Math is Figure-Out-Able with Pam Harris

Ep 127: The Over Strategy - Division

November 22, 2022 Pam Harris Episode 127
Math is Figure-Out-Able with Pam Harris
Ep 127: The Over Strategy - Division
Show Notes Transcript

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.
Talking Points:

  • A Problem String to develop relationships for the Over strategy for division
  • Great teacher move that we call "the trail off method" to help students generalize and justify the relationships they are using
  • Calling out the relationships vs direct teaching
  • Where does the Over strategy fit in the hierarchy for division?
  • Advantage of using a ratio table to model Partial Products vs 'lucky 7' or 'magic 7'
  • Over strategy is helpful for addition and subtraction, but it is essential for efficient multiplication and division         

Be sure to get the BIG download with all of the major strategies! https://www.mathisfigureoutable.com/big

Plus, get the download specifically for building the Over strategy for all four major operations:
https://www.mathisfigureoutable.com/over

Pam Harris:

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim Montague:

And I'm Kim.

Pam Harris:

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 y'all rotely repeating steps actually keeps our students from being the mathematicians they can be.

Kim Montague:

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 gonna heavy in to talk division.

Pam Harris:

So don't leave if you didn't catch the first three, you know, hanging, 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. There's like, oh, it has the 'Ooh factor'. Yeah, cool. So Kim,

Kim Montague:

Yip. Yes.

Pam Harris:

Let's start today, where I'm going to give you a division problem, I'm gonna 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 (click) really nice Over Okay. 36 divided by four, 36 divided by four. strategy. So you might be thinking about the problem: 396 divided by four, 396 divided by four. 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?

Kim Montague:

Nine.

Pam Harris:

I mean, that's, like, just how they didn't..

Kim Montague:

Just a fact. Yeah. Ninety.

Pam Harris:

Yeah, you didn't probably have to think about it too much. I'm just gonna note that we're kind of thinking Because? Because on my ratio table, I have nine to 36. And I thought about fours. So I'm going to draw, I just drew a horizontal ratio table, and I said, we're thinking about four. So I put one to four. So like, it almost looks like 1/4, on this horizontal ratio table, kind of one to four. And I asked you for that 360 was just 10 times bigger than 36. So my answer is 36 of them. So next to that four, I drew a 36, drew, I wrote a 36. And you said that that was nine. So like to get to 36 you multiplied by nine. So we kind of have two equivalent ratios now. I've got one to four, and nine to 36. And that's a way to represent 36 divided by four is nine. Cool. Next problem: 360 divided by four? gonna be 10 times bigger than nine, which is... Ten times nine is, sorry didn't mean to interrupt.

Kim Montague:

No, it's ok.

Pam Harris:

So since 360 is 10 times 36, then 90 is 10 times nine. Cool. Next problem, how about 356 divided by four?

Kim Montague:

That's just four less than 360. So it's one group of four less, so it's going to be 89.

Pam Harris:

So 89 fours would give you 356.

Kim Montague:

Yep.

Pam Harris:

Cool. So right now, on my ratio table, I've got one to four, nine to 36, 90 to 360, 89 to 356. Yeah. All right. Next

problem:

400 divided by four?

Kim Montague:

One hundred.

Pam Harris:

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 four, 36, it's all the dividends are going on the bottom. And then you said that was 100 fours? Yeah? And then how about, so the hundreds on top. So now that that 100 to 400 is an equivalent ratio in the same table. How about 392 divided by four?

Kim Montague:

So 100 fours was 400. And this is two fours less than 400. So it's going to be 98.

Pam Harris:

So I actually just added two fours or the ratio two to eight in the ratio table, and then I wrote down 392, and you said it would be?

Kim Montague:

Ninety-eight.

Pam Harris:

Ninety-eight, because you took away two groups from the 100. So I'm kind of looking at the 100, and the two and the 98 are next to each other and the 400 and the eight and the 392. So now I have this equivalent ratio of 98 to 392. Yeah?

Kim Montague:

Yep.

Pam Harris:

Cool. Next problem. How about 200 divided by 4?

Kim Montague:

Fifty.

Pam Harris:

How do you know?

Kim Montague:

Because 100 fours is 400. So it's only 50 fours to make 200.

Pam Harris:

Oh nice. So you could go from the ratio of 100 to 400 to 50 to 200? Cool.

Kim Montague:

Yep. And actually have a divided by two in my ratio ratio.

Pam Harris:

A hundred divided by two is 50. Four hundred divided by two is 200. Nice. So 200 divided by four is 50. Next problem. How about 196 divided by 4?

Kim Montague:

Forty-nine. It's going to be one less group of four than the previous problem.

Pam Harris:

How about 198? Oh, this must be a typo. Can you do 198? I'm kidding. (laughs)

Kim Montague:

You're so funny. 48.5 or 48 and a half. Oh gosh, 198 divided by four?

Pam Harris:

Yeah. How do you know?

Kim Montague:

Because now I have to think about instead of, well, I'm still thinking in terms of fours. But half of four is two. So I wrote half on top of my ratio table and two at the bottom.

Pam Harris:

Before you go on... Yeah.

Kim Montague:

Oh, did you ask me 198? Good gravy. I just did 196. So it's 49 and a half. And now you're asking me 198?

Pam Harris:

Ah, because you said 48 and a half.

Kim Montague:

Yeah, I did.

Pam Harris:

But now you're thinking 49 and a half?

Kim Montague:

Yeah. Forty-nine and a half.

Pam Harris:

Because? Keep talking.

Kim Montague:

Because that half of four, which is two, I need to add to the 49 and the 196.

Pam Harris:

Gotcha. So on the ratio table, right, now I have an equivalent ratio of point five to two.

Kim Montague:

Yep, yep.

Pam Harris:

And then and I have it right next to the 49 to 196. And if I just add the 49 plus point five, and then 196 plus two, then I get the ratio of 49.5 to 198?

Kim Montague:

Yep.

Pam Harris:

Which is a way of solving 198 divided by four 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. Oh.

Kim Montague:

Like, so I wonder if that...

Pam Harris:

Well, don't worry, because I'm about to run out of room on mine.

Kim Montague:

Yeah, mine's getting smaller and smaller.

Pam Harris:

I guess I write slightly smaller than you do or something like that. Okay. Last problem, what is 199 divided by four?

Kim Montague:

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 gonna be another fourth of a four.

Pam Harris:

Okay.

Kim Montague:

So I want it to be 49 and three fourths, or 49.75.

Pam Harris:

Forty-nine point seven five. So I do have room on my second layer of my ratio table. So right next to the point five to two, I've written point two five to one. It's so funny that we have this two, like two is two. Like, I'm just aware that since we're audio, and when I say, oh, point five to two, it's the ratio of point five to the number two.

Kim Montague:

And you know, what I wish I would have done?

Pam Harris:

Well, hang on a second. Let me finish my sentence. So I have the ratio of point five to the number two. And I have the ratio of point two five to the number one. And you were saying that you added one to 198. So you added point two five to 49.5. That's what you said. Right? But now you have a better idea?

Kim Montague:

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.

Pam Harris:

It's almost like you could have used Over.

Kim Montague:

Yeah.

Pam Harris:

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?" 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 four to help you with the 199 divided by four. 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 four. Did anybody, you know if you know that there's 50 fours and 200? And I don't know, did anybody use that to help you think about 49? Or to help you think about 199 fours, anybody?" Anyway it's a way of doing that. What were you just gonna say?

Kim Montague:

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 use the 90 to get to 89. And they use the 100 to get to 98. And they use the 50 to get to 49. And now I'm thinking about the 50 to get to 49.75. And I use 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 ratio, in my, 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. Love it.

Pam Harris:

And by calling it out, this is not where all of a So after that, we would expect kids to think about sudden I start direct teaching. This is where I pulled it out of division as equivalent ratios. We would want to bring in, well, students. Like what relationships did you find, 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. This problem is 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 the 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 or early strategy that we want kids to think about, even when they're thinking about single digit facts. And most missed facts is seven times eight. We might say, "Well, do you know eight eights?" Well often students do, often they'll know eight eights, "Well, can you use eight eights to help you think about seven eights?" 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 I didn't mention 5 is Half a 10. And we'd also want them to use 5 seven, 57 divided by seven... I'm dying here. I was going for sevens. If they're thinking about 56 divided by seven. And you're like, "Well, you.." No I'd have to use eight to do the Over. Holy I'm not doing this very well. This is off the cuff y'all. Can you tell? If I'm thinking about 56 divided by eight. And I say to myself, "I don't know how many eights are in 56." I might say, "Well, I know how many eights are in 64. So If I know how many eight turns 64, can I use that to help me think about how many eights 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. is Half a 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 fours are in, that's what we just did with this Problem String. How many fours are in 36? How many fours 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 the 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.

Kim Montague:

Yeah.

Pam Harris:

One thing I want to really point out, that's super

Kim Montague:

Make sense to them. 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, we do that. 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

Pam Harris:

But yeah, like, so if I were doing a problem, like, letting students use quotients that mean something to them. I don't know, help me Kim, 356 divided by four. They would put the 356 in the house top, and they would put the four outside. And then they would say, "Okay, what do you know?" And if a kid goes, "Well, I know four 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 one to four, it was equivalent to the nine 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 get the four 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', and not so much. 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.

Kim Montague:

I just totally agree. I think it's really problematic when that's the model, 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.

Pam Harris:

It's a major reason that we would say, "Good try but nope."

Kim Montague:

Yeah.

Pam Harris:

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 do 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 y'all, 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 times that 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 1188 divided by 12, 1188 divided by 12. Pause the podcast, if you wanna 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 1188, 1188 is just 12 below that. Bam. So it's 99. One thousand, one hundred eighty-eight 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.

Kim Montague:

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 that over strategies. And you can find that at mathisFigureOutAble.com/over.

Pam Harris:

Yeah, Kim, I was just talking with some of the leaders and 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. All right, y'all, 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