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- [Sal] I'm here with Bill McCollum. We've already done a few videos, and we're going to continue delving into the spirit and intent of the Common Core. What we have here is, I've put down most of the articulated standards on fractions that students see in elementary school. I was hoping to work with you, Bill, just making sure we have a good sense of the narrative and how this might be different from the way fractions have traditionally been approached. - Okay, sure. One of the big differences, I think, is really pretty firmly moving towards understanding fractions as numbers. So, you'll see that in the second grade standard there, there's a standard about partitioning geometric figures into halves or thirds, and understanding that language. That's a very standard sort of thing to do, understanding fractions as sharing. That's a good introduction to the idea in grade three. Again, the first standard in grade three is also about dividing a quantity into equal parts. But quickly we move to that second fraction standard, which I think is all-important for the approach in the Common Core: understand a fraction as a number on the number line. A lot of people come out of school, even adults, somehow never quite thinking of fractions as numbers. They think of them as mysterious things, complicated to add them, complicated to divide them. But, somehow, a firm sense of that as a quantity, being able to place 3/2 or 5/3 on a number line, that's what we're trying to establish early on in grade three there. - [Sal] This is a departure from the way it's traditionally done? Traditionally, the idea of a fraction as a part of a whole? - Exactly, and they're not losing that idea. That is an important notion, of course. But sometimes you never get beyond that idea. One symptom of this is people thinking fractions always have to be less than one. If you're always thinking of parts of wholes, you think that fractions bigger than one, like 3/2, you tend to think of them as sort of weird. In fact, traditionally, in school mathematics, those fractions are called "improper." When there's nothing improper about them really. They're just fractions. - [Sal] So that's an interesting point, and what we'll talk about are a couple of other. There are these things that we all remember, like improper fractions. So, is it improper for people to use the word improper fractions? (laughs) - No, I'm in favor of people using whatever words they want to use. I'm not an authoritarian. On the other hand, mathematically, there is no difference between ... Fractions bigger than one are no less fractions than fractions less than one. They're all just fractions. I think that's an important idea to understand. So as long as that term is not emphasized in a way that makes kids think they somehow are different, I guess there's no harm using it. It's actually not used in the Common Core because we wanted to emphasize the unity of the system of numbers on the number line. - [Sal] That makes sense. Second grade, that's the very first introduction. There you are, thinking about splitting things into equal shares and fractions of equal shares, things like that. Third grade, you don't lose that, but you start thinking of it as a number on the number line, this 3 NF A.2. - Right. - [Sal] Then, we also start thinking about equivalents of fractions, which I guess could be a combination of both. You could think about them, where they sit on the number line. You could also think about them as fraction of shares. - Right. Now, the full-blown equivalent of fractions comes in grade four. But as soon as you start putting fractions on the number line, or even if you're just dividing up rectangles or something, you are gonna discover that 2/4 refers to the exact same point as 1/2. It's the same number. So, really, those are two different ways of saying the same number. That comes out a little bit in grade three, but in grade three it's examples like the one I just gave. Then in grade four, you move on to the general rule for equivalence of fractions, which is that if you have any fraction and multiply a numerator and denominator by the same number, you'll get an equivalent fraction. Again, you can see that visually, actually for that, a good model is the rectangle model, where you divide a rectangle up into fourths, say, vertically. Then, maybe three of those sub-rectangles would represent 3/4. Then if you wanted to see why that was the same as 6/8, you could draw a line down the middle there. Suddenly you've divided your rectangle up into eight equal parts, and get six of them and it's the same fraction, right? - [Sal] You've multiplied both your numerator and denominator by two. - Yeah, and that's enacted in this representation by that act of dividing the thing vertically into two equal parts, which multiplies both the total number of parts and also the number of shaded parts by two. - [Sal] So one way, maybe you want to tell me if I'm interpreting this wrong, this 3 NF A.3, when we're talking about explaining equivalence of fractions and special cases, you could have a student comparing 3/5 to 3/7 or comparing 2/7 to 5/7, or 1/2 to 2/4, and they're discovering when things are equal or when something is larger. And this is comparing. - Right. There's equivalents and there's also comparing. So that includes cases where they're not equal. - [Sal] Right, so this is almost discovering it through depicting, conceptualizing it on the number line or even through some type of a diagram. But it's in fourth grade that they learn to put it all together,to think about it a lot more systematically. - Right, exactly. And in grade three, I think you're going to be dealing with pretty small denominators. Halves, thirds, maybe quarters and fifths, but not much more than that. - [Sal] Hmm, okay, that's an interesting ... Well, obviously, once you start doing this, you can get fairly large denominators in your-- - Yeah, and I think the prep or discovery phase in grade three is a preparation for understanding this general rule in grade four. - [Sal] Yep, yep, now that makes sense. - So you also have the comparison there in 4 NF A.2. You have a more systematic approach to it. - [Sal] So 4 NF A.2, just to compare it again to what happens in the third grade, you start to introduce the notions of greater than, equal, or less than. Just to remind me, the students have already seen these symbols in the context of comparing integers? - Correct. - [Sal] Or whole numbers I should say, not integers at this point. So they're familiar with it, but now they're applying these ideas to fractions. While in grade three, you're not necessarily using those symbols, you just say hey, which one's larger or which one's ... - I think that's right. One of the things I always want to say is that just because something isn't mentioned it's not forbidden. There's no harm using those symbols in grade three. They will have already seen them even in grade one, actually, with comparing whole numbers. But part of the emphasis in grade three is less about the formal symbolism. It's about understanding the symbolism of a fraction itself and just grokking what they are. And playing around on the number line or also with area models and getting an introduction and a firm feeling for them as numbers. And comparing them and seeing when two different fractions are equivalent and therefore name the same number. That's part of getting a feel for them as numbers. - [Sal] I see. At least on a first reading, this standard feels similar to this standard right over here. It has the obvious difference of this is explicitly calling out the symbols, but it sounds like comparing, let me say, 5/11 to 6/13. This is something that's probably closer to the fourth grade. - Exactly, because at that point, how are you gonna do that? You're gonna try and find a common denominator and see which one has more. Eleven times 13, what's that? A hundred and forty-one or something? One forty-three? So you're gonna put them both as fractions over 143, and then you're going to see which one has more. That's like a formal algorithmic thing. If you've had experience comparing 3/5 to 3/4, or maybe 2/3 to 3/5 or something, in grade three, and you've played around with how to see, with small denominators you could actually divide the number line. Then what I just said, 2/3 and 3/5, you could divide something up into fifteenths and count. But once you get up to the bigger fractions, you're not gonna actually do that with manipulatives, you're gonna do that more with an understanding of common denominators. - [Sal] But in third grade, we wouldn't necesssarily expect them to divide into fifteenths. I could imagine them making a very precise diagram. - Yeah. - [Sal] The fifteenths wouldn't necessarily be expected. It would be a great discovery for a student. - Exactly, exactly, yeah. - [Sal] Okay, great. Well, we'll just continue this conversation in the next video because I think we're getting close to our time limit. - Sure, alright. Nice talking to you, Sal. - [Sal] Same here.