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-[Voiceover]So here with Bill again. We're just going to continue our conversation of essentially the progressions of fractions in the common core through the grades. And one thing that you wanted to clarify, Bill. -[Bill] Ya, well so, I'm glad you made that comment at the end of the last video about when I was talking about the example where you compared two thirds and three fifths in grade three, and you were saying "ya, that would "be more of a discovery thing." And you're absolutely right that. Actually dividing into fifteenths is beyond the grade three standards. So, there's a little footnote in the standards that says in grade three you can subring denominators of two, three, four, or six, or eight. So, I just wanted to point that out. Now that doesn't mean you couldn't do that example in grade three, but it's beyond the grade three expectations. -[Voiceover] I see so these would be, since you have five as a denominator, that's getting us more into the grade four-- -[Bill] Right. -[Voiceover] into the grade four standard. Okay-- -[Bill] Grade three is really -[Bill] meant to be focused on pretty gentle introduction, really anchoring the idea of a fraction as a number. -[Voiceover] Okay, after your video is, maybe it will get unfrozen. We can continue to chat, we can hear you just fine. -[Bill] Okay. -[Voiceover] I'll move it off here. -[Bill] So that we don't have me looking soulfully into the distance. -[Voiceover] (laughs) That's right. -[Voiceover] Why's he looking away? Continuing with the kind of progression as we go through B dot four NF B dot three, understand a fraction A over B, with A greater than the sum of fractions one over B. So this is one over B plus one over B plus one over B, you have three of these. It's just three over B. -[Bill] Right, and this is the beginnings of fraction addition. So, one of the other things about the common core that I think is important is that it's really focused on unit fractions and thinking of a fraction as a certain number of a certain unit like three-fifths, the unit is one-fith and you're taking (mumbling). And then you can use that, or three-quarters is quarter plus a quarter, plus a quarter. You know, that's something you discover. I mean the definition of three quarters was that you put those three quarters together and count them out along the number line and you say, oh well that's putting three quarters together, so that's a sum. And moving to thinking of that fraction as a sum of unit fractions is preparation for adding fractions. Then you just add like three quarters and five quarters, you going to add the same way you add three and five. You've got quarter plus a quarter plus a quarter. And you've got quarter, plus a quarter, plus a quarter, plus a quarter, plus a quarter. And you put them together -[Voiceover] ya. -[Bill] eight quarters adding up together. And you've got eight quarters So, you want kids to just see fraction addition, with like denominators as just and extension of the operation of whole number addition. -[Voiceover] Right, right and all these substandards of B dot three are really kind of building that-- -[Bill] Um hum -[Voiceover] Oh I guess, you actually are starting to add and subtract mixed numbers with like denominators with equivalent fractions. This is essentially decomposing a fraction to a sum of fractions of the same denominator, understanding it. So this is kind of that core meat of it, exactly what you just-- -[Bill] That's right, and -[Bill] this is a sort of conceptual build up. One of the problems with fractions for a lot of people is that they think that they're completely different from whole numbers. The way you add them looks different, right? - [Voiceover] Um hum. -[Bill] The algorithm for adding two fractions looks very different from adding two whole numbers. But conceptually the addition is the same, it's just a certain bunch of a number of units and you're putting these together on the number line. So building that unity of fractions as part of the whole number system, and it's making sure that the operations of addition and then later on multiplication and division. Making sure that all those operations just grow naturally as extensions of whole number operations. That really builds a sense of number as a unified number system. -[Voiceover] You hit it not just with a traditional decomposing it into fourths, so to speak, if we're thinking about four as the denominator, or the addition of subtraction, but also that it could be, if we're thinking about fifths, it could be a one-fifth plus four-fifths. It could be one fifth, plus two fifth, plus two fifths. All of the different permutations really hits that- -[Bill] Exactly. -[Voiceover] conceptual idea. -[Bill] (connection breaks up)thing here is in the equivalence in the grade three equivalence standard, this is another thing I didn't mention before, but one important equivalence is understanding that, one a whole number is equivalent to a fraction. That is to say like three-thirds is one. Six thirds is two. -[Voiceover] Right. -[Voiceover] Okay, so that happens as early as third grade. -[Bill] that starts to happen in third grade and then once you have the general rule for equivalence of fractions in grade four it's an important instance of that. -[Voiceover] Right, right, and also in grade four, you start to think about multiplying, you know, here you see understanding fraction A over B as a multiple of one over B. It kind of ties to what we just talked about where if you-- - [Bill] Right. -[Voiceover] if you already know that if one-fourth plus one-fourth, plus one fourth is equal to three fourth. The students already know multiplication and so they can say, "well this is literally "three one fourths." -[Bill] Right, right. -[Bill] And that's an important, they've understood multiplication earlier on in grade three as equal groups. Three fours is three equal groups of four. Three fourths is three equal, well, you maybe don't call them groups anymore, but three equal portions of a quarter. And again, the idea is to link back to previous understandings of multiplication. -[Voiceover] Ya, and there's multiply a fraction by whole numbers and solve word problems. And then we start to go into things that are, I guess, laying the foundation, or, I guess, not even just foundation, being able to realize that decimals are really just a representation of fractions. -[Bill] Exactly. -[Voiceover] That's what we see right over here. -[Bill] They're not (computer breaking up) like another sort of number, they're a way of writing a number. -[Voiceover] Right, right, in fact there's kind of two ways of representing any number, decimal notation or fraction notation. -[Bill] Right. -[Voiceover] And then in fifth grade, is when you start to kind of really fill out your full fraction arsenal where you're starting to think about unlike denominators, including mixed numbers, equivalent fractions, and obviously word problems, mixed numbers and you start to think about division. -[Bill] Ya, and again, division progression. So, you'll notice what's happened there with the addition progression is we had grade four addition with like denominators, grade five addition with unlike denominators. It's sort of carefully paced out. And the same thing happens with division. In grade five, you just have division in the cases where there's really easy, intuitive interpretation. If I want to know how many fifths there are in two, I can see five fifths in each of the ones that make up two, and I can see ten of them. And so that sort intuitive notion of how many fifths are here in a two. Which is an extension of your notion of division, you know, how many threes are there in twelve or something like that. And then there's also the sharing interpretation of division. You know if I shared twelve things among three people, how many does each one get? And that extends to fractions when you're dividing by whole numbers. Like if I want to divide three-fifths by four. I'm going to break each of the fifths into four pieces so I have twentieths and then I'll have three twentieths. So, we're not doing the whole full blown division formula in grade five. That actually waits till grade six. Because you first want to build up again this idea of extending the previous understanding of an operation to figure out what that operation is with fractions. -[Voiceover] I see, and then you start to think apply previous understanding of multiplication to multiply fractions or whole number by fractions. At least on the fraction side, this is where you do start to think about one half times-- -[Bill] Right -[Voiceover] three eights or something like that. -[Bill] Right. -[Voiceover] And one thing -[Voiceover]that I think is interesting, I know that we've talked about this before offline, which is, I think an interesting conversation for everyone, we've talked about adding and subtracting and multiplication and dividing. But one thing I've found interesting is the whole notion of simplifying fractions. I know this is something we've kind of talked about at Khan Academy. -[Bill] Ya, right. -[Voiceover] Should you be forced to simplify fractions, are, as you pointed out, are fractions truly simplified? What's your point of view on that? -[Bill] Well, my point of view there, is that I don't want kids to think that simplifying is a mathematical imperative. Like there's something mathematically wrong with writing six-eights, there isn't. It's perfectly legitimate way of writing three quarters. It's a housekeeping thing. -[Voiceover] Um hum. -[Bill] Sometimes, you might want to notice that it's three-quarters, for various reasons, but it should be viewed as mathematically necessary. It's not a part of the fundamental mathematics of fractions. And sometimes, in schools, there's this imperative, it's like you must be tidy. You must always keep your room clean, or something, it's a housekeeping imperative, not a mathematical imperative. And it fights against the fundamental idea, you want kids to have, which is these are all the different names for the same number. -[Voiceover] Um hum. -[Bill] As soon as you start insisting on simplification, you might be giving the false impression that somehow you're not even really there yet, until you simplify it. So we don't require simplification. This come out, I think a particularly good example of this is fraction addition. A lot kids think that the algorithm for fraction addition is find the least common denominator, then add, then simplify. Those two first steps were not necessary. Any common denominator will do and simplification is not necessary for you to have added the fractions. So, I think we wanted to separate the underlying mathematical, and in fact, gets in the way of the understanding of the underlying operation. If you just say, you add two fractions by subdividing each unit in a way that is reflected by the denominator of the other fraction, in other words, just apply the denominators and add, then you've added the fractions. That has a nice clean link back to the previous grade's work of adding fractions with like denominators. Which in turn has a nice impact to whole number addition. If you complicate that algorithm with these housekeeping imperatives, then you've somehow muddied the mathematical waters. -[Voiceover] So what I'm hearing is, you're not opposed for students to kind of maybe learn the housekeeping eventually, but when they're first, especially their first exposure, the whole idea of least common multiple being more important than just a common multiple, does muddy why you're doing it and what it's all about. -[Bill] Exactly. -[Voiceover] So, if we can separate them out in an interesting way, but you're not opposed to, especially later on it is useful from a housekeeping point of view to have smaller numbers often and be able to reduce your computation and whatever else. -[Bill] That's right. -[Bill] And sometimes it's just convenient to notice the fact (mumbling) that's only three-fifths. Okay, I know what it is. I would say though that even there, you have to be a little careful. If you're adding three-tenths and five-tenths, you get eight-tenths, converting that eight-tenths to four-fifths is sort of weird, right. I mean, because these are tenths and, like, the decimal way of writing it would be point three plus point five equals point eight. -[Voiceover] Right. -[Bill] Now if you insist -[Bill] on simplifying the eight-tenths, you've actually lost the connection with, the parallel with the decimals a little bit. -[Voiceover] So what I'm hearing is there's a little more artfulness than we're used to in making sure that students don't just think that this is the answer and not this, but at the same time it is useful, maybe later on, when they're doing more computationally intensive things with fractions to be able to reduce things when appropriate. -[Bill] It should be a -[Bill] judgment thing, and most importantly, kids shouldn't be marked wrong for not simplifying fractions. Cuz they're not wrong. -[Voiceover] Right, right. -[Voiceover] Very interesting, well we're out of time for this video, but I think this was a really helpful clarification on a lot of the points on common core fractions. Thanks Bill. -[Bill] Thanks a lot Sal. Bye.