We're wrapping up our series on the Over strategy. In this episode Pam and Kim reason through a Problem String to develop the Over strategy for division.
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Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
And I'm Kim.
And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but ya'll rotely repeating steps actually keeps our students from being the mathematicians they can be.
If you're just tuning in, you have just missed the first three episodes of a series that we're going to wrap up today about a super important strategy that can be used for all four operations. Not only can it be used, but we argue that it should be used. But don't worry, you can always go back and listen to the other three episodes, and check out how we talked about the Over strategy for addition, subtraction, and multiplication. But for today, we're going to heading in to talk division.
So, don't leave if you didn't catch the first three. You know, hang in, you can listen to division. But yeah, you'll want to check out those others. And one of the reasons that you'll want to check out all of the episodes on the Over strategy is because the Over strategy is so cool, and it's so wonderful to maybe start with. Like, if you teach higher grades, you might be like, "Oh, why am I listening to all this arithmetic stuff?" Well, let me tell you. It's one way that you can get your students starting to think about mathematizing. Because the Over strategy is so... I don't know, there's a snap to it. There's like this pop. It's like, "Ooh!" it has the "Ooh" factor. Yeah, cool. So, Kim?
Let's start today, where I'm going to give you a division problem. I'm going to give everybody a division problem to think about, and then we are going to do a Problem String and see if we can get people's brains firing in a certain way. So, that division problem kind of gets a really nice Over strategy. So, you might be thinking about the problem: 396 divided by 4. 396 divided by 4. Alright, so think about that, everybody. Maybe pause the podcast, solve it a little bit, and then come on back. And we are going to work on the Over strategy with division. All right, Kim, you ready?
Okay. 36 divided by 4. 36 divided by 4.
I mean, that's, yeah. Like, you probably didn't...
Just a fact. Yeah.
Yeah, you didn't probably have to think about it too much. I'm just going to note that we're kind of thinking about fours. So, I'm going to. I just drew a horizontal ratio table, and I said, we're thinking about fours. So, I put 1 to 4. So, like, it almost looks like one-fourth, on this horizontal ratio table, kind of 1 to 4. And I asked you for 36 of them. So, next to that 4, I drew a 36. Drew. I wrote a 36. And you said that that was 9. So, like, to get to 36, you multiplied by 9. So, we kind of have two equivalent ratios now. I've got 1 to 4, and 9 to 36. And that's a way to represent 36 divided by 4 is 9. Cool. Next problem: 360 divided by 4?
Because on my ratio table, I have 9 to 36. And I thought that 360 was just 10 times bigger than 36, so my answer is going to be 10 times bigger than 9, which is...
10 times 9 is? Sorry, didn't mean to interrupt.
No, it's ok.
So, since 360 is 10 times 36, then 90 is 10 times 9. Cool. Next problem. How about 356 divided by 4?
That's just 4 less than 360, so, it's one group of 4 less, so it's going to be 89.
So, eighty-nine 4s would give you 356.
Cool. So, right now, on my ratio table, I've got 1 to 4, 9 to 36, 90 to 360, 89 to 356. Yeah?
Alright. Next problem: 400 divided by 4?
I mean, that's (unclear). Right? So I'm just going to put that in the ratio table, the 400 is on the same line as the 4, 36. It's all... The dividends are going on the bottom. And then, you said that was 100 4s? Yeah? And then, how about... So, the 100 is on top. So, now that 100 to 400 is an equivalent ratio in the same table. How about 392 divided by 4?
So, one hundred 4s was 400. And this is two 4s less than 400. So, it's going to be 98.
So, I actually just added two 4s, or the ratio 2 to 8 in the ratio table. And then, I wrote down 392, and you said it would be?
98 because you took away two groups from the 100. So, I'm kind of looking at the 100, and the 2, and the 98 are next to each other and the 400, and the 8, and the 392. So, now I have this equivalent ratio of 98 to 392. Yeah?
Cool. Next problem. How about 200 divided by 4?
How do you know?
Because one hundred 4s is 400. So, it's only fifty 4s to make 200.
Oh nice. So, you could go from the ratio of 100 to 400, to 50 to 200? Cool.
Yep. And I actually have a divided by 2 on my ratio table.
100 divided by 2 is 50. 400 divided by 2 is 200. Nice. So, 200 divided by 4 is 50. Next problem. How about 196 divided by 4?
Forty-nine. It's going to be one less group of four than the previous problem.
How about 198? Oh, this must be a typo. Can you do 198? I'm kidding. [Pam laughs]
[Kim laughs]. Oh gosh. 198 divided by 4?
(unclear). 48.5 or 48 and a 1/2.
How do you know?
Because now I have to think about instead of... Well, I'm still thinking in terms of 4s. But half of 4 is 2. So, I wrote half on top of my ratio table and 2 at the bottom.
Before you go on...
Oh, did you ask me 198? Good gravy. I just did 196. And now you're asking me 198?
So it's 49 and a 1/2.
Ah. Because you said 48 and a 1/2.
Yeah, I did.
But now you're thinking 49 and a 1/2?
Yeah. 49 and a 1/2.
Because? Keep talking.
Because that half of 4, which is 2, I need to add to the 49 and the 196.
Gotcha. So, on the ratio table, right now, I have an equivalent ratio of 0.5 to 2.
And I have it right next to the 49 to 196. And if I just add the 49 plus 0.5, and then 196 plus 2, then I get the ratio of 49.5 to 198?
Which is a way of solving 198 divided by 4 is 49.5.
Yep. And mine's not actually next to mine. I ran out of room on my ratio table, so I had to write it back at the beginning again.
Like, so I wonder if that...
Well, don't worry because I'm about to run out of room on mine.
Yeah, mine's getting smaller and smaller.
I guess I write slightly smaller than you do or something like that. Okay. Last problem. What is 199 divided by 4?
Okay, so then I wish I had more room for an entry because I would say that that is a... it's just one more, so it's going to be another fourth of a 4.
So, I want it to be 49 and 3/4. Or 49.75.
49.75. So, I do have room on my second layer of my ratio table. So, right next to the 0.5 to 2, I've written 0.25 to 1. It's so funny that we have this 2, like two is two. Like, I'm just aware that since we're audio, and when I say...
...0.5 to 2, it's the ratio of 0.5 to the number 2.
And you know what I wish I would have done?
Well, hang on a second. Let me finish my sentence.
So, I have the ratio of 0.5 to the number 2. And I have the ratio of 0.25 to the number 1. And you were saying that you added 1 to 198. So, you added 0.25 to 49.5. That's what you said, right? But now you have a better idea?
Yeah because I already have 50 fours is 200. And I'm just 1/4 of a four away. So I could have gone down from 50 to get to 49.75.
It's almost like you could have used Over.
And so, if you wouldn't have shared that with this Problem String, knowing that my goal is to develop the Over strategy, I would have asked for it. Now, sometimes teachers are like, "What do you mean, you would have asked for it?" Oh, that's such a good question because this is such a...what's a good word...a subtle teacher move in this point. So, I might say, "Oh, nice strategy, Kim, to go from the 198 divided by 4 to help you with the 199 divided by 4. I'm a little curious. Did anybody use, I don't know, the 200? Maybe? I don't..." Like, I'll just sort of lob it out there, and if it's within the zone of proximal development of students, somebody will grab it and run with it. And then, I'll say, "Oh, look. Like you could? Well how could you?" And then, it's almost as if they sort of had it to begin with. So, we kind of call it the "trail off method" where you just kind of say enough, and then wonder if somebody can kind of finish. And maybe I have to add a little bit more before they... "You know, did anybody use the 200 divided by 4. Did anybody? You know, if you know that there's fifty 4s in 200? And I don't know. Did anybody use that to help you think about one hundred and ninety-nine 4s? Anybody?" Anyway, it's a way of doing that. What were you just going to say?
I'm chuckling at myself because. You know, I know that we're talking about Over. But I'm looking back at my ratio table and I'm realizing, without thinking about in the moment, that I used the 90 to get to 89, and I used the 100 to get to 98, and I used the 50 to get to 49. And now, I'm thinking about the 50 to get to 49.75. And I used the Over strategy without it being like a major undertaking. It was just... It was so naturally embedded into the Problem String that I wasn't saying to myself, "Okay, I'm going to use the Over strategy today. And I'm going to focus on it." And it just... Like, I used it multiple times in my thinking of these problems, and it's clearly on my ratio table that represented my thinking, but I was not even focused on it. So, I love that you're saying at the end of your Problem String, you're going to call out some of those relationships and ask kids to think about what they've used, even if they aren't aware of it, even if they haven't specifically noted it.
And by calling it out, this is not where all of a sudden I start direct teaching. This is where I pull it out of students. Like, "What relationships did you..." And it's kind of what you did. "What relationships did you find yourself using over and over in this Problem String? Oh, sure enough." Like it's... We kind of use this problem. It was a little too much, so we just backed up a little bit. And this problem was a little too much, so we just hacked off that little extra. And as we pull those words out of students, they gain clarity by stepping out of it and looking back at their work and generalizing and discussing and creating. It's almost like they're creating arguments. They're sort of constructing viable arguments. It doesn't have to be an argument to be a generalization, to be... Or, you know, in other words, you don't have to be arguing to construct a viable argument. And then, as students say what they say, and then they add on, or they repeat, or they help clarify, that's critiquing the reasoning of others. That's one of the things that we're trying to do. And one of the reasons they can do that is because we were just modeling with mathematics. We were making the thinking visible on a ratio table, so the students could point to things. They could look back at the problems and see the relationships they were using. So, there's a couple of things to point out. Nice, super nice. Let's talk a little bit about where the Over strategy for division fits in with the hierarchy of division strategies. So, a first thing to ask would be, what property is the Over strategy for division kind of based on? Is it based on the associative property? The commutative property? The distributive property? And often, a few of them are used, but there's kind of a main one to focus on. And we would suggest that, or not we would suggest, that it would be true that the Over strategy for division is based on the distributive property. You kind of have to mess with that a little bit because it's really the distributive property of multiplication over subtraction. And that's brilliant. And that is one of the first things that we want kids to think about as they are solving division problems. First, we want kids to do partial quotients. But really quickly, we want them to think about smart partial quotients. And one of the smart partial quotients that we want them to consider is this idea of, can I find a smart partial quotient that's a bit too much? And then, I have to back up. I have to adjust a little bit from that quotient that gave me kind of too big of an answer, and I have to adjust back. And so, Over division would be a fairly early strategy that we want kids to think about, even when they're thinking about single digit facts. A most missed facts is 7 times 8. We might say, "Well, do you know eight 8s?" Well often students do. Often they'll know eight 8s, "Well, can you use eight 8s to help you think about seven 8s?" That's an Over. Oh, I'm doing multiplication not division. But I can do. Let me do the same division idea. If they're thinking about 57 divided by 7. 57 divided by 7. I'm dying here. I was going for sevens. If they're thinking about 56 divided by 7. And you're like, "Well, do you..." No, I'd have to use 8 to do the Over. Holy, I'm not doing this very well. This is off the cuff ya'll. Can you tell? If I'm thinking about 56 divided by 8. And I say to myself, "I don't know how many 8s are in 56." I might say, "Well, I know how many 8s are in 64. So, If I know how many 8s are in 64, can I use that to help me think about how many 8 are in 56?" Ah, that's an example of the Over strategy for division coming in pretty early. And you can tell how connected multiplication and division are in my head because I just sort of went down the multiplication route. And that's okay because I have multiple connections, and it's alright for me to go. Kim, you like I'm justifying my slip there? Alright, cool.
So, after that, we would expect kids to think about division as equivalent ratios. We would want to bring in. Well, I didn't mention 5 is half a 10. And we'd also want them to use 5 is half of 10 in division strategies. But then, we would also want them to have that other interpretation of division where they're not just thinking quotatively, about how many 4s are in. That's what we just did with this Problem String. How many 4s are in 36? How many 4s are in 360? In a big way, we were kind of thinking that way. We would also want them to have the other interpretation, the partitive interpretation of division, where they can find equivalent ratios. So, we could have thought about ratios to solve all these problems. But we didn't this time. But we would want students to do that. So, that then they can find equivalent ratios, which would be the most sophisticated division strategy.
One thing I want to really point out, that's super important about one of the reasons we emphasize... No, let me say it differently. A lot of teachers will say. They'll do a Problem String with us where we're thinking quotatively, how many of the divisors are in the dividend? And they'll say, "Oh, yeah, yeah yeah. We do that. We do that thing." And I'll say, "Great, like, tell me more about that." And then, they'll do, they'll show me the "magic seven", or the "lucky seven", or this way of it kind of looks like long division. But they're letting students use quotients that mean something to them.
Make sense to them.
Yeah. Like, so if I were doing a problem like... I don't know. Help me, Kim. 356 divided by 4. They would put the 356 in the house top, and they would put the 4 outside. And then they would say, "Okay, what do you know?" And if a kid goes, "Well, I know 4 times 10 is 40." It's a terrible one to choose. Then, they'll kind of write the 40 down there, and then somewhere they'll keep track of that 10. And then, they'll subtract, and then they'll keep going. And forever and ever they'll just keep subtracting off these partial quotients. And they'll kind of keep track of how many of them there were. So, the idea of using partial quotients that you know is brilliant. Recording it in this "lucky seven" and having kids do these really inefficient chunks, is not super great. They will do inefficient chunks when we start, but if we record them on a ratio table, we can get kids more efficient much faster because we can scale in the ratio table. And so, just like when Kim used the. We started off with a 1 to 4, it was equivalent to the 9 to 36. And then, when I said, "Well, what's 360?" Then bam, she could just scale up. And then, how that 356 was related to that 360. Bam, she could just like use relationships in there. Similarly, kids can do that when they start with a division problem in a ratio table. They can use relationships in and out that they know and use fairly sophisticated relationships. But maybe most importantly, they can go over in a ratio table. You cannot go over in that "lucky seven", or that "magic seven". There's no opportunity for you to say, If I've drawn that, put that 356 inside that house top, and I've get the 4 out front." There's no opportunity for me to go, "Well, I know 100 of them, that's 400." Like, it's not part of that repertoire. And so, that's probably the biggest reason we would say, "Good try. Good try in your 'magic seven', 'lucky seven'. Eh, not so much. Let's put it on a ratio table. Do smart partial quotients on the ratio table, and then we can help get smarter, more clever, partial quotients on the ratio table. Kim, it sounded like you want to add something.
I just totally agree. I think it's really problematic when that's the mode...if you will even call it a model...that teachers choose. I love the fact that they're letting their kids think, but the not being able to go over is really troublesome.
It's a major reason that we would say, "Good try, but nope."
Yeah. Excellent. So, you might find it interesting that we would consider the Over strategy. Well, so notice, we've just done the Over strategy for addition, the Over strategy for subtraction, the Over strategy for multiplication, and today we did the Over strategy for division. You might find it interesting that we consider the Over strategy in addition and subtraction super cool but not as essential, not as important, not as sort of mandatory as it is in multiplication and division. Like, you could get away with being fairly efficient adding and subtracting, using relationships you know, if you don't have the Over strategy. You can Give and Take, you can use constant difference, and you kind of don't... It's not as essential. It's still cool. We still want to do it. There are good reasons to do it in addition and subtraction because we build really good place value, but it's not as essential. But ya'll, in multiplication and division, thou shalt work with your students to build the Over strategy because without the Over strategy in multiplication and division, we're not efficient enough. We're going to run into problems like times 99 where we just can't be efficient enough without, and we're so efficient with the Over strategy with multiplication and we're so efficient with division. I can give you a problem. Like, one of our favorites is 1,188 divided by 12. 1,188 divided by 12. Pause the podcast, if you want to think about that for a second. Think about those numbers, those relationships. So, stinking efficient with the Over strategy. If I can just think about 1200 divided by 12 is 100. And now, I just have 1,188. 1,188 is just 12 below that. Bam. So, it's 99. 1,188 divided by 12 is just 99. It's so, so efficient to have that Over strategy with division that it's essential in multiplication and division. So, as much as we want you to play with it in addition and subtraction, please consider, please be invited, please join in with the fact that we need the Over strategy for students to be efficient enough to think multiplicatively enough in multiplication and division, so that we don't have to have the algorithm and that kids are really reasoning and building their brains to be more dense. And it just works so well. It's just... It's really quite essential.
Yep. So, thanks for tuning in for the last few weeks as we share about one of our favorite strategies. Maybe it'll become one of your favorite strategies too. And also, we want to make sure that you download some resources that we are hoping that you get a lot of use out of. The first one is our big ebooklet. And it is from... Gosh, we keep hearing all over the place about how essential it is, how important it is, how useful it is. You can find that at mathisfigureoutable.com/big. And then, another resource that you'll want to grab is the over strategies. And you can find that at mathisfigureoutable.com/over.
Yeah, Kim. I was just talking with some of the leaders in our leader support group, and they are loving both of these resources to use with their teachers to really help teachers parse everything out. So, especially if you're a leader, make sure that you check that out, but it's definitely for teachers as well. Alright, ya'll, thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!