May 03, 2022
Pam Harris
Episode 98

Math is Figure-Out-Able with Pam Harris

Ep 98: If Not the Division Algorithm, Then What?

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Math is Figure-Out-Able with Pam Harris

Ep 98: If Not the Division Algorithm, Then What?

May 03, 2022
Episode 98

Pam Harris

Division is figure-out-able! In this episode Pam and Kim discuss the four main strategies students need to develop to solve any division problem that's reasonable to solve without a calculator.

Talking Points:

- Pam's personal harmful experience from the long division algorithm
- A precursor strategy: Partial Quotients
- Smart Partial Quotients
- Over-Under
- 5 is Half of 10
- Equivalent Ratios
- Combining strategies

Get the BIG download of all the major strategies for addition, subtraction, multiplication, and division here! https://mathisfigureoutable.com/big

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Division is figure-out-able! In this episode Pam and Kim discuss the four main strategies students need to develop to solve any division problem that's reasonable to solve without a calculator.

Talking Points:

- Pam's personal harmful experience from the long division algorithm
- A precursor strategy: Partial Quotients
- Smart Partial Quotients
- Over-Under
- 5 is Half of 10
- Equivalent Ratios
- Combining strategies

Get the BIG download of all the major strategies for addition, subtraction, multiplication, and division here! https://mathisfigureoutable.com/big

**Pam Harris **00:01

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

**Kim Montague **00:08

And I'm Kim.

**Pam Harris **00:09

And we make the case that mathematizing is not about mimicking steps or rote memorizing facts. But y'all it's about thinking and reasoning about creating and using mental relationships. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what are you teaching?

**Kim Montague **00:39

Y'all we're in the middle of a fantastic series that we actually hemmed and hawed about for quite some time.

**Pam Harris **00:44

Little bit.

**Kim Montague **00:45

Because of what we believe about teaching and learning, and we think it's all about gaining experiences. So we cried uncle a little bit, and we're doing the thing that Pam keeps getting asked about. And that is, outlining what are the major strategies for the four operations. So we've tackled addition and subtraction and multiplication, and today, we're going to address division.

**Pam Harris **01:06

Division, everybody's fav, right? Division, whoo. Yeah, we've helped people everywhere construct these relationships in the in person and the online workshops that we give, and in journey our online implementation support system. That's our favorite way to construct these relationships to actually have experiences where we all get a chance to really create neural connections in our brains. But we've decided to put them out there for you, our wonderful listeners. And more than that, more than just talking about them in the podcast, we are also giving you our big download, our resource. Y'all, it is really an ebook, like we have created a free ebook, multiple pages where you can find all of these strategies. In fact, you could download it right now, pause the podcast, download it at mathisFigureOutAble.com/big, b-i-g. It's big. Download that free eBook. And you can like learn right along with this podcast. You can be looking at what we're talking about with these different strategies right along as we dive into this week, as we dive into division. And y'all next week, on May 11, we're gonna have our next You Can Change Math Class Challenge, where we continue to build all these strategies, where you can gain experience with these strategies. And you can register for that now. Registration is open for the You Can Change Math Class Challenge. You can register at mathisFigure-Out-Able.com/change. MathisFigure-Out-Able.com/changes is where you can register for the next You Can Change Math Class Challenge starts on May 11. If you are listening to this episode, at some other time, you can check out what we're currently doing. Because let's be clear, we are always doing something. you can check out what we're currently doing at mathisFigure-Out-Able.com. Okay, let's dive into division. Yeah, let's do it. I have to be honest with you, division is an interesting thing for me, because I have some polar opposite memories. I have a very strong memory of me in fourth grade getting it and feeling so cool. "Getting it" that's like in parentheses, no in quotes, like, meaning I could mimic the steps and get the answer. And there was this odd satisfaction in me being able to do it mainly, and I'm not proud of this, but mainly because nobody else did.

**Kim Montague **03:25

Yeah,

**Pam Harris **03:25

So there was like this pride, there was this thing. It wasn't, like now that I look back on it. I'm like, "Ah, pride in this like, weird, can I mimic the teacher thing." Okay, whatever. But I also have this polar opposite memory. Kim has no idea I'm about to tell this story. Opposite memory of my sister and I have four sisters. So I'm not going to tell you which one. But I have a sister who cried through all of fourth grade. And would throw up-

**Kim Montague **03:51

Ahh.

**Pam Harris **03:51

-when my mom was like, "It's time to go to school," and she would vomit. And my mom would be like, "We're going to school." And it has so much to do with the fact that she couldn't do long division. It was so sad to watch, like so much of her. Now, she might be listening to this right now going, "Pam, you have a story a little bit wrong." But that's my memory of my sister that had so much to do with the fact that there was this stupid thing that she felt like she couldn't do.

**Kim Montague **04:16

Yeah.

**Pam Harris **04:16

And my mom drilled it over and over and over with her. And she finally you know, like, made it through and she's a successful, wonderful adult, but against those odds. And so why don't we like stop that torture? Instead like open the doors of division to everyone? And we can do that with these main division strategies.

**Kim Montague **04:37

Yeah.

**Pam Harris **04:38

So let's kick right off. Okay, so a division strategy that's kind of a precursor that we need. It's not one of our main four, we're going to call four main strategies we need to develop in students. So this is a precursor, just like in addition, we had Partial Sums. And in multiplication, we had Partial Products. And division, we're going to have Partial Quotients. That's going to be a thing that kids are going to try. They're going to start there. So it's kind of this precursor thing. Let's sort of eliminate it. And then let's get into the four that we need to develop, the four that are going to be really important. So firstly, Partial Quotients. Kim, if I give you the problem, 192 divided by 12.

**Kim Montague **05:17

Okay.

**Pam Harris **05:17

What might we see kids do that's kind of less sophisticated, but it's kind of a first shot? What might kids think about?

**Kim Montague **05:25

Okay, so they might think about, "I've got some twelves." Right? So they might think about it, like 120 divided by 12 is 10.

**Pam Harris **05:39

That's accessible.

**Kim Montague **05:40

Okay, yeah. 10. And then they might think, Okay, "So let's see, I've got 72 away from the 192." What else would a kid think? 12? Some more twelves? How about?

**Pam Harris **05:55

I've seen a lot of kids just start marching up by twelves at this point.

**Kim Montague **05:59

Yeah, I was gonna say maybe two 12s. Like maybe two 12s, they would know is 24. And so 24 divided by 12 is two. And so then they're thinking about how far away I am. And so maybe another 24 divided by 12 is two. And so let's see how far away they - 48. What's 168. So maybe another 12 divided by 12 is another one.

**Pam Harris **06:26

And I've lost track a little bit. So are we at sixteen so far?

**Kim Montague **06:30

So 15 so far, right? Because at some point, they're thinking about: Gosh, I'm getting close to 192. So maybe instead of two twelves, they're gonna slow down a little bit. Okay. And then another 12 divided by 12. So I've got a whole bunch of chunks here. So I've got ten 12s, and two 12s, and two 12s, and another 12, and another 12. So that gets me to 182. So that would be sixteen 12s.

**Pam Harris **06:55

Gotcha to 192. And so these Partial Quotients, you sort of used what you knew, and you kind of built up and you had to kind of keep collecting them to see how much you had. And then a really important move is first to nudge kids to ask, "How much more do you need?" And you were kind of asking that naturally. That's not so natural sometimes for kids. They start adding all these little chunks, and then they get kind of lost in like, I don't even know where I am. And then they don't even think about where am I going. So I can sort of see how much I still have to do. And so that's a really important move that as teachers is as we encourage students to use what they know and use the smaller chunks to build up to where they need to go, is to help them go, "Oh, so where are you? And how much more do you need?" Oh, that can be like for some weird reason, some kids don't think about that. And so we're like, "How much more do you need? Oh, yeah." And then they can kind of keep tacking on amounts. That's fantastic. His kids are using what they know. They're solving problems. It's not efficient enough.

**Kim Montague **07:52

No.

**Pam Harris **07:53

But I don't think anywhere out there we've done a good job of helping teachers to know what to do next, unless it's just the algorithm. Like some teachers like Okay, so we'll use, you know, they'll sometimes they'll call it the Lucky seven. They'll draw this sort of thing, and kids just kind of use what they know. But what we haven't done is help teachers know, where do we nudge kids next, so that they can be more efficient? Because all of these little, little chunks of numbers, like Kim was just using, I was getting lost. It was like, "Okay, how many have you done to Oh, then you just add one more to, okay, but when we need two more?" Yeah, like how many of these little guys. So what would be, what would we want to do next with with students? We call that Smart Partial Quotients. Like, let's try to get bigger, fewer chunks, bigger, fewer chunks. So, Kim, you were just trying to think like a kid.

**Kim Montague **08:43

Yeah.

**Pam Harris **08:44

What if you were thinking more like yourself?

**Kim Montague **08:45

Well, it is so funny, because I was trying so hard not to think like myself. You heard it. I love that you said that, "How far away am I?" Because it's a huge question that leads into Smart Partial Products. I think it's kids who are not asking, 'How far away am I?' get caught up in the little tiny pieces, because they don't want to get-

**Pam Harris **09:05

You just said products. But you meant quotients.

**Kim Montague **09:07

Oh, sorry. Quotients. Yeah.

**Pam Harris **09:08

You're good.

**Kim Montague **09:08

So a question that I would ask myself is like, "What do I know about 12s?" So if I've got to get to one, maybe two, what do I know about 12s? And I know and for whatever reason, lots of kids know, twelve 12s. It's, I don't know. Is it because it's the - big - fact? I'm not really sure. But for whatever reason.

**Pam Harris **09:27

Square numbers? A lot of kid's know squares: eight times eight, six times six.

**Kim Montague **09:30

Yeah. So I would say 144 divided by 12 is 12. And then I would have to ask myself, "How far away am I?" And at that point taking stock of where am I in relation to my total?

**Pam Harris **09:43

Because you are at 144.

**Kim Montague **09:44

I'm at 144.

**Pam Harris **09:45

And I need to get up to 192.

**Kim Montague **09:47

Yep, yep, that's a huge, huge question. And so I would say to myself, "Like I'm 48 away. So what do I know about 12 and 48. Ah, I do know that 48 divided by 12 is four." So those two chunks give me sixteen 12s is 192.

**Pam Harris **10:06

So you have the chunk of 144 divided by 12. That's twelve 12s.

**Kim Montague **10:10

Yeah.

**Pam Harris **10:10

And then that leftover 48 divided by 12. That's four more 12s. Yep. So twelve 12s and four 12s is a total of sixteen 12s.

**Kim Montague **10:19

Yep.

**Pam Harris **10:19

Nice, nice chunks. And if a kid doesn't know that 144 divided by 12, that's not something sparks for them, they could have done your start of 120 divided by 12. And then said, "Let's see, now I still have 120. What do I have left to get it to get up to 192? Is that 72?" And then they ask themselves, "Do I know 72 divided by 12?" And if they don't, then they take a chunk they know in that but we try to encourage bigger, fewer chunks as we're trying to help them develop smarter Partial Quotients. Not just any random take any, like little tiny but, but we're always saying, "Oh, like, I see what you did. Check it out. Let's compare to this student who did bigger, fewer. Do you know those chunks?" And we just sort of make a point that we're trying for bigger, fewer chunks.

**Kim Montague **11:08

And sometimes sometimes just having a student who is using Partial Products that are-

**Pam Harris **11:14

Partial Quotients?

**Kim Montague **11:16

I'm gonna do that's a whole episode.

**Pam Harris **11:21

No problem.

**Kim Montague **11:22

Sorry. They're using a whole bunch of little chunks, sometimes just asking them to look back at the pieces that they found, and wonder if they could have collected several of them together. Once the pressure is off, they've got you know, their answer or whatever. Having them look back and say, "Could you have put a few of those together? Like you had 12?" For me, I had 12 divided by 12, and another 12 divided by 12. Supplying that 'How far away am I?' is the nudge they need.

**Pam Harris **11:50

Oh, yeah, it just would have been two twelves. And I knew that. It was just 24. I would have known that was two 12s. Yeah, that would be really nice. And you know, I'm pushing on you when you say 'product' instead 'quotient', but -

**Kim Montague **12:01

But you know why I'm saying it?

**Pam Harris **12:03

I totally do! That's what I was just gonna say. Yeah, but I won't put words in your mouth. Tell everybody why.

**Kim Montague **12:09

Well, because I love to think about division multiplicatively.

**Pam Harris **12:16

Absolutely, yeah. In fact, the first one when we were just doing these Partial Quotients. And you said 120 divided by 12, I had written down '12 times what?'

**Kim Montague **12:25

Yeah.

**Pam Harris **12:26

Like I was recording stuff in a ratio table, because I was thinking about multiplying up and I had to stop myself and do the Partial Quotients.

**Kim Montague **12:34

Yeah.

**Pam Harris **12:35

So one of the things that we are encouraging, teachers, is that as students think about a problem, like 192 divided by 12, that they think of that as 12 times something, that missing factor, is 192. And they could do what Kim just did thinking about was Partial Quotients. But they could also think about it as Partial Products and literally multiply up by 12s until they get to 192. And that missing factor that they found is the same as the quotient of the division problem. Yeah, totally. Okay, so we have kind of that baseline strategy of Partial Quotients. And then we just talked about that we want to get our - so of our four main division strategies, one of them is Smart Partial Quotients. Yeah, bigger, fewer chunks. Let's get at the other three main division strategies.

**Kim Montague **13:20

You want me to tackle some?

**Pam Harris **13:21

Yes. You want to give me one next?

**Kim Montague **13:23

Sure. Yeah. So our next strategy is, I'm just going to tell you the next strategy and give you a problem. Okay.

**Pam Harris **13:30

Yep. Yep.

**Kim Montague **13:30

Alright. So the next strategy is Over/Under. And here's your problem-

**Pam Harris **13:34

Which is one of our favorites, right?

**Kim Montague **13:35

Yeah. Yeah.

**Pam Harris **13:36

Let's be clear, it is your favorite.

**Kim Montague **13:38

I do lpve to Over.

**Pam Harris **13:38

You live, dream, breathe, eat Over.

**Kim Montague **13:43

It's so weird.

**Pam Harris **13:44

And I do now. It took a hot minute to develop it. And now I do. Okay.

**Kim Montague **13:48

Okay, ready? Alright, your problem is 266 divided by 14.

**Pam Harris **13:56

Alright, so let's see how we can think about 266 divided by 14. When I look at this problem, I am literally thinking to myself, "14 times something is 266." And I'm thinking about 14s. And I'm gonna say to myself, "Let's see, I know that," like I literally in my head really quickly thought about ten 14s. And I said, "That's not enough, 140. Because I'm trying to get to 266." But I will say I did think about it.

**Kim Montague **14:22

Okay.

**Pam Harris **14:23

And so then I said, "Well, let me think about 20." Do you know how I found 20? I could have just doubled that 140, right? But I actually thought about two 14s. That's 28. And then I scaled that times 10 to get twenty 14s is 280.

**Kim Montague **14:38

Yeah.

**Pam Harris **14:38

So now I have kind of a pigeonhole. Now I can say, "Well, 140 is too small and 280 is too big, but 266 is really close to 280. So I'm gonna work from that." Now I'm going to ask that all important question: How close am I from 266 to 280? Oh, bam, that's just 14. Sweet, so I don't need twenty 14s, that's 280. I just need 14 less than 280. That's just one 14. So I don't need twenty 14s. I only need nineteen 14s. So 266 divided by 14 is 19.

**Kim Montague **15:12

Very nice. You know what I was just doing while you were talking?

**Pam Harris **15:15

Tell me.

**Kim Montague **15:16

I was just thinking about what the traditional algorithm would look like for that problem.

**Pam Harris **15:22

Oh, not too bad for that one.

**Kim Montague **15:25

Um, I mean, if you know how many 14s are in 126.

**Pam Harris **15:31

Oh, well, okay. The first move wasn't too bad but the second one would be yucky.

**Kim Montague **15:35

Yeah, that's when they go off to the side, and they guess and check a bunch of stuff, right?

**Pam Harris **15:38

Totally a ton of '14 times' problems. And they get half of those wrong. Yeah. You're right. Granted. Nice.

**Kim Montague **15:45

Alright, my turn. Give me one.

**Pam Harris **15:47

Okay. So that was the Over/Under strategy. Yes?

**Kim Montague **15:49

Yep.

**Pam Harris **15:49

So where we look, before you turn, just real quickly, where we go a little too big. And then hack off, like sort of adjust from the two big.

**Kim Montague **15:57

Yeah.

**Pam Harris **15:58

Okay. Cool. Alright, so one of our major strategies is Smart Partial Quotients. One of them is this idea of can I grab a nice friendly multiple, and they go a little over a little under. Alright, so next, our next strategy that we want to exemplify, in fact, before I say it, is anybody noticing how similar the strategies are to the multiplication strategies? Multiplication, we had Smart Partial Products. And then we had Over and Under. And similarly in multiplication, we had 5 is Half of 10. And sure enough, in division, we're going to also have 5 is Half of 10. So Kim, here is your problem to exemplify 5 is Half of 10: 1,224 divided by 24.

**Kim Montague **16:44

Hmm, okay, so I'm thinking, "Want do I want to do?" Oh, I want to do 2,400 divided by 24. Can I just talk about multiplicatively?

**Pam Harris **16:53

Absolutely.

**Kim Montague **16:54

Okay, so I'm gonna go a hundred 24s is 2400, which is way too much.

**Pam Harris **17:01

Way too much, Kim.

**Kim Montague **17:02

But I love the fact that then that gets me to fifty 24s is 1200. So if I know 100 of them, which is pretty nice to find, then I can just half as many to find half the product. So then I have fifty 24s is 1200. I'm looking at how far away I am. And that's just one more 24. So my answer is 51.

**Pam Harris **17:27

Because you're supposed to get to 1224.

**Kim Montague **17:30

Yep.

**Pam Harris **17:30

You're at 1200.

**Kim Montague **17:32

Yep.

**Pam Harris **17:33

One more. So you're saying 1224 divided by 24 is 51.

**Kim Montague **17:37

Yep.

**Pam Harris **17:39

And that would exemplify really nicely this idea of 5 is Half of 12. 5 is- 5 is Half of 10. I looked at my paper where I had a 12 written down. Sorry. 5 is Half of 10. Anybody worried a little today? Five is what? Yeah. So I'm also thinking about the fact that as you were trying to think about the relationship between 24 and sort of 1200, because that was 124 divided by 24. I'm a little curious as you thought, well, 24 times what can get you up to 1200. I'm thinking you might have thought 24 times something is close to 12.

**Kim Montague **18:20

Yeah, yeah.

**Pam Harris **18:23

Yeah, does that feel like a half to you?

**Kim Montague **18:24

Yep. Yep.

**Pam Harris **18:25

So is it conceivable that somebody might be thinking about 24s and think of half of 24? So I just literally wrote down point five. Yep, And half of 24 is 12. Now scale up to the 1200. To get from 12 to 1200, I've multiplied by 100. So then I would have to multiply the point five times 100. And that's the 50. Now I've got fifty 24s, are 1200. And then I still add the one to get 51. I don't know if I did that too fast. But that's a version of 5 is Half of 10, where instead of what Kim did was scale up to one hundred 24s, cut the 100 in half to get fifty 24s. And that's a version of 5 is Half of 10, 50 is half of 100. But you could also instead of scaling up to 100, and then cutting in half, you could also cut it in half, cut the 24 in half to get 12 and then scale times 100. And I, Kim, I just wrote down point five times 100, or times 100 divided by two.

**Kim Montague **19:27

Right.

**Pam Harris **19:27

And those are equivalent, right? I can do those because of the commutative property. I can either find a half times 100 or times 100, then find a half. And those are equivalent. And sometimes it's easier to do one or the other. So we kind of want to develop both and students. Now, we don't emphasize that really young, in young fourth grade students. That's more of a fifth grade, sixth grade strategy when really start thinking about - let me say that again. 5 is Half of 10 as in find the 100, cut that in half to get 50, that yeah, we should be doing that the end of third grade, all through fourth grade for sure and up,. But this idea of an equivalent strategy, being finding half of it and then scaling it up, that comes a little later. We will want to build that a little bit later, that version of 5 is Half a 10.

**Kim Montague **20:12

Alright.

**Pam Harris **20:13

Okay.

**Kim Montague **20:13

Ready for another one?

**Pam Harris **20:14

Yes.

**Kim Montague **20:15

Okay. The final strategy that we're going to talk about today is-

**Pam Harris **20:20

Oh, wait, sorry. I want to interrupt you.

**Kim Montague **20:22

Okay.

**Pam Harris **20:23

Because I just had this flash, before we do the final strategy. Could we like do a combo? Because often when we do division strategies, well in fact, let me be clear, the three strategies, the three main strategies we've talked about so far, Smart Partial Quotients, Over and Under, 5 is Half of 10, are all based on the distributive property. And we're about to switch. That last strategy is going to be based on the associative property. So before we leave the distributive property, let's be clear, as we are sort of thinking multiplicatively about division. Often, it's not so clearly just one of these strategies. More often, it's a mixture, it's a combo. So Kim, I'm gonna give you a combo problem, oh, I'm gonna give you a problem and see how you would solve it. But I'm gonna bet you're gonna combo it. I'm making a prediction. Alright, so Kim, what if I gave you 1152 divided by 24? 1152 divided by 24.

**Kim Montague **21:16

1152 divided by 24?

**Pam Harris **21:20

Hey, as you're thinking about that, can you think while I talk? Kim's thinking. So I just saw a quote the other day that I'm gonna start using more and more that Einstein said something like 'paper is to record', golly now I'm forgetting it, 'paper is to write down the stuff because your brain is needed for thinking'. Oh, 'papers is to write down the stuff you need to remember, your brain is needed to think' or 'use your brain to think'. So that's what Kim's doing. She's writing down the stuff she needs remember, so she doesn't have to hold that in her head. That frees up her brain to think. I will totally find that quote, and will quote it better in the next episode, but something like that. Alright, Kim, could you think while I was talking?

**Kim Montague **21:58

Nice paraphrasing. Okay, yes, I'm good. So I thought about one hundred 24s is 2400. And then I thought about 5 is Half of 10, or 50 is Half of 100. So I went 50 times 24 is 1200. And then I asked myself, "How far away am I?" and-

**Pam Harris **22:18

How far is that 1152 from 1200?

**Kim Montague **22:21

Yep, and I realized that because I play I Have, You Need, I know 52's partner is 48. So 1152, needs 48 to get to 1200. So I asked myself, "What do I know about 24 and 48?" And that's just two 24s. So then I was at fifty 24s is 1200. And I'm going to get rid of two 24s, which is 48. And that's going to leave me with forty-eight 24s to get to 1152.

**Pam Harris **22:54

So we kind of had a couple of 48s flying in that problem. But when you subtract it out those extra two 24s, you got to your target of 1152. And that was 48 of them. Nice, nice. So that's a combo. If I can just spell it out, you used 5 is Half of 10 to get to the fifty 24s. You realized you were too much. 1200 was too much. And so then Over and Under you went under that 1200. Took off those extras to find-

**Kim Montague **23:21

I kind of went way over to start with.

**Pam Harris **23:23

Oh, yeah, true. True that, when you found the 100, you went way over, but that's kind of 5 is Half 10 to get back to the 50. And then it was still too much. So Over/Under you kind of took off the, yeah. So combination of using 5 is Half 10 and Over and Under. We do that all the time, right? All the time, we're using combinations, well in fact, you subtracted two. So that's kind of a Smart Partial Quotient right there. Smart Partial, you did just take off one 24 and then another 24. It was smart because you took off all the two and ask yourself, "Hey, what's the difference?" And I'm playing I Have, You Need, like all these things kind of coming together, little bit of a combo strategy. Okay. Now I'm ready for our last major division strategy.

**Kim Montague **24:04

Yeah. And like you said, this takes advantage of a different property, the associative property. And this one is Equivalent Ratio.

**Pam Harris **24:12

Ratio? Division, Kim, we're talking about division. What? Division. Alright, what's my problem?

**Kim Montague **24:17

Okay, you've got 288 divided by 32.

**Pam Harris **24:22

Okay, so now I want to tell you what I just wrote down. Okay. I just wrote down 288, the division symbol with the line and the two dots, divided by 32. And then I said to myself, "But I'm going to think about that-" So then I wrote equals the equivalent sign, equals to 88. And then I wrote the division bar. So sort of over 32. In other words, it looks like the fraction 288 thirty-seconds or 32s. Yeah. And as soon as that looks like that fraction, bam, I get different urges. Now, all of a sudden when I look at that fraction looking thing or a ratio, I look at the ratio of 288 to 32, I get this urge to find common factors and simplify that ratio. So then I say to myself, "Well, there's obviously some twos in there." So 288 divided by 32, I could think of half of 288 as 144, and half of 32 as 16. I see some more twos in there, now I'm starting to go, "I probably could have pulled a four out of both of those, oh well." But 144 divided by two is 72. And 16 divided by two is eight. So I had 144 divided by 16, that's equivalent to 72 divided by 8, bam. seventy-two divided by 8 is just 9.

**Kim Montague **25:43

Nice.

**Pam Harris **25:44

And I was just able to find equivalent ratios in order to solve a division problem.

**Kim Montague **25:51

Yeah.

**Pam Harris **25:52

And we have the connection between division and fractions popping out. And that really is a really, really important connection that we would want to build as we build division. And as we build fraction sense that there is this quotient connection, that we can think of 'a divided by b' as 'a/bths'. I never say that very well. But as something like 12 divided by three, I can think of as 12 thirds. Or as two divided by five, I can think of as two fifths. And so as I can think about division as a ratio, or that quotient as a ratio, then I can simplify that ratio. And bam, we have our last most important - ah, most important - our last most, maybe most sophisticated division strategy. Yeah.

**Kim Montague **26:39

Very nice.

**Pam Harris **26:40

Hey, and real quick, I know this is going long. This is the longest episode I think we've ever recorded. But division is important. It's interesting to note that often Kim and I will run into a division problem. And we will start solving it using Equivalent Ratio. We'll like instantly sees some common factors. And we just divide those out. And then we'll get to one where it's not, and then we'll turn it into a distributive property. And then we'll sort of use one of the Smart Partial Quotients Over/Under, or 5 is Half of 10 strategies from there. That's kind of interesting.

**Kim Montague **27:09

Yeah.

**Pam Harris **27:09

So Kim, if we went back to that problem that we started off with 192 divided by 12. Can you give us an example of the switchy thing?

**Kim Montague **27:18

Yeah, sure. So 192 divided by 12, I'm going to actually record the way that you just mentioned. I wrote 192 fraction bar 12, or division bar.

**Pam Harris **27:29

192 twelfths.

**Kim Montague **27:31

Yep, I'm going to simplify that to 96 divided by six.

**Pam Harris **27:39

To cut both of them in half.

**Kim Montague **27:40

Yep. And then at that point, instead of cutting them in half again, I'm actually looking at that six. And I'm thinking to myself, I know ten 6s is 60. So I'm left with 36. So that's just six more 6s. So ten 6s and six 6s is sixteen 6s.

**Pam Harris **27:59

Bam, so very nice. Sort of used Equivalent Ratio first. And then you said, "Well, let me look at some chunks of that, that I know."

**Kim Montague **28:06

Partial Products.

**Pam Harris **28:07

And interesting - what?

**Kim Montague **28:09

I said products again. Geeze.

**Pam Harris **28:10

And I didn't even catch it that time. Hey, so when you said 96 divided by six, You know what I was thinking of?

**Kim Montague **28:18

What?

**Pam Harris **28:18

Was how brilliantly, you didn't want to cut them in half. And I'm okay with that. But how brilliantly you could have cut those into thirds. Like 96 divided by three is so brilliantly 32. And six divided by three is two. And now I have 32 divided by two, which is 16. And then we're gonna get 16. Super, super cool. Y'all, it is about choices. It is about empowering students by building these relationships in their brains so that then they can solve any problem that's reasonable to solve without a calculator. If you want to know more about these strategies, remember that you can download our free ebook at mathisFigureOutAble.com/big and you can join in the You Can Change Math Class Challenge where we build all of these strategies. That's happening on May 11, 2022. You can register at mathisFigureOutAble.com/change. If you're listening to this episode some other time, check out what we're currently doing at mathisFigureOutAble.com. And if you want to learn more math and refine your math teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that math is Figure-Out-Able.

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