Pam: 0:00

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

Kim: 0:07

And I'm Kim.

Pam: 0:08

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 students to think and reason like mathematicians. Not only are algorithms really not helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim: 0:35

So, in the last two episodes, we've been answering your question from a follower of our Math is Figure-Out-Able, teacher Facebook group, and we've been diving into fraction division. So, today, we're going to hopefully finish and continue to answer Beth's question.

Pam: 0:51

So, I'm going to be honest. It might be our last episode that we do on fraction division for a minute because we have some other things we're going to do. I don't know that we're going to finish fraction division today, but I think we'll give you some really good things to think about and continue your journey, at least about thinking quotitively about fraction division.

Kim: 1:08

Yeah.

Pam: 1:09

So, what about fractions divided by fractions, especially fractions where their denominators are not factors of each other? I say that because in last week's episode, we did a bunch of fraction problems, fraction division problems, things like thirds divided by sixths, and thirds divided by twelfths. And so thirds and sixths. 3 and 6, and 3 and 12, 4 and 12. Those are all related.

Kim: 1:36

Right.

Pam: 1:36

Their factors of each other. And so, there's maybe some different thinking that happens. And so today, we kind of wanted to dive in. What if the fractions are not quite as nice? And let's do it. Are you ready, Kim?

Kim: 1:49

Yep.

Pam: 1:50

Alright. So, your first problem today... I'm getting my pen out? Do you have your pencil out?

Kim: 1:55

I do, of course. (unclear) eraser?

Pam: 1:58

Yeah, I have no eraser because I have a pen. I have a gel pen today. Let's see if it smears on my... Actually, I have a really nice gel pen. I don't think this one smears. Anyway, moving on. What is 3 divided by 2? And I know we said fraction division, but bear with me. What's 3 divided by 2?

Kim: 2:13

One and a 1/2.

Pam: 2:15

Okay, and I was going to write 1.5 until you said 1 and a 1/2, and then, I wrote 1 and the fraction 1/2. But it could be either, right? 1.5, 1 and a 1/2. How do you know?

Kim: 2:25

Because I can think about it like I can fit 2 into 3, one time. Like, there's room for 2. But then, I can only fit 2 into 3, one-half again. So, there's already one and a half 2s in 3. Yeah,

Pam: 2:44

One and a half 2s in 3. So, you're really thinking quotitively, which I kind of asked you to.

Kim: 2:48

Yeah.

Pam: 2:49

How many 2s are in 3? And you're like, "Well, I can fit a whole 2 in 3."

Kim: 2:52

Yep.

Pam: 2:53

And I've only got 1 leftover. And how many 2s are in that 1 leftover? Just a half. Cool. Okay, so you can think about 3 divided by 2 is 1 and a 1/2. Listeners, if you've never thought about something like that, 3 divided by 2, quotitively. If you've never thought about, "How many 2s can I fit into 3." You might want to pause the podcast and think about that maybe a little bit more before we go on. Maybe. Maybe think about how many. Like, if I said, "5 divided by 2," you might think "how many 2s are in 5?" Kind of see if you can kind of make some generalizations about that. Okay, so next problem. What about three-sixths divided by one-sixth. Three-sixth divided by one-sixth? What do you got?

Kim: 3:35

Three.

Pam: 3:36

Because?

Kim: 3:37

Because there's one-sixth, three times in three-sixths.

Pam: 3:43

Cool. And so, I might could draw. In fact, I should have maybe talked about 3 divided by 2. I might have had 3 candy bars. And in this case, I'm going to say, "If I wanted to share that with 2 people..." No. That's not how I want to say that. Nope,

Kim: 3:59

Yeah.

Pam: 3:59

Nope, nope. I want to say if, "I'm going to give..." No. How would I say that? I hadn't thought about a model for this one Kim ahead of time, and that's not... That wasn't smart on my part.

Kim: 4:13

Well, I was just thinking about like of a 1 size candy bar is 3 pieces long. and another size candy bars 2 pieces long. How many of the 2 piece candy bars would fit in the 3 piece candy bar? You know like, 3 bars long. If it's an extra large candy bar.

Pam: 4:33

The pieces would have to be the same size, right?

Kim: 4:35

Yeah.

Pam: 4:36

So, how many 2 piece candy bars can fit in a 3 piece candy bar?

Kim: 4:39

Yeah, (unclear).

Pam: 4:40

That's perfect. Perfect. Nicely done. Cool. Alright, so we've got 3 divided by 2 is 1 and a 1/2. 3/6 divided by 1/6 is 3. What's three-sixths divided by two-sixths?

Kim: 4:53

Oh, you just asked me three-sixths divided by one-sixth, and that was three. But now the pieces are twice as big, so there's going to be half as many of them. So, it's going to be 1 and a 1/2.

Pam: 5:07

Because half of 3 is 1 and a 1/2. So, you kind of used the relationship with the problem before. So, if I say 3/6 divided by 1/6 is 3, then three-sixths divided by something twice as big has to be half as... The answer's half as much?

Kim: 5:26

Yeah.

Pam: 5:26

So, 3/6 divided by 2/6 is 1 and a 1/2?

Kim: 5:28

Mmhmm.

Pam: 5:29

Can I say that in a slightly different way?

Kim: 5:31

Sure.

Pam: 5:31

If 3/6 divided by 1/6. How many 1/6 are in 3/6? Is 3. Then, when I asked 3/6 divided by 2/6, how many 2/6, how many something twice as big are in the same total? That would have to be half as many. 1 and a 1/2.

Kim: 5:47

Yeah. So, I actually was just thinking while you were talking about how I thought about the first problem. Like, how many 2s are in 3?

Pam: 5:56

Mmhmm.

Kim: 5:57

And so, for this problem, you could think about how many two-sixths fit inside three-sixth? So, there's going to be one 2/6 inside that 3/6. But I can only fit half of the two-sixths the second time.

Pam: 6:15

Like, in the leftover ones?

Kim: 6:17

In the leftover. Mmhmm.

Pam: 6:20

So, that's interesting. What I'm hearing you say is. I've given you three problems. To do the third problem, first you thought about the second problem to help you.

Kim: 6:28

Yeah.

Pam: 6:29

But then you also said,"But I can also think about the first problem to help me."

Kim: 6:33

Yep.

Pam: 6:34

So, listeners, this might be. If you've never actually pulled out a pen and pencil while you're listening to the podcast, this one might be one that you want to. Yeah.

Kim: 6:42

(unclear). Pull over on the side of the road.

Pam: 6:44

So, if we can just focus on how you use the first problem. It's almost like I hear you saying...the first problem you thought about 3 divided by 2....you thought about how many 2s are in 3. And the third problem, three-sixths divided by two-sixths, you thought about how many two-sixths are in three-sixths. It's almost like you were thinking about how many two-somethings could fit into three-somethings.

Kim: 7:06

Mmhmm.

Pam: 7:07

And in the first problem, you were like 2 wholes. How many 2 wholes could fit into 3 holes? And then, the third problem you thought, how many two 1/6 could fit into three 1/6? And both times the answer was 1 and a 1/2.

Kim: 7:19

Yep.

Pam: 7:20

1 and a 1/2 of those 2 whatevers could fit into those 3 whatever's. Interesting. Okay, next problem. Three-fifths divided by one-fifth?

Kim: 7:32

Three 1/5s.

Pam: 7:36

Fit into 3/5s.

Kim: 7:38

Fit into three-fifths.

Pam: 7:39

Okay, (unclear).

Kim: 7:40

Did I say that wrong?

Pam: 7:41

No, you said it, right. It's just depending on how you're... Yeah, okay. So, 3/5 divided by 1/5 is 3. Cool. Next problem. Three-fifths divided by two-fifths?

Kim: 7:51

So two-fifths can fit inside of three-fifths, one time, and then half again. So, it's going to be 1 and a 1/2.

Pam: 8:01

Again.

Kim: 8:02

Oh, and that was like the first problem, Pam.

Pam: 8:04

Say more.

Kim: 8:05

You said 3 of something, divided by 2 of something. It has been 1 and a 1/2. So, three 1/5s divided by two 1/5s is still going to be 1 and a 1/2.

Pam: 8:20

Still 1 and a 1/2. So, I wonder... Like, I'm adding a problem late in the game here. So, if I said something like three-sevenths divided by two-sevenths?

Kim: 8:34

Yeah.

Pam: 8:35

You think it still might be?

Kim: 8:36

One and a 1/2.

Pam: 8:37

One and a 1/2?

Kim: 8:39

Yep.

Pam: 8:39

What if it was 3/29 divided by 2/29?

Kim: 8:46

Yep.

Pam: 8:47

Still 1 and a half?

Kim: 8:47

Still 1 and a 1/2. Yep.

Pam: 8:49

Because 2 whatevers can fit 1 and a 1/2 times into 3 whatever's.

Kim: 8:54

Yeah.

Pam: 8:54

Whatever they are. Interesting.

Kim: 8:56

Yep.

Pam: 8:56

Alright, next next problem of the string. 10 divided by 9.

Kim: 9:02

Well, you're asking me to think quotitively, so 9 can fit into 10, one time. And then...

Pam: 9:10

But there's some stuff leftover.

Kim: 9:11

There's some stuff leftover. And 9 can fit into the stuff leftover only 1/9. So, 1 and 1/9.

Pam: 9:22

You're saying 10 divided by 9 is 1 and 1/9?

Kim: 9:24

Yeah.

Pam: 9:26

Say that... Why again?

Kim: 9:29

Because 9 can fit...

Pam: 9:32

How many 9s are in 10? You're thinking about how many 9s are in 10. Sorry to interrupt.

Kim: 9:35

Yeah. That's okay. So, 9 can fit into 10, one full time.

Pam: 9:40

And then there's what leftover?

Kim: 9:41

There's a tenth leftover.

Pam: 9:45

I think there's one leftover, yeah? A tenth of 10 is 1.

Kim: 9:50

Sorry. So, it can fit into 10, one time, and then there's 1 leftover. And then, 9 can only fit into that 1 leftover, 1/9(unclear).

Pam: 10:05

Yeah.

Kim: 10:05

It can't fit the whole time. It's just one-ninth.

Pam: 10:08

We can only get one-ninth. Yeah.

Kim: 10:10

Yeah.

Pam: 10:10

There's only... Yeah. That's crazy kind of reasoning, right? That's interesting. So, I have on my paper written 10"division sign" 9, "equals" 1 and 1/9.

Kim: 10:21

Yes.

Pam: 10:22

I also want to remind us that sometimes we can write 10 divided by 9 as 10 "fraction bar" 9.

Kim: 10:29

Yeah.

Pam: 10:30

10/9.

Kim: 10:31

Yep.

Pam: 10:32

And is 10/9 equivalent to 1 and 1/9? Like, you just reason quotitively.

Kim: 10:36

Yes.

Pam: 10:37

Oh, sure enough, sure enough. So, a couple different connections that we can make. Now, I kind of forced you to think quotitively, and you were like, "Alright, I can think about how many 9s fit into 10." And well done doing that. But we can also sort of... That connection between division and fractions, we can think about 10 divided by 9 as 10/9. Cool. Nice thinking. Alright, ready? Next problem. Ten-twelfths. So, the fraction ten-twelfths divided by nine-twelfths. Ten-twelfths divided by nine-twelfths.

Kim: 11:06

Nine-twelfths fits inside 10/12, one time.

Pam: 11:11

With some stuff leftover.

Kim: 11:12

With some stuff over. And there's a twelfth leftover. And nine-twelfths can fit into one-twelfth, a ninth.

Pam: 11:29

Like, it really doesn't fit, right. (unclear).

Kim: 11:30

It doesn't fit. It can only fit (unclear).

Pam: 11:33

(unclear).

Kim: 11:33

But you know what would have been like a little bit easier for me?

Pam: 11:36

Okay, mmhmm, go ahead.

Kim: 11:37

If I had thought about the problem before.

Pam: 11:40

Mmhmm?

Kim: 11:40

And what we kind of just talked about.

Pam: 11:43

Yeah?

Kim: 11:43

And 10/12. The thing is the twelfths. So, 10 something, 10/12, divided by 9 something, 9/12 is going to be the same. It's an equivalent answer to 10 divided by 9. So, it's still 1 and 1/9.

Pam: 12:01

So, you're saying 10/12 divided by 9/12 is equivalent to 10 divided by 9?

Kim: 12:09

Mmhmm.

Pam: 12:09

Because it's almost like 10 things divided by 9 things?

Kim: 12:13

Yep.

Pam: 12:13

It's 10 divided by 9, which is 1 and 1/9 or 10/9.

Kim: 12:16

Yep.

Pam: 12:18

Fascinating. Okay. Okay. So, similar to what we were doing before. If we can kind of think about 10/9, 10 divided by 9, we can think about 10 anythings divided by 9 anythings. There's still going to be equivalent to 10/9.

Kim: 12:31

Yep.

Pam: 12:31

Cool. Alright. Next problem. How about five-sixths divided by three-fourths? Now, this isn't related to anything we've done, so just random problem. Five-sixths divided by.

Kim: 12:41

Well, I'm glad I've been writing the problems down because.

Pam: 12:45

So, listeners, you might want to actually take the advice we have, (unclear) writing stuff down.

Kim: 12:49

So, I'm comparing the problem you just gave me and the problem prior to that. So, five-sixths is equivalent to ten-twelfths from the previous problem.

Pam: 13:06

Okay.

Kim: 13:06

And three-fourths is equivalent to the nine-twelfths in the previous problem.

Pam: 13:12

Okay.

Kim: 13:13

So, the answer is going to be the same. 1 and 1/9.

Pam: 13:17

Or, 10/9. So, Kim, what did you think, though? Like, when you saw five- sixths divided by three-fourths, is there a first thought that you thought, "Okay, how many three-fourths are in..." Or did you just go, "It's a Problem String. I'm going to see if they're equivalent."

Kim: 13:30

I totally did. I'm not going to lie. I totally did. I was like, "There's got to be something here because that problem is funky."

Pam: 13:35

Well, so when you said that problem is funky, then I actually want to pause that brief second because I think you really quickly were like, "That problem is funky. It's a Problem String. I'm going to look for a pattern. I'm going to look for something."

Kim: 13:48

Yeah.

Pam: 13:48

And I think we could potentially kind of skip over the fact that you actually considered five-sixth divided by three-fourths first. Like, mathematicians consider that. You don't just look to(unclear).

Kim: 13:58

Oh, yeah, yeah. I looked at it. And I thought about it. And I was like, "Ugh."

Pam: 14:02

And then, you said to yourself, "What else do I know?" Oh, great mathematical thinking. And you're like, "Hey, sweet! It's equivalent to the one before, and so the answer is going to be equivalent."

Kim: 14:11

Yeah.

Pam: 14:12

Let's maybe parse out a little bit. Five-sixths divided by three-fourths? You could think about how many three-fourths are in five-sixths. And then, that's like. Wow, like fourths and sixths. And those are not delightfully related.

Kim: 14:28

Right.

Pam: 14:28

(unclear) in a super, you know like, "If I just cut them in half, I get the other one," or whatever. So, then, you kind of said to yourself, "Since they're not, I wonder what else I know." And yeah, the problem before was just sitting there. But in a way, the problem before when you said, "What else do I know?" you kind of found a problem that had the same sort of pieces.

Kim: 14:49

Mmhmm.

Pam: 14:50

Like, often when we add and subtract fractions, we say to ourselves, "Oh, I can't really add sixths and fourths, so let me find equivalent fractions that have the same kind of pieces." Maybe twelfths. And then, "Hey, I could add them together." It's almost like this time you said to yourself, "I wonder if I could find the same number of pieces, so that I could divide?" I don't know that we typically have done that, teachers. I wonder, you know like, when you said,"Five-sixths divided three-fourths... Hey, but they both can turn into twelfths. And if I can turn them both into twelfths, then I could just think about the twelfths."

Kim: 15:20

Yeah. So, I think there's two things that happened too that I'm aware of that I want for our students. Like, in this moment is, One, what I was going to say to you is, I think this is the funkiest problem that we've done in the last couple of episodes.

Pam: 15:37

Yeah.

Kim: 15:38

And I want for kids to think about the fact that there are kind of different... I don't want to say "levels of problems", but like, some are a little funkier than others. And so like, I want them to be a little bit more discerning about, "I have this variety of strategies, and this problem is one that I might have to think about a little bit differently." But also, I know, you just gave me the ten-twelves divided by nine-twelfths, but I looked at it and was like, "Oh, those are equivalent," so I had to know that those were equivalent to be able to make use of it.

Pam: 16:15

Absolutely. Hey, one other thing that we should probably mention is, in this Problem String if I was doing it with students and not Kim, I think that I... Well, I guess I could have done it with you too. But I might have said, "Hey, let's actually talk about five-sixths and three-fourths. How do they relate? If I'm asking how many three-fourths fit into five-sixths, am I going to have a lot of three-fourths and five-sixths? Am I going to have... Is the answer a fraction? Three-fourths doesn't fit? Is three-fourths less than five-sixths? Like, how do those relate?" And, Kim, how would you answer that? How do those relate?

Kim: 16:46

You know, the first thing I'm thinking about...

Pam: 16:49

(unclear). I was just going to say just relative size. But go ahead. What was the first thing you were thinking?

Kim: 16:53

Well, so I know that I know a lot about three-fourths, and so I was actually thinking about what's 3/4 of 6 to see if 5/6 is more than 3/4 or less than 3/4.

Pam: 17:05

[Pam laughs] Okay. Alright. Let's go there in a minute. Can we slow that down a little bit?

Kim: 17:09

[Kim laughs] Sorry.

Pam: 17:10

I want to slow you down just a handful. So, if we were just to compare five-sixths and three-fourths, which one's bigger?

Kim: 17:19

Five-sixths.

Pam: 17:20

Five-sixths.

Kim: 17:20

But I had to know that because I had to think about what I know about three-fourths.

Pam: 17:27

Sure, sure. And in that way, we would want to do some work with students. Five-sixths is just one unit fraction away from six-sixths. It's just one-sixth away from six-sixths. And three-fourths is just one unit fraction, one-fourth away from four-fourths. And so, I can think about what's bigger? One-fourth or one-sixth? And that can help me think about what's closer to 1. Since one-sixth is smaller than one-fourth, five-sixths is closer to 1 than three-fourths. And so, now, I have this comparison. Three-fourths is smaller than five-sixths. So, when I ask "How many three-fourths are in five-sixths?" I should get an answer of "Well, at least one. One and some extra." Right?

Kim: 18:11

Mmhmm.

Pam: 18:11

Did I say that right? Okay, cool.

Kim: 18:12

Yeah.

Pam: 18:13

Then, I could go to what you were talking about that(unclear).

Kim: 18:15

Well, I think I compared as well. I just compared it in a different way than you did. I think it's the same. Like, we were both seeking to compare.

Pam: 18:23

Oh, interesting. What is 3/4 of 6? I think your way is more multiplicative. I'll just say that. Mine was less multiplicative. Mine was more based on the definition of fractions, and I think yours was a bit more multiplicative. So, you said three-fourths of six. Why would you ask three-fourths of six?

Kim: 18:41

Because I know I have five, 1/6s, and I want to know if that's more than three-fourths or less than three-fourths. So, I thought about what is 3/4 of 6, and I know that it's 4 and a 1/2. So, if four and a half of 6 is 3/4 than 5/6 is going to be more than 3/4.

Pam: 19:04

I love it. That's brilliant thinking. Well done. Alright, cool.

Kim: 19:08

Moving on.

Pam: 19:09

So, the upshot of this problem is, we can look at a cranky problem like five-sixths divided by three-fourths, and we can say to ourselves, "I wonder if we can find the same kind of pieces, and then deal with that problem." So, you said, "Yeah, I know ten-twelfths divided by nine-twelfths. Bam! I can think about that like 10 divided by 9." Woah!

Kim: 19:25

Yep.

Pam: 19:25

Cool. Last problem set.

Kim: 19:28

Okay.

Pam: 19:28

9 divided by 4. What's 9 divided by 4?

Kim: 19:33

I just took a deep breath(unclear) a lot of talking. Okay, so I know...

Pam: 19:37

A lot of thinking.

Kim: 19:38

Yeah. So, I'm going to say that that is... I can fit two 4s in 9. Two 4s in 9. So, then I'm going to say 2 and a 1/4.

Pam: 19:54

2 and 1/4. Because two 4s is 8, so you've got 1 leftover.

Kim: 19:59

Yep.

Pam: 19:59

And then, you're thinking about how many 4s are in 1?

Kim: 20:02

Yep.

Pam: 20:03

And you're like, "That's a fourth, one-fourth."

Kim: 20:04

Yep.

Pam: 20:05

Okay, cool. So, 9 divided by 4 is 2 and a 1/4. Could I also remind you that we can think about that as 9/4?

Kim: 20:12

Yep.

Pam: 20:13

So, 9 divided by 4 is 9/4, 2 and a 1/4. Those are all equivalent. We want all that going on.

Kim: 20:18

Yep.

Pam: 20:18

Cool. Next question. What's nine-twelfths divided by four-twelfths.

Kim: 20:25

That's 9. It's similar to 9 divided by 4. So, it's still going to be 2 and a 1/4.

Pam: 20:33

Because?

Kim: 20:35

Because I have 9 somethings divided by 4 somethings, so that's the same as 9 divided by 4. I could fit four 1/12s into nine-twelfths, twice. And a fourth again.

Pam: 20:51

Another fourth leftover. Cool. So, similarly, if I know something about 9 divided by 4, then I can reason about 9 anythings divided by 4 anythings, would also have that equivalent 9/4 solution. Nice. Last problem. Three-fourths divided by one-third. How many one-thirds are in three-fourths?

Kim: 21:14

That's nice.

Pam: 21:17

[Pam laughs].

Kim: 21:17

No, it really is because I'm thinking about how I could...I can scale both those up, and I could use twelfths. I could use a couple of different things, but I'm going to go with twelfths. And if I scale the three-fourths, that would be equivalent to nine-twelfths. And if I scale the one-third to twelfth, that would be four-twelfths. And then, I land back in the same place where it's 9/12 divided by 4/12, or 9 divided by 4, which is 2 and a 1/4.

Pam: 21:45

And so, you're saying the answer to 3/4 divided by 1/3 is 2 and a 1/4 because it's equivalent to the two problems that we did before?

Kim: 21:52

Yeah.

Pam: 21:53

Or 9/4. Either one.

Kim: 21:55

I'm going to go looking for problems like these.

Pam: 21:58

Nice. So, ya'll, fraction division is figure-out-able, and we want to spend time helping students reason about fractions, reason about partitive and quotitive. And we've only been kind of focusing on quotitive division. We will do more episodes later where we focus on partitive division for fractions. But I wanted to kind of think about this Problem String that we did today, and I want to take a slightly different what we just kind of ended with. If I have fractions that are kind of gnarly like five-sixths divided by three-fourths or three-fourths divided by one-third. We said to ourselves, "Mmm, I wonder if I could find equivalent, if I create an equivalent problem where I could think about the same denominator, same kind of pieces. Then, I could kind of reason about it." Like that last one. Three-fourths divided by one-third. Bam! I can think about that as equivalent to nine-twelfths divided by four-twelfths. Oh. Well, that's equivalent to 9 divided by 4 anything. 9 anythings divided by four anythings. And so, we just sort of think about that. If I could just maybe leave you with, what we just did was thinking about finding common denominators in order to divide. But what we didn't do was tell kids that, "Hey, when you divide, find common denominators." It's not a rule that we're telling kids. It's relationships we're developing as we actually reason through the math. And you might be like,"Pam, why do we want to stress kids out? Can't we just tell them what to do and let them do it? It's so much easier." Well, what's your goal? If your goal is to get answers in the easiest way possible, why are you not just using ChatGPT, or Google, or a calculator, or... Like, if your goal is to get the answers the easiest way possible, then go for it. But if your goal is to develop reasoners and mathematicians and help kids think more and more sophisticatedly, then division of fractions is figure-out-able. Y'all thanks 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. Thanks for joining us, and let's keep spreading the word that Math is Figure-Out-Able!