Pam  00:00

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

Kim  00:08

And I'm Kim. 

Pam  00:10

And you're Kim! 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. Oh my goodness! Noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only our algorithms not particularly helpful or fun in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. Why are you laughing at me?

Kim  00:41

I don't know. I don't know what's wrong with me. We've spent a lot of time together recently.

Pam  00:47

Yeah, that's true. It's been a week. Okay. So, Kim, in today's episode, we thought, we realized that we could maybe have had a... Well, let me tell you what happened. I was in a Twitter conversation, which was super cool. It was fantastic. Somebody asked... Oh, I should have probably looked up who started it. Somebody asked Tad Watanabe and me about some standards for fractions. And they asked Tad about standards in Japan and me about standards in the US. And Tad is quick on the draw. Because before I can answer, he had answered for both of us. Which, thanks, by the way because I didn't want to look it up. I mean, I kind of knew, but I would have checked. So, thanks for that, Tad. Appreciate that. And then, proceeded to have a conversation back and forth about maybe something was missing. There's lots of things we could talk about today. But part of the conversation, I think specifically, Dr. Laurie Jacques was involved and Professor Smudge was involved. Neither of whom I have met in person, although I would love to because they have very respectful and intelligent, thoughtful conversations on Twitter. I (unclear) thoroughly enjoying the conversations that we're having. But along the way, there was some conversations. So, it was about fraction standards, right? There were some conversations about fraction division. And there was a conversation about unit fractions divided by a whole number. And as we went back and forth, someone said something about "Well, but you know like, 1/25 divided by 4? Like, we can't expect students to reason through that." Then I was like, "Yeah, we need to." Like, "We need students to reason through something like 1/25 divided by 4." Because to me, that's like the definition of what fractions are. Like, when you divide fractions, then you have to think about the relationship to the whole and how that. Anyway, and back and forth, we kind of went. And somebody in the conversation said, "I don't think a typical primary pupil would intuitively know this." Okay, real quick. Can you tell they're British? Pretty sure. Pretty sure most of the people in that conversation were British. Not Tad. He's from Japan, now lives in the States. But "I don't think a typical primary pupil would intuitively know this." And Kim and I had a conversation about that.

Kim  03:06

Yeah, yeah. I remember seeing it on Twitter, actually. And, you know, I'm not a super Twitter engager. But I was struck by the idea that kids "couldn't"...I'm air quoting...(unclear). And I kind of wanted to chime in, but you know, I don't really as much. But I wanted to say, "What does that mean?"

Pam  03:26

You would if it was on Facebook, huh?

Kim  03:28

Probably. 

Pam  03:29

(unclear) Facebook or Instagram? Which one? Facebook? Instagram?

Kim  03:32

Facebook, probably more. 

Pam  03:34

Okay.

Kim  03:34

I just have all the words, and I can't. I struggle with the character count. I just really do. I struggle.

Pam  03:43

That's awesome. So, I'll hang on Twitter. You hang on Facebook. 

Kim  03:46

Okay, that sounds good.

Pam  03:46

We make a whole person that way.

Kim  03:48

Well, so I was struck by the idea that there's a thinking that kids can intuit things, and so I wanted to say like, "What does that mean?" And you and I agree that we need to give kids experiences, so that they can.

Pam  04:05

Yeah. And so, like maybe. Maybe, "I don't think a typical primary pupil would intuitively know this" means that "I don't think they would know it without experiences first." 

Kim  04:15

Yeah, yeah. 

Pam  04:16

But we absolutely think kids can use intuition and the budding understanding they're gaining about fraction to reason through what is 1/25 divided by 4. I think they could reason through that. And we got really kind of interested in that. But part of what we realized was, in our series episode 142, 143, and 144, all about fraction division, that we probably should have had a precursor. Like...

Kim  04:45

Yep.

Pam  04:45

...I don't know 141 and a 1/2 or something should have been the episode we're going to do today. So, this is kind of a precursor conversation that has everything to do with building the idea of what a fraction is. Like, how do you name a fraction? And how do you make sense of naming that fraction? So, we thought. What do I do when I want to help develop an idea? I write a Problem String. 

Kim  05:10

Yep.

Pam  05:11

My go to. I write a Problem String, so I understand it better. And then, I think through experiencing that Problem String, I hope that whoever is experiencing it can also learn to think that way. Like, use those relationships to solve problems in that manner. So, ya'll, you're going to need a pencil for this one. If you are driving in the car, all the best with this. Maybe pause and come back to this one. Or feel free to listen to it knowing you're going to want to re-listen to this one with a pencil. Maybe not everybody, but.

Kim  05:43

I'm going to point out that you said pencil because that is the correct.

Pam  05:47

Oh, gosh. What in the world?! 

Kim  05:49

That's the correct tool.

Pam  05:50

I have no idea why I said pencil today because I actually have a pen in my hand.

Kim  05:53

Okay.

Pam  05:54

I don't even know if I have a pencil in my desk. I'm looking. I do have a pencil. Hey, and it's a Ticonderoga because you convinced me that was the best. I do have a mechanical pencil too, but I think it's (unclear).

Kim  06:06

No! No, no, no. 

Pam  06:07

Yeah, see that's how often I use that one. Alright, so I have a pen in my. Whatever. Pick something up to write with. That's funny. It must have been because I was talking to you maybe. Who knows. Okay, so alright here we go. Kim.

Kim  06:21

Mmhmm?

Pam  06:22

I know you're not a big candy bar fan, but we're going to use candy bars anyway. Ya'll, if we go to the snack aisle. I'm getting a candy bar and Kim's... What are you getting? A gummy something? Gummy? Salty?

Kim  06:33

Sour. Well, yeah, if I have a choice of no candy, it's chips and salsa every time. But if I have to candy, then it's going to be a like chewy sour.

Pam  06:40

Someone forces you to go candy?

Kim  06:42

Chewy sour, yeah.

Pam  06:44

Yuck! Yeah, no. Chewy sour. Like, I don't think you could pay me to go. Anyway, so think candy bar if you(unclear).

Kim  06:53

Okay, I get it.

Pam  06:54

Okay, so there's only a half of the candy bar left. You and I are somewhere. There's only a half. You wouldn't eat it. It can't be you and me. My daughter and I. That would work. My daughter and I are hanging around. There's only half a candy bar left for whatever reason, and so she and I are going to split that half of candy bar. So, what is 1/2 of a candy bar divided by 2? In other words, I'm going to say that again. Half of the candy bar, shared among 2 people. And I've written on my paper 1/2, the division symbol that looks like a subtraction sign with the two dots, divided by 2 people. So, 1/2 divided by 2 people. How much candy bar would each of us get?

Kim  07:33

You would each get 1/4 of that candy bar. 

Pam  07:35

How do you know?

Kim  07:38

Half cut in 2 pieces is a fourth. I mean...

Pam  07:44

Okay (unclear)

Kim  07:45

That's kind of hard for me to think about yeah.

Pam  07:46

It's almost hard to say because it's.

Kim  07:47

Yeah.

Pam  07:48

Yeah. So, if I drew kind of a whole candy bar, and then I took half of it, and it went away. It's gone. And I said, "So, you only have this half left." And we cut it into 2. How many pieces would have been in that whole candy bar? 

Kim  08:02

Yeah (unclear).

Pam  08:03

(unclear) There would have been 4. I cut both halves into 2. And I kind of on my paper, I've got this whole candy bar, and I split it in half, and I kind of scratched out half of it. But it's kind of still there, you know like, kind of a ghost image. And so, you only have this half image that's sort of solidly there left. We cut it in half. Well, if I cut thae whole candy bar like that. You know, 4 total pieces, but I only get 1, Abby only gets 1. And so, we get 1 of those 4 total pieces. 1/4. Is that a way to think about that? 

Kim  08:32

Yep, yep. 

Pam  08:33

Okay, next problem. Tomorrow, we're eating a different candy bar, and there's only a third of it left. Man, who keeps eating all the candy bars! So, there's only a third of the candy bar left, and Abby and I are going to split it. So, I've got 1/3 divided by 2 people. A third of a candy bar divided by 2 people. How much would each of us get?

Kim  08:51

You would each get a sixth. 

Pam  08:54

Why?

Kim  08:54

And this time because you talked about it, I sketched a candy bar, and I split it into thirds. And I focused just on one of the thirds, and I cut that third in half. And so, those pieces would be a sixth because you would need six to make the whole candy bar in that size piece.

Pam  09:16

In that size piece. Nice. Nice. 

Kim  09:18

Yeah. 

Pam  09:18

Cool. So, you would have six. So, six pieces. You need 1 of those. So, your answer, your final answer, is one-sixth?

Kim  09:23

Mmhmm.

Pam  09:26

1 of those 6 equal sized pieces. Cool. Nice. Alright, how about one-half of a candy bar. It's a different candy bar now. Let me go back to the first candy bar. A half of a candy bar. We got half of a candy bar. But two of my sons came along because they also like candy bars. Two of my sons came along, and so now there's four of us that are going to share that half of candy bar. So, we got half a candy bar, but there's 4 of us. So, 1/2 divided by 4. What's 1/4 divided by 4.

Kim  09:56

Not very much candy bar. It's going to be an eighth of a candy bar. 

Pam  10:03

How do you know? 

Kim  10:03

So, I actually thought about the first problem. And when you had a half divided by 2, that was a fourth.

Pam  10:10

Mmhmm.

Kim  10:11

But I know that divided by 4 is kind of like divided by 2, and then divided by 2 again. So, that would be an eighth. Because you would need eight of those size pieces to fill the whole candy bar. 

Pam  10:25

Yeah. And so, am I right? You kind of thought about that fourth? If it was half of a candy bar shared by 2 people, everybody got a fourth. And you said, "What's half of that fourth?"

Kim  10:35

Yep. 

Pam  10:36

Yeah. How do you know what half of a fourth is? How do you know half of a fourth is an eighth? 

Kim  10:42

Because you would need twice as many. So, it's half as big, and you would need twice as many to fill the candy bar.

Pam  10:55

Nice. I will never forget the day when I came home one day, and I said to my son Cameron, who is the inspiration for all of this. And I'm always asking him how he's thinking about stuff. And I came home one day, and I said, "Hey, dude, how do you think about equivalent fractions?" Because at that point, in my mathematical career, I only knew equivalent fractions as this formula, where you wrote down the sort of common denominator thing, and it was this goes into that times that, and then you could write an equivalent fraction. It was totally steps that I did to find equivalent fractions. And I said, "What do you know about equivalent fractions?" And he goes, "If you cut the pieces in half, then you have to have twice as many." I will never forget that day. I had to totally think about what he said. But it's exactly what you just said. Like, how do you know that a half of a fourth is an eighth? Well, if you cut those fourths in half, then you have to have twice as many to fill the whole thing. So, you'd have 8 pieces, right? If you had 4 pieces. So, if you had 4 pieces, then I've got one-fourth, you've got one-fourth, fourth, and a fourth. But if you cut those in half, each of those in half, you've got 8 total pieces. You have to have twice as many to have the same whole candy bar, right? Hopefully, I say that right. I can't hear you. I need to turn my mic up.

Kim  12:15

Am I not speaking loud enough?

Pam  12:15

Hello, hello, hello.  I don't know.

Kim  12:17

Sorry.

Pam  12:19

Okay, cool. Next problem. How about... What if I had a fifth of a candy bar. Let's say it was a really big candy bar. I had a fifth of a candy bar, and 4 of us want to share that fifth. So, it's a giant sized candy bar. We've only got a fifth, but it's big. But 4 of us are going to share it. Now, what kind of piece do we get?

Kim  12:39

You get a twentieth. 

Pam  12:41

And how do you know? 

Kim  12:44

So, if you...I'm drawing...if you cut a candy bar into fifths, and you divide 1 of those fifths into 4 pieces, then you would need 4, and 4, and 4, and 4, and 4. So, twenty pieces to fill the whole candy bar. Each of those fifths would need 4 pieces.

Pam  13:04

Each of those fifths had four pieces. And so, if each fifth has 4 pieces, it's almost like you then have to take those 4 pieces times those five 1/5s? Five, 1/5s.

Kim  13:16

Each of those 5 pieces times (unclear).

Pam  13:20

Yeah, each of those five 1/5s...

Kim  13:22

Yes. 

Pam  13:23

...has 4 pieces in them. Yes, each of those five 1/5s has 4 pieces, and so you would have 20 total of those pieces. Ah, therefore they are a twentieth. Cool. And the last problem. What if I had 1/25 of a really, really large candy bar.

Kim  13:38

Okay.

Pam  13:39

I've got the super, like picture this long candy bar, that goes across my whole kitchen. 

Kim  13:43

Okay.

Pam  13:44

And maybe we could say it was like... Have you ever seen those those gutter Sundays? We did this once at a family reunion. It's like you take a rain gutter that's really clean. Though, it didn't feel like it. Anyway, it was a really clean rain gutter, and then we put ice cream in it, and all the toppings, and stuff. And then, we all ate. You've never seen it? It's like a Guinness Book of World Records thing. 

Kim  14:04

I feel like (unclear). 

Pam  14:04

I have a delightful sister-in-law that, yeah, she thought it would be fun. And it was oddly fun. But, anyway. So, picture this really super long thing, right? Super long thing. Can you picture 1/25 of that really, really long candy bar? 

Kim  14:17

Yep. 

Pam  14:17

1/25? 

Kim  14:18

Yep. 

Pam  14:19

Tell me about that 1/25. What does it look like compared to that whole candy bar?

Kim  14:24

It's a section that you can repeat, and there will be 25 of those sections to fill the whole thing.

Pam  14:34

So, I would get one. You would get one. 25 People would get an equal, a fair share. Cool. 

Kim  14:39

It's like a whole classroom. 

Pam  14:40

So, if I'm just looking at that. Hey, so that would be a great way of thinking about it. Like, if I had a class of 25 kids, whatever the piece would be that each kid would get, that would be 1/25.

Kim  14:49

Yep.

Pam  14:49

They're even, equally shared pieces. Cool.

Kim  14:52

Yep.

Pam  14:52

Okay, now take that 1/25 that one kid got. And I'm kind of picturing that 1/25 actually the size of my clipboard right here. Like, it's a really long candy bar, and so when I got my chunk, my 1/25, it's the size of my clipboard. Here it sits. (unclear) 3 of my friends didn't come up. And so, you know, I'm nice. I'm going to share it with them. It's the size of my clipboard. That's a pretty big size thing, right? And so, that 1/25 is going to be shared with 4 people. 1/25 divided by 4. 

Kim  15:22

Yeah.

Pam  15:23

How much of the whole candy bar did each of the 4 of us get, if we took that 1/25 and divided it into 4 equal shares?

Kim  15:31

You would each get one-hundredth.

Pam  15:36

Of?

Kim  15:37

Of the whole great entire. 

Pam  15:40

Of the whole Candy bar

Kim  15:40

Candy bar, whatever.

Pam  15:42

Whatever we're sharing. Yeah. 1/25 of that whole thing. So, it's almost... Well, do you want to keep talking, or do want me to kind of?

Kim  15:49

Well, when you were talking, I was just actually thinking about how, if you're sharing, if you're dividing each of those 1/25 into 4 parts. Because we were talking about class, I was also picturing like you can feed now 4 times as many people. So, it would be like you get one-hundredth because there can be 100 people that can have their portion. 

Pam  16:17

Ah, it's almost like you said, "Hey, everybody in the class, you each got 1/25 of this very large candy bar. But there's 4 classes total, our class and 3 other classes. Hey, everybody, go share your clipboard size piece with somebody in each of the other 3 classes." 

Kim  16:33

Yep.

Pam  16:34

And like if we, "Hey, we could feed all 4 classes with 100 pieces." 

Kim  16:39

Yep.

Pam  16:40

100 equal sized pieces. Cool. So, it's almost like you sort of took that 25, and it's kind of scaled it by 4 a little bit.

Kim  16:49

Mmhmm.

Pam  16:49

Like, when you when you said 1/2 divided by 4, there was kind of this thought about like this 2 times 4, and the answer became 1/8. And when it was 1/5 divided by 4, there's this 5 times 4, and the answer became 1/20. And when it was 1/25 divided by 4, there's this relationship of 25 times 4, and the answer became 1/100. Seems like there's some kind of scaling going on here.

Kim  17:16

Mmhmm. 

Pam  17:17

It's almost like when you cut the section, the fraction into more pieces, it's almost like you can then scale how many more kids can eat. 

Kim  17:29

Yeah.

Pam  17:29

And if more kids are eating from the same thing, their piece got that much smaller by that same scale factor?

Kim  17:37

Yep. 

Pam  17:37

That was really general how I just said that.

Kim  17:39

Well, and I was looking back at the previous problems because I happened to write them down. So, when when you got a fifth divided by 4, you got 1/20. But when... Let's see if I can say this. When you got a fifth of the size of the piece. So, going from 1/5 to 1/25, then your portion went from 1/20 to 1/100. Your portion got (unclear)

Pam  18:10

Yeah. Sorry, I shouldn't have interrupted you. Yeah, because I kind of got excited. if you have done as we asked and get a pen or pencil, and write this down, then you might also have had on your paper, 1/5 divided by 4 equals 1/20. And below that, 1/25 divided by 4 equals 1/100. And then, you can sort of see, "Ooh, when that fifth got cut into 5 chunks to make 1/25, then the 1/20 was also cut into 5 chunks to make 1/100. So, we can not only sort of scale inside the problem, but we can scale between the problems. Nice. Nice relationship. So, I don't think at that point we're done with... Well, let me say one more thing about that Problem String. Ya'll, I think that Problem String is the definition of "fraction". Like it is if I have a fraction like 1/3, and I cut it in half, then I've got to think about how do I name that new piece I've come up with. And I do have to reason about, well, if I've cut a third in half, then how many pieces are in the whole? If I cut all the thirds in half, how many total? Because I'm trying to name what that piece is. But that is the understanding of what a fraction is. That we name a fraction, we name that piece, by its relationship to the whole. And as long as we've shared fairly, what is the relationship of that piece to the whole is the understanding that we need kids to have. Kids need to understand that is the definition of "fraction". Yeah.

Kim  19:48

And I think we can have these experiences earlier than traditional fraction division (unclear).

Pam  19:56

Oh, I'm so glad you brought that up. Yeah, yeah. Because that was the original part of the conversation right? Was where does this standard a fraction divided by a whole number come into the standard somewhere. And fifth grade is way too late for these more simple questions to come up. Now, probably the 1/25 divided by 4, I'm not having that conversation in third grade.

Kim  20:17

Right, right. 

Pam  20:18

But I am having the conversation about 1/2 divided by 2, and 1/3 divided by 2, and even 1/2 divided by 4, and I think 1/5 divided by 4. I'm absolutely having those conversations in third grade because I want to get at what fractions actually mean. Yeah, I think it's super important.

Kim  20:36

But I think it's far less about, "Let me show you how to record this notation and what to do," and more like, "You get kids involved in a thing that has them do this dividing, and then you tag notation afterwards". Like, they're doing some work about actually dividing things, and we just, "What could that be called?"

Pam  21:02

Hey, when you say that, do you mean... I'm asking because this came up in the Twitter conversation. Do you mean that you're going to give kids a bunch of cubes or candy bars and knives or scissors? When you say "and actually do the dividing"?

Kim  21:21

I don't think it has to be an actual object, but I think if you give them a context where they can make sense of sharing items, sharing things, then we use models that represent their thinking. I mean, we were in a third grade class a couple years ago, where kids were really grappling with naming the things and like renaming the things. And I don't think they needed like, you know, an actual brownie in front of them, as long as the representation they could make sense of.

Pam  22:00

Mmhmm. I think we should do a whole episode on that some other time. I'll just drop in, we don't really like, and research supports this, that circles are not fantastic to have kids start to divide themselves because circles are super hard to divide evenly. So, one of the reasons we use candy bars is because a rectangle is much more easy for kids to get some sense of dividing. We do like kids to do things like divide like a strip of paper by folding it and making sense of kind of how they're thinking about. So, we do want kids to have some experiences actually dividing things and creating fractional pieces. But I don't know that we would suggest that, in this case, trying to think about 1/25 divided by 4, I don't think we would give them 100 cubes... 

Kim  22:46

No.

Pam  22:48

...and create 1/25 as 4 cubes each. It's almost like we've done the work for them, in order to hand them that 100 cubes and divide it into 25 sections of 4 cubes each, and then say, "Okay, here's your 1/25 divide it by 4." Then they're looking at. That reunitizing that's happening? You've done the work for them. So, I don't think we would recommend using objects in that way.

Kim  23:13

Yeah.

Pam  23:14

Okay, cool. I'm looking at the time, and I'm aware. Kim, do you think we can do this next string in an episode? I'm wondering if the listeners are like, "Do it! Do it!" I think we can. Let's dive in. 

Kim  23:25

Let's do it. Let's do it.

Pam  23:26

I won't spend as much time explaining. Alright, here we go. Because I kind of did a lot on that last one. Alright (unclear) first problem of the next string 12/25 divided by 4. 

Kim  23:38

Oh, okay. Non-unit fraction. 12/25 divided by 4 is 3/25

Pam  23:47

Okay, and did you just do some Keep, Change, Flip? Did you multiply by the reciprocal.

Kim  23:53

Ew, no.

Pam  23:54

Did you invert and multiply? No. What did you do?

Kim  23:58

So, I know that 12/25 means twelve 1/25s. And so, twelve 1/25s divided by 4, then I'm really just thinking about the 12 things divided by 4, which is 3 of the things. So, 3/25.

Pam  24:12

three 1/25s, and so that's why you answer was 3/25 

Kim  24:18

Yeah. Yep. 

Pam  24:20

And that's, that's interesting. You're saying that you can think about 12/25 as twelve 1/25s, 12 things. 

Kim  24:27

Yes. 

Pam  24:27

So, therefore, you can just think about those 12 things divided. I'm just repeating what you said. Sorry, but. 

Kim  24:31

Yeah, yeah. Well, I mean.

Pam  24:33

Yeah.

Kim  24:34

Well, and I... At the risk of taking a long time. That's a struggle, right? Like, when kids are seeing the fraction 12/25. Like, do they think that's 12 of 1/25s? I think all throughout we've been talking about. We say 12/25, but we also say twelve 1/25s just to kind of like hint, and nudge, and remind that you're talking about 12 of the unit fraction.

Pam  25:05

Yeah, and ya'll if you've only ever thought about fractions in a part/whole sense. Which is the way the education system is set up is to typically introduce fractions as 12 out of 25. And the only thing you can picture is 25 things and 12 of them, and you're thinking about counting those 12 things out of counting 25 things. We would really encourage a much more multiplicative view of 12/25, which is twelve of those 1/25s. I've got this 1/25, and I got twelve of them. That's very much more multiplicative. And so, if you can kind of set aside what the thing is and just think about, "Well, I got 12 of these things." So, 12 whatevers divided by 4 is going to be 3 of those things. "What were the things? Oh, yeah, they were 1/25." 

Kim  25:50

Yep. 

Pam  25:50

Alright, cool. Next problem. How about 16/25 divided by 4? 

Kim  25:55

Sixteen of the 1/25s divided by 4 would be 4 of the 1/25s.  4/25, mmhmm.

Pam  26:05

Sixteen anythings divided by 4 is going to be 4 of those things. Four 1/25s. 

Kim  26:10

Yep.

Pam  26:10

How about 14/25 divided by 4? 

Kim  26:14

14/25 divided by 4. Ah, is smack dab in the middle of the 12/25 and the 16/25, so it's going to be smack down in the middle between 3/25 and 4/25. Which were my answers before. So, I'm going to go 3.5/25. 3.5/25. Hey, I'm curious how you wrote that because I just wrote 3.5/25. I did too. I wrote 3.5 "division line" 25

Pam  26:42

Yeah, same here. (unclear).

Kim  26:43

Which people will not be happy with.

Pam  26:46

Well, they will have to get over that. If you wanted to write that as an equivalent fraction that didn't have a decimal in the fraction. Do you have a thought about that?

Kim  26:57

Yeah.

Pam  26:58

(unclear) is equivalent to?

Kim  27:00

7/50.

Pam  27:03

And you divided? Or you like 1/25s are equivalent to 1/50s and so you're just like, "Oh, I can just think about 7/50." 

Kim  27:12

Mmhmm.

Pam  27:14

Nice. Cool. Next problem. How about 13/25 divided by 4?

Kim  27:19

13/25 divided by 4. So, I could take the 12/25 that I had before divided by 4, and then what I would have left is one more 1/25 divided by 4. Or, I could see that it's nestled between the 12/25 and the 14/25. But either case, I'm going to say it's 3.25/25. 3 and a quarter 1/25s.

Pam  27:52

Nice. But if you had to find an equivalent fraction?

Kim  27:56

That would be 12... Oh, 13/100. 

Pam  28:01

Why did you say 12? 

Kim  28:03

Because I thought about 6.5/50, and I forgot about the half. (unclear).

Pam  28:12

Oh, I thought you were just talking out loud. I thought you were like, "12...and 13."

Kim  28:14

Yeah, but you know what? Here's the thing, though. I absolutely, when I wrote 13/100 I smacked myself on the head. Like, literally went "Ugh". Because 13/25 divided by 4, I could have said, "I have thirteen 1/25s. But I want to think about... Instead of 1/25, I'm going to divide that by 4, and go 1/100, but I still have the 13 of them." Does that make any sense?

Pam  28:42

I think you're going to have to say that again for (unclear)

Kim  28:43

Okay, yeah, yeah. Okay. So, when I landed on 13/100, I looked over at (unclear). 

Pam  28:50

As the answer.

Kim  28:51

Yes, as the answer. I looked at the initial problem, and I went, "Ugh, Kim. 13..." Instead of thinking about it like 13 divided by 4, but stay in 1/25 land, I could have said, "I still have 13, but mess with the 1/25 divided by 4, which is 1/100." So, I was focused on dividing how many of the things that I had. But I also could have thought about scaling the denominator. Like, thinking about the 1/25 divided by 4. In other words, I could have scaled down either how many I had or the size of the pieces.

Pam  29:36

Yeah, nice. And another way of saying that is, you could mess with the numerator or you could mess with the denominator. 

Kim  29:42

Yeah. 

Pam  29:43

And if I could point out, every one of the problems... So, when you did 12/25 divided by 4...which was brilliant...you thought, "Well, I can do 12 anythings divided by 4. I can think about that." You were messing with the numerator, and so you're like, "Well, 12 divided by 4 is 3. Well, what were the things? Oh, they were 1/25s, 3/25." But what if you'd scaled the denominator? What if you thought about 12/25 divided by 4, and ignored the 12 for a hot minute, and said, "What are 1/25s divided by 4?" 1/25s divided by 4 are? Go ahead.

Kim  30:09

Hundredths, and so I would have had 12 of those hundredths, so be 12/100.

Pam  30:13

12/100. Yeah. And you had gotten 3/25. Is 3/5 equivalent to 12/100? 

Kim  30:19

Yeah, yeah. 

Pam  30:20

Sure enough, sure enough. Yeah, nice. Nice. And we could go back and do that with each one of them, right?

Kim  30:25

Yep.

Pam  30:25

So, we want to build that facility in students to be able to think sort of about these problems both ways. Which one makes more sense in the moment? And when you got down to the 13/25, you know, on purpose, I kind of gave you a problem where you're like, "Ugh, do I really want to think about 13 divided by 4? I mean, I can." And it's in between. And that was, you know, we had some other problems for you to mess with to do it. But then, when you saw the answer as 13/100, that's a ping to go, "Oh, man this whole time, I could have been thinking about denominators as well." Super. Ya'll, it's not about one and only one way to do something. It's about dense relationships. And how do we build those dense relationships? It's experiences. Every day that you don't give experiences like these Problem Strings is a day where students will flounder with fractions. So, ya'll three important things. The definition of fractions. We want to have kids always go back to, can I think about this problem based on what I know. How to name the piece. If I think about these pieces, can I reason and then name the piece. So, the definition fraction. Second, use context. You might have noticed that the candy bar was super helpful. Use context as much as you can. And third, teachers, before you worry about getting to cranky stuff where it's fractions divided by fractions and all these cranky numbers. Instead, work with fractions divided by whole numbers and whole numbers divided by fractions. That was in our other episodes we told you about earlier. Work with those first, even though they're not part of your standards because you're going to get at your standards to help students reason about fractions from the get go.

Kim  32:07

Yeah, and I want to go back to the idea that kids can't intuit that. And if we (unclear) these experiences, if they have a good sense of division and a good sense of fractions because we've given them those experiences, they absolutely can intuit it.

Pam  32:23

Absolutely. Ya'll. Thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able... You know what? I am going to. Hang on. Kim, I wanted to say one quick thing, and I forgot about it till just now in the middle of that ending. I got to tell you. After I had this Twitter conversation, I was at church, and I grabbed a bunch of kids. I had a kid that was... I think I had a 9 year old, up to a 14 year old. And there's about 5 or 6 of them. And I said, "Ya'll, can you think about 1/25 divided by 4?" And they kind of looked at me, and I said like, "Do you know something like..." and I did, basically, a brief kind of, "You know, do you know 1/2 divided by 4." I did like two easier problems with them. And at first they said 1/25 said divided by 4 was 1/75. And I said, "Why?" You know, with that good neutral response. And very quickly, in fact, the youngest kid was the one that went, "No, no. It's 1/100. 1/25 divided by 4 is 1/100." And all the rest of them were like "Oh, yeah!" And then, we have this great conversation. Ya'll, kids can they reason through it? Yes. Do they know it right away? Maybe not. But can they reason through it? Absolutely. Alright, ya'll. Thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!