# 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

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  00:00

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

Kim  00:07

And I'm Kim.

Pam  00:08

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.

Kim  00:36

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.

Pam  01:05

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?

Kim  01:46

Yep.

Pam  01:46

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?

Kim  02:22

Yes.

Pam  02:23

Okay. 36 divided by 4. 36 divided by 4.

Kim  02:28

9.

Pam  02:29

I mean, that's, yeah. Like, you probably didn't...

Kim  02:31

Just a fact. Yeah.

Pam  02:32

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?

Kim  03:17

90.

Pam  03:18

Because?

Kim  03:20

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...

Pam  03:33

10 times 9 is? Sorry, didn't mean to interrupt.

Kim  03:36

No, it's ok.

Pam  03:36

So, since 360 is 10 times 36, then 90 is 10 times 9. Cool. Next problem. How about 356 divided by 4?

Kim  03:46

That's just 4 less than 360, so, it's one group of 4 less, so it's going to be 89.

Pam  03:53

So, eighty-nine 4s would give you 356.

Kim  03:56

Yep.

Pam  03:56

Cool. So, right now, on my ratio table, I've got 1 to 4, 9 to 36, 90 to 360, 89 to 356. Yeah?

Kim  04:06

Mmhmm.

Pam  04:06

Alright. Next problem: 400 divided by 4?

Kim  04:10

100.

Pam  04:12

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?

Kim  04:40

So, one hundred 4s was 400. And this is two 4s less than 400. So, it's going to be 98.

Pam  04:49

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?

Kim  04:59

98.

Pam  04:59

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?

Kim  05:14

Yep.

Pam  05:15

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

Kim  05:20

Fifty.

Pam  05:21

How do you know?

Kim  05:22

Because one hundred 4s is 400. So, it's only fifty 4s to make 200.

Pam  05:27

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

Kim  05:32

Yep. And I actually have a divided by 2 on my ratio table.

Pam  05:35

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?

Kim  05:46

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

Pam  05:53

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

Kim  06:10

[Kim laughs]. Oh gosh. 198 divided by 4?

Pam  06:14

Yeah.

Kim  06:14

(unclear). 48.5 or 48 and a 1/2.

Pam  06:16

How do you know?

Kim  06:18

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.

Pam  06:31

Before you go on...

Kim  06:32

Oh, did you ask me 198? Good gravy. I just did 196. And now you're asking me 198?

Pam  06:37

Yeah.

Kim  06:38

So it's 49 and a 1/2.

Pam  06:40

Ah. Because you said 48 and a 1/2.

Kim  06:42

Yeah, I did.

Pam  06:42

But now you're thinking 49 and a 1/2?

Kim  06:44

Yeah. 49 and a 1/2.

Pam  06:45

Because? Keep talking.

Kim  06:47

Because that half of 4, which is 2, I need to add to the 49 and the 196.

Pam  06:56

Gotcha. So, on the ratio table, right now, I have an equivalent ratio of 0.5 to 2.

Kim  07:01

Yep.

Pam  07:02

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?

Kim  07:14

Yep.

Pam  07:14

Which is a way of solving 198 divided by 4 is 49.5.

Kim  07:19

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.

Pam  07:24

Oh.

Kim  07:25

Like, so I wonder if that...

Pam  07:27

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

Kim  07:30

Yeah, mine's getting smaller and smaller.

Pam  07:33

I guess I write slightly smaller than you do or something like that. Okay. Last problem. What is 199 divided by 4?

Kim  07:42

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.

Pam  07:56

Okay.

Kim  07:56

So, I want it to be 49 and 3/4. Or 49.75.

Pam  08:00

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...

Kim  08:20

Oh (unclear).

Pam  08:21

...0.5 to 2, it's the ratio of 0.5 to the number 2.

Kim  08:26

And you know what I wish I would have done?

Pam  08:28

Well, hang on a second. Let me finish my sentence.

Kim  08:30

Okay.

Pam  08:30

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?

Kim  08:47

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  08:59

It's almost like you could have used Over.

Kim  09:01

Yeah.

Pam  09:02

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?

Kim  10:11

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.

Pam  11:21

Kim  15:24

Love it.

Pam  15:25

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.

Kim  16:13

Yeah.

Pam  16:13

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.

Kim  16:49

Make sense to them.

Pam  16:50

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.

Kim  19:01

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.

Pam  19:17

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

Kim  19:21

Yeah.

Pam  19:21