
- Now we ask ourselves, how many times does 16 go into 588? How close can we get to that.
- And let's just assume that we only know the stuff right over here, or we can multiply 16
- times a multiple of 10. So 800 would once again be too big.
- Let's just go with 320 right over here. We know that 16 times 2 is 32
- so 16 times 20 is going to be 320. I just multiplied 2 times 10 which would give our product to times 10.
- So we can subtract this right over here. 8-0 is 8, 8-2 is 6 and then 5-3 is 2.
- Now I'm left with 268 and we say, how many times does 16 go into 268.
- Let's see, 800 is too big, even 320 is now too big. Well, we could say, let's see 10 times 16
- results in 160. Let's just try that out. We do not even have to get the right exact answer.
- We do not have to get the highest multiple that is less than 268. We just have to make sure
- that we are still within 268. If we multiply, we do a new color over here, 16 times 10 we get 160.
- 160 we subtract again. 8-0 is 8, 6-6 is 0 and 2-1 is 1. We are left with how many times does 16 go into 108.
- And we can go back to..., we know 16 times 5 is 80. So let's just try out 5.
- 16 times 5 is 80, we subtract right over here. 8-0 is 8, 10-8 is 2, so we are left with 28.
- Now it is pretty simple. How many times does 16 go into 28? Well it only goes into it one time.
- Then when you subtract 16 from 28, 8-6 is 2 and 2-1 is 1. You are left with a remainder of 12.
- We might say how do we know how many times does 16 go into 1388?
- Well it goes 50 times plus 20 times plus 10 times plus 5 times plus 1 time.
- That is going to be, you could just add up all of these things on the right hand side.
- That is going to be 50 plus 20 is 70, plus 10 is 80, plus 5 is 85 plus 1 is 86. So there we have it.
- It went into it 86 times with a remainder of 12.
- What's cool about this method is that every step, I could put 60 over here
- and I could do the math correctly. Or I could have picked my two multiples to be 16 times 6 and 16 times 3
- and I would get different results here, but at the end I would still have gotten the right answers.
- So what it does is give us some methods so that we are always thinking about,
- we are kind of biting away chunks of what we are dividing into. So first we bit off an 800 piece chunk.
- Then we bit off a 320 piece chunk, and we keep going until we cannot divide by 16 anymore.
- Hopefully you found that kind of interesting.
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