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FRACTIONS AS DIVISION ANCHOR CHART (1) (1)

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Dividing a Whole Number by a Unit Fraction - Video

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Dividing a Whole Number by a Unit Fraction - Video

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Video transcript

- Lets think about what three divided by one fourth is equal to. Pause this video and see if you can figure it out on your own. And I'll give you a hint, take three wholes and divide it into pieces or sections that are each one fourth of a whole and then think about how many sections you have. Alright now lets work through this together. Now let me draw three wholes. And so lets say this is... I'm gonna hand draw it, so its not gonna be perfect. But lets say that is one whole right over there. This is two wholes right over there. And then I have my third whole just like that. And now I'm going to divide it into pieces or sections that are each one fourth of a whole. So this first whole is four pieces that are each one fourth each, of four fourths. And then I have another four fourths right over here. And then I have another four fourths right over here. Another way to think about it, I took the three wholes, and I divided it into fourths. So if you divide it into fourths how many pieces am I going to have? Well this is pretty straight forward, you have one two three four five six seven eight nine ten 11 12 pieces. And I didn't even have to count that, each whole is gonna be four fourths. So three wholes is going to be three times four fourths, or 12 pieces or 12 sections. So that is going to be equal to 12. Another way to think about it is if you take three and you divide it into equal sections that are each a fourth you are going to have 12 equal sections.

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Dividing a whole number by a unit fraction (video) | Khan Academy

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Divide a Unit Fraction by a Whole Number - Video

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Divide a Unit Fraction by a Whole Number - Video

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- [Instructor] So let's see if we can figure out what 1/3 divided by five is. And I'll give you a hint. Try to draw out 1/3 of a whole and then divide it into five equal sections. Pause this video and try to do that. I always try to work through it together and to help us as I promised or I suggested I guess. We said let's draw 1/3. I will represent a whole by that square right over there. And now let me split up into three equal sections. So this is all hand drawn with the aid of a computer. So it's not going to be perfect. But let's say that that is three equal sections. It's roughly three equal sections. I didn't do it perfectly but you hopefully get the idea. And so 1/3 would be one of those three equal sections. So that's a third right over there that I have just shaded in and I want to divide it into five. I want to divide it by five, I should say. So let's do that. So to divide it by five, I'm gonna divide it into five equal sections. And if I'm dividing that one into five equal sections, let me just divide all of the thirds into five equal sections. I'm essentially just going to make five rows here. One, and I'm gonna eyeball it, so it's going to be approximate. Two, three, and then four and five equal sections. I now split this whole into one, two, three, four, five rows of equal height. Now if I go to my original third and I divide it by five, I would be left with this right over here. But what fraction is this of the whole? Well what I've done now is I've split my whole into 15 equal sections. How do I know that? Well I could count them. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. Or you could just say look I had one, two, three thirds and now each of those have been split into one, two, three, four, five equal sections. So three times five is 15. So each of these is a 15th and so the 1/3 divided by five is just one of those 15ths. So that right over there is one of those 15ths. So this is going to be equal to 1/15 and we are done.

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Dividing a unit fraction by a whole number (video) | Khan Academy

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Instructor] So let's see if we can figure out what 1/3 divided by five is. And I'll give you a hint. Try to draw out 1/3 of a whole and then divide it into five equal sections. Pause this video and try to do that.

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Understand Fractions as Division - Video

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Understand Fractions as Division - Video

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Understanding fractions as division (video) | Khan Academy

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Sal shows how a/b and a÷b are equivalent. That is, the fraction bar and the division symbol mean the same thing.

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We've already seen that a fraction like 2/9 can be interpreted as 2 divided by 9. Let me do the 2 in the same color. That if we have a fraction, it could be interpreted as the numerator divided by the denominator. And this leads to all sorts of interesting conclusions, some of which we've already seen, and some of which are a little bit new. So, for example, if I had the fraction 7/7, this can be interpreted now as 7 divided by 7, as our numerator divided by our denominator. And 7 divided by 7 is of course equal to 1. And this is consistent with what we've already seen. 7/7 would get us to a whole, and a whole is the exact same thing as one. But we could do things a little bit more interesting as well. We could take something like 18/6 and realize, wow. This is the same thing as 18 divided by 6, which we know is equal to 3. And we should do a little reality check. Does this make sense, that 18/6 should be equal to 3? Well, we could rewrite it. We could rewrite 18/6. Let me make the sixths that same orange color. That's going to be the same thing. 18 is 6 plus 6 plus 6. And then all of that over 6, and then that's the same thing. That's the same thing as 6/6 plus 6/6 plus 6/6. And I could make this right over here in orange. And we've already seen, or we've seen many, many videos ago, that 6/6, just like 7/7, these are each equal to a whole. These are each equal to 1. And we can now view this as 6 divided by 6, which is the same thing as 1. So this is 1 plus 1 plus 1, which is, of course, equal to 3. But this starts to raise an interesting question. This all worked out just fine because 18 is a multiple of 6. 6 divides evenly into 18. But what happens if we start having fractions where the denominator does not divide evenly into the numerator? Let's say we have a fraction like 23 over 6. Well, we know that we can interpret this as 23 divided by 6. And if we actually divide 23 by 6-- let's do that. So we divide 6 into 23. We know 6 goes into 23 three times. 3 times 6 is 18. And then when you subtract, you end up with a remainder of 5. So we might say, hey, 23 divided by 6 is equal to 5 remainder 3. But that's not that satisfying. What do I do with this remainder? This really isn't a number here. This is just saying that we're going five times, and then we have a little bit left over. What we can do now is manipulate this a little bit so that we can realize that this is a number, and in particular, a mixed number. So for example, we could start with a 23 over 6, and we could divide it into-- or we could decompose the numerator into one part that is divisible by 6, evenly divisible by 6, and the remainder. So for example, 23 over 6 we can rewrite as 18 plus 5 over 6. Notice I decomposed the 23 into one part that is a multiple of 6, and it's the largest multiple that essentially fits into 23, or that is less than or equal to 23, and then the remainder. When you divide 6 into 23, you get a remainder of 5. You could view it as, I divided it into the remainder and everything else. And the reason why this is interesting is because we know that this is going to be equal to 18 over 6 plus 5 over 6. Well, we already know that 18/6 is the same thing as 18 divided by 6, or 3. So this is the same thing as 3. So we know that 23 over 6, which is the same thing as 18 plus 5 over 6, is the same thing as 3 plus 5/6. Or, if we want to write it as a mixed number, we could write it as 3 and 5/6.

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Creating mixed numbers with fraction division (video) | Khan Academy

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We've already seen that a fraction like 2/9 can be interpreted as 2 divided by 9. Let me do the 2 in the same color. That if we have a fraction, it could be interpreted as the numerator divided by the denominator.

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Multiply Fractions and Whole Numbers - Video

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Multiply Fractions and Whole Numbers - Video

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Mixed Numbers and Improper Fractions Resource

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Mixed Numbers and Improper Fractions Resource

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Write Mixed Numbers as Improper Fractions

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Write Mixed Numbers as Improper Fractions

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Write 5 and 1/4 as an improper fraction. An improper fraction is just a pure fraction where the numerator is greater than the denominator. This right here, it's not a pure fraction. We have a whole number mixed with a fraction, so we call this a mixed number. So let's think about what 5 and 1/4 represents, and let me rewrite it. So if we're talking about 5 and 1/4, and you can literally think of this as 5 and 1/4 or 5 plus 1/4, that's what 5 and 1/4 represents. So let's think about 5. Five is 5 wholes, or if you're thinking of pie, we could draw literally five pies. Let me just cut up the pies from the get go into four pieces since we're dealing with fourths. So let me just cut up the pies right over here. So that's one pie right over there. Let me copy and paste this. Copy and paste. So I have two pies, and then I have three pies, and then I have four pies, and then I have five pies. So this is what the 5 represents. 5 literally represents-- so let me circle all of this together. That is the 5 part right there. That is what 5 literally represents. It represents five whole pies. Now, I have cut up the pies into four pieces, so you can imagine each piece represents a fourth. Now, how many pieces do I have in these five pies? Well, I have four pieces per pie. Let me just right it here. 4 pieces per pie times 5 pies is equal to 20 pieces. Or another way to think of it, since each piece is a fourth, this is also equal to 20 times 1/4, or you could just write this as being equal to 20/4. So we have 5 whole pies is equal to 20 fourths. Let me write it like that. 20 fourths. Or we could write it as 20/4. I've kind of done the same thing twice. So that's what the five pies represent. 20/4 or 20 pieces, where each piece is 1/4. Now, the 1/4 right here represents literally one more fourth of a pie or one more piece of a pie, so let me draw another pie here. So that is another pie. Cut it into four pieces. But this 1/4 only represents one of these pieces, right? This is one of the four pieces. The denominator tells us how many pieces. The 1 tells us how many of those pieces we're dealing with, so it's just this one piece over here. That right there is the 1/4. Now, if we write 5 and 1/4, we just saw that the 5 right here is 20/4. So we could rewrite this. Let me write it like this. 5 and 1/4 can be rewritten as the same thing as 5 plus 1/4, which is the same thing as-- we just saw that five whole pies is the same thing as 20/4. And to see that these are the same thing, you literally just divide 4 into 20. You get 5, and nothing is left over. So 5 is the same thing as 20/4, and then this plus 1/4 is the same thing as plus 1/4. So if I have 20 fourths and I add one more fourth to it, how many fourths do I have? Well, I have 21. I have 21 fourths. Or another way of thinking about it, this 5 is-- so this right here is 20 pieces of pie. You can even count it. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. But a quicker way is to say, well, we have five pies. Each of them have four pieces. 5 times 4 is 20. This 1/4 right here represents one piece plus one piece, so total we're going to have 21 pieces. So we have 21 pieces, where each piece is 1/4, so we could say we have 21 times 1/4 or 21 fourths pieces of pie. However you want to think of it, but we've solved the problem. We're at an improper fraction. We've written 5 and 1/4 as an improper fraction. Now, I've gone through great pains to give you the intuition of what 5 and 1/4 means, but there is a fairly straightforward process for getting straight to the improper fraction. Let me color code it. So if you have 5 and 1 over 4, to convert it into an improper fraction, you're going to keep the same denominator, so you're going to have the over 4 there. But your numerator is going to be your numerator of the fraction part before. So it's going to be 1 plus your whole number times your denominator. So 1 plus-- or actually, let me do it the way I tend to think of it. What I do is I take 4 times 5. So let me write that down and I want to color code it. 4 times 5, and then to that, I add this numerator. So I literally do 4 times 5 plus 1, which is-- so this is equal to 4 times 5 is 20, plus 1 is 21, and then that's over 4, so it's 21/4. And all of this is kind of a fast way to do it. We're literally doing the exact same thing that we did here in kind of a slower way. We're saying, OK, 5 wholes is the same thing as 20 fourths, so you take 5, and I figure that out, 5 times 4, and then I have one more fourth there, so 4 times 5 plus 1 gives 21.

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Writing mixed numbers as improper fractions (video) | Khan Academy

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