Ep 206: Fraction Multiplication Part 1

Pam  00:00

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  00:09

And I'm Kim Montague, a reasoner, who now knows how to share her thinking with others. At Math is Figure-Out-Able. We are on a mission to improve math teaching.

Pam  00:18

We know that algorithms are amazing historic achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  00:33

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:40

We invite you to join us to make math more Figure-Out-Able. (unclear)

Kim  00:45

Hi, Pam.

Pam  00:46

I'm getting that down. I think. Finally. The new intro is finally kind of.

Kim  00:49

It's still new. It's been a month and half. 

Pam  00:52

I mean, just a few. It's just a few episodes. Fine. Hey, Kim.

Kim  00:58

Hi, so we're excited. We're just coming off a challenge which was super fun, right? It's (unclear).

Pam  01:05

Super energizing. Yep.

Kim  01:06

But it's good. And there's always so much learning, which is cool. (unclear).

Pam  01:10

So, many people from around the world (unclear).

Kim  01:12

Yeah, and lots of first time people, which was so fun. 

Pam  01:15

Yeah. 

Kim  01:16

And in it, of course, we did some Problem Strings, which is...

Pam  01:19

(unclear).

Kim  01:19

Your favorite.

Pam  01:20

My favorite. 

Kim  01:22

So, I wanted to share that Judith Griscom said... She was a fraction workshop participant, and I'm mentioning her because she was also in the challenge. And she said, "I privately tutor an upper level math student who has been mimicking teacher procedures her entire math life." I mean, many are, right? She said, "I've started to do little things like Problem Talks with her, where we look at one or two particular problems, stew in them for a bit, and talk about efficient ways to approach each one. It's been eye opening for sure." So, I bring this up because we talked a little bit about Problem Strings and Problem Talks in the challenge. But then she said, "Kind of related, but I finally listened to the MathStratChat for this week on the podcast, and it was 144 minus 36." I don't know if it was this week. But anyway she said, "I know, I know. It wasn't fractions." But she wanted to share her thoughts with us. And she's says, "For first time I saw this as a multiplicative problem." Meaning, she saw it as 12 times 12 minus 4 times 12.

Pam  02:23

Oh, mmhm.

Kim  02:24

And she said, "It's taken me hearing this on the podcast probably three or four times to finally have it be something I see on my own, and it just reinforces the idea that you can't give up on kids if they don't get it the first time. Even as a math person, it took me a few times to get it to click." And so, I love that for all the reasons.

Pam  02:43

Nice, yeah. 

Kim  02:44

But you know what? I love that she said, "It took me a few times. Which, like that's kind of why we do series of problems in MathStratChat, right? Like, nobody... If anyone thinks, "Hey, I'm going to see something one time, and I'm going to like get it." "It." I'm air quoting "get it". That put some pressure on you that, you know, there's something wrong with you if you don't see what people are seeing the first time. 

Pam  03:08

Totally, yeah. 

Kim  03:09

And, you know, just it supports what we love about Problem Strings. And she mentioned the podcast, so.

Pam  03:15

Well, it supports what we know about learning. Let's be clear. Like, maybe you can rote memorize something and be able to retrieve it again. But to actually move something down deep, that takes your brain traveling that path a few times. And then not just having it be like one solitary lonely path, but with connections. 

Kim  03:35

Yeah.

Pam  03:35

It's like it's more of a map of the terrain, and not just like this one path. 

Kim  03:40

Yeah.

Pam  03:41

Owning something, isn't... I don't know if it's ever a one time thing. So, yeah. Really learning Absolutely. Way to go. Thank you for sharing that. That's (unclear).  Yeah,

Kim  03:50

Yeah, and it was kind of fun because you actually did a Problem String a little bit towards this in the challenge as well. And it's like we like doing a series of problems. Okay, so today we're going to kick off a short series that people have been asking for. And we do love getting suggestions from people, so you can always email me if you are dying to hear something on the podcast. That's kim@mathisfigureoutable.com.

Pam  03:51

totally email Kim. Not me.

Kim  03:57

It won't get anywhere.

Pam  04:01

I mean, I'll probably read it. It's just that I'm so terrible answering. That's the problem. Yeah, all the things, all the things. I try. I try. I try.

Kim  04:27

So, today, we're going to chat about fraction multiplication. 

Pam  04:32

(unclear). 

Both Pam and Kim  04:32

Yeah.

Kim  04:33

And you actually had some fun with that recently.  Okay. Yeah. 

Pam  04:35

Yeah, totally. So, one of the things that we do for our online community called Journey is that we video Problem Strings and classrooms with... I was going to say "real kids", and then I know you were going to laugh at me. Not fake kids. Not fake kids. Real kids. In real classrooms with real teachers. And then we, you and I, voiceover them, so the teachers can hear our reflection about what's going on and everything. So, we recently filmed in an eighth grade class. Shout out to Wallace Middle School and Ms. Castro. Had a fantastic time there. What a great group of kids. So, it was eighth grade. I had a blast. And in one of the class periods, I did this particular Problem String. So, Kim, what I'm going to do today, maybe a little differently than we've done on the podcast is I'm going to say a problem and instead of asking you how you would solve it yet. I do want you to think about it. So, I want you to kind of get in your head how you would think about it. Then I'm going to kind of tell you the experience that I had in the class, and then we'll come back and ask you how you think about it. Is that okay? Okay, cool. So, in this particular class, I had done... When we film, I usually go in the day before. You taught me this actually. I go in the day before, so I can kind of get to know the kids a little bit. They get to know me a little bit. We put their names on name tents, so I can pronounce them correctly. Because I'm sure we've said this on the podcast. I think names are important. We think it's important for kids to hear their name and, you know like, be seen. Anyway, so I had been in the day before. This group is great group. They were a little energetic I'll say. Yeah, and it's right before lunch, so. And they lunch late, so they were like hungry. It was funny because a couple of them were coming in like stuffing snacks. The teacher was like, when the bell rings, you got to put those away. And so they were like getting them in. It was great.

Kim  06:15

Eighth graders.

Pam  06:16

You can see that this great rapport with her because they're smiling at her with their cheek cheeks all out, you know. It was great. Okay, so I started the Problem String, and I just wrote on the board one-fourth times one-fifth. But I said, "What is one-fourth of one-fifth?" And then I kind of paused a little bit, and I said, "Alright, you guys..." I kind of... I looked to see if there would be like this instant, "Oh, yeah." And no. There was not an instant. There was kind of like stopped look on some of the kids faces. On some of the kids faces it was a little bit more of a panicked look. 

Kim  06:52

Yeah. 

Pam  06:52

Now, let's be clear. This is eighth grade, right, so this is not their content. This is not something they do. It's not something they were responsible for this year. It's probably something they haven't necessarily seen very much for a while. And a couple of the kids kind of looked at each other, and I thought, "Okay, we're going to need a minute on this." And so I said, "Alright, you know like, think about it. Turn to the person next to you. What are you thinking about one-fourth times one-fifth or one-fourth of one-fifth?" So, I walked out into the group, and I'm listening to a few students. A couple of kids were talking, and based on what they were talking about, I kind of drew a rectangle on the paper, and I said, "So, if this is a candy bar, what are you talking about?" And they sort of talked about what a fifth of the candy bar would look like. And as I was talking to this group. There's two partnerships were kind of getting in on it. The coach at the school, he was also in the classroom. And everybody was talking, right, because I said, "Turn and talk to your partner." And he kind of laughs loud enough. And I look up over at him, and he's like, "You have got to hear this." I said like, "What's going on?" So, I kind of walked over there, and he goes, "This is like proving everything you've been saying." And I said, "Say more about that." And he goes, "These group of kids over here are arguing between these three algorithms." And I was like, "What do you mean?" And he said, "They're trying to decide if they are supposed to cross multiply and divide or if they're supposed to..." Gal, what were the other two? Cross multiply and divide... Mmm. Now, I can't even remember. But none of them, none of the three. So, what would they be? Invert and multiply. That was one of them. Cross multiply and divide. They called it Keep, Change, Flip. I can't remember the third one. But the three? None of them were multiply across. Which is sort of like the rule if you were going to memorize a rule for multiplying fractions, and so he was cracking up. He's like, "They're over here arguing between these three, and they're pretty sure it's one of these three, and it's not even the right one." And he and I just kind of smiled a little bit. So, then I walked up to the front of the room, and I said, "Hey, this group over here. They were kind of talking about a candy bar. We were talking about sharing. And so, can you guys just start us off?" And so, that group said, "Well, if you had a candy bar..." So, then I drew on the board a big, long rectangle. So, it's a long, flat. Not tall, but like a wide. It's a horizontally long rectangle. 

Kim  06:53

Hey, Pam, should I be drawing?  I'm thinking...

Pam  08:05

Maybe, maybe. We're you thinking back to recent experience where we totally got lost in each other's visuals? Yeah, yeah, yeah. Maybe that's why I was like describing this rectangle so much.

Kim  09:18

I'm making a long rectangle.

Pam  09:20

So, it's just like horizontal, right?  (unclear) fat. Okay. It's not tall.

Kim  09:22

Yep. Okay.

Pam  09:24

Alright, so this is very long. You know like, Hershey bar kind of rectangle. 

Kim  09:28

Yeah, I got you. 

Pam  09:28

And I said, "So, if we want a fourth of a fifth, what is a fifth of the candy bar? What does that look like? One-fifth of the candy bar?" And that wasn't too bad. Though, there were definitely some kids that were still kind of. It was almost shell-shocked was the look. Like, "Please don't ask me about fractions." And I said,  something like, "Alright, one-fifth. What is it?" And somebody said, "Well draw four lines." Which I thought was super interesting because this kid's thought about drawing things into fifths or into into pieces. Because typically a very first answer for a kid might be draw five lines, right? Fifths, draw five lines. But if you draw five lines, then you've cut it into six pieces. If you haven't tried that, listeners, you want to try that real quick. Like, how many lines do you draw in order to separate something into pieces? So, the kid said draw four lines, and somebody else was like, "No, not a fourth, a fifth." And I was like grinning a little bit. And I was like, Okay, so like what is one-fifth mean?" And some kid said, "Well, it's like 1 out of 5." He didn't say equal pieces, but something that meant that. Like, 1 out of 5 if you're sharing with... There's 5 of us sharing. And I was like, "Okay, so let's draw that up here. And I kind of cut it in. Hey, you know, cutting something into 5 pieces is not easy, right? 5 equal pieces? Because you can't like to cut it in half, and then cut again. I don't know. You have to kind of eyeball 5 equal pieces. Anyway, so I kind of eyeballed 5 equal pieces. I said, "Is that good enough?" One kid goes, "Make that line go over a little bit." I'm like, "Okay." (unclear). Yeah, I know. So, we should have shifted a little bit. And then I just drew one-fifth right above one of the pieces. In hindsight, I almost wonder if I wish I would have put one-fifth above all 5 of the pieces. Like one-fifth, one-fifth, one-fifth... But I didn't. I just put one-fifth above the first kind of left-ish piece. And I said, "Alright, does everybody agree this is like one-fifth of the candy bar? If we had a candy bar, and we cut it into 5 equal pieces, each of us have got a piece, and what would that piece be called?" And they were like, "Okay, one-fifth." I'm like, "We got that. But the problem was we want one-fourth of that fifth." Again, kind of the shell-shocked look like, "Ah, don't ask us to do stuff with the fractions." Or for sure, don't ask us to like understand stuff with fractions. Which was equally interesting. So, I said, "Let's go back to that sharing thing. Like here we've got this. It's almost like we've got this piece here, this one-fifth, and that's what we're focused on. But now more people came in, and now we're not. I don't get that whole one-fifth. Now, it's like I got to share it with some more people. Like, how many more people?" And we had a little conversation about that. Like, is it for more people and me? Or is it me and 3 more people? Okay, so alright, there's 4 of us. There's 4 of us that are going to share this piece right here. And I'm like, "What do you want me to do?" And so we cut that fifth vertically into 4 pieces. That was a little easier to do. We took that little one-fifth, and we cut it in half, and then we cut those halves in half. And so, we had 4 equal pieces of that one-fifth. Does that makes sense?

Kim  10:51

Nice. Mmhm. 

Pam  12:20

Alright, Kim, so I kind of want you to tell me what your rectangle looks like right now.

Kim  12:23

Yeah. Okay, so I have the long rectangular, the original Hershey's bar. And I have cut it into those five really not equal pieces. 

Pam  12:31

They should be though, right?

Kim  12:32

It should be. It should be. And I did go ahead and label above each of those chunks, one-fifth, one-fifth, one-fifth, one-fifth, one-fifth. 

Pam  12:39

Okay, okay. 

Kim  12:40

And so, then I kind of zoomed in on the left one-fifth, and I kind of ignored the other first side. But on that left one-fifth, then I drew 3 lines.

Pam  12:53

In which direction?

Kim  12:55

Like columns. So, I drew them vertically.

Pam  12:57

Vertically. So, in that moment, the coach and the teacher in the back of the room. If I remember correctly, both of them looked at me like I was off. And I don't remember who, but I think one of them said something about, "Why didn't you draw those lines horizontally?" We're going to talk about that in a second. So, that was kind of an initial reaction. It was like, Why are you breaking it in kind of the same way I had broken the fifths, it's almost like I was snapping the candy bar into fifths. And here we have this one-fifth, and then I snap that one-fifth kind of the same way into 4 equal pieces, and so now we have those fourths. And then I said, "Okay, guys, so here is a piece." And I shaded in that far left little guy, that little one-fourth of that one-fifth. And I said, "Here is this little piece. This is the one we're talking about. This is one-fourth of one-fifth. What is it called?" Kim, I got to tell you. In that moment, I fully expected everyone in the class to go, "Oh, okay. It's 1/20." Like, I just had this. I was like, "Now, that we're clear what it looks like, and it's right there, then everyone's going to be like, oh, duh. Okay." Nothing. Now, the kid that drew that said draw four lines. That kid was smiling. That kid was like, "Yeah, I got it." Maybe there were a couple of other kids that were like, "Okay." But for the most part, most of them were like, "What?"

Kim  14:17

Yeah.

Pam  14:18

I mean, not in a bad way, right? But like in an honest like, "I don't know. How could we know that?" And so, we kind of went back a little bit and we chatted a little bit about why did we call these one-fifths? Oh, because they were 5 equal pieces. Why did we call these one-fourths? Because they were 4 equal shares of this little piece? And so, what is this? What is this 1 out of? This is an equal piece out of..." And one of the kids said something about, "Well, if you cut all of those one-fifths..." Which is, Kim, why I kind of had wished maybe I'd labeled all of the one-fifths.

Kim  14:50

Mmhm.

Pam  14:51

So, if you cut each of the one-fifths into those 4 equal pieces, if you cut it, so the whole candy bar we've now snapped each of those fifths fits into 4 pieces. If we cut them, then how many of those little tiny, snappy pieces would there be? And then a kid was like, "Oh, I think there would be like a lot of them." "Right. Good. How many?" And they're like. And then the kid literally started adding. "We would have 4, and 4, and 4..." And I'm like, "Okay, you're thinking additively. Got it. I don't know if anybody was counting them all. Though, I wouldn't be surprised if they were counting. So, we could have had counting strategies. We definitely had additive where they were like, "There's 4 pieces in that fifth, and 4 pieces in that fifth." And then we definitely had a few kids that were like, "Well, there's five 1/5s, and each of them has 4 pieces, so there would be 20 total pieces." And I was like, "Oh, okay, so if there's 20 total pieces..." So, then right next to the problem one-fourth times one-fifth. I had written that on the board. I wrote equals. You said there's 20 total pieces. I drew the fraction bar, and I put the 20 underneath. And I said, "So, there's 20 total pieces. How many pieces are we talking about right here?" And they were like, "Oh, 1 of them." And I said, "Okay, so that's one-twentieth. It's 1 out of those 20 pieces. Kim, I swear, most of the kids that class were like, "Yes. Okay. Yes." It's like they were reasoning maybe for the first time or at least for a long time. "Yes. That makes sense. Yes. One-twentieth." 

Kim  16:17

Yeah.

Pam  16:17

Alright.

Kim  16:18

Okay, so, like, there's so much to unpack here. And, you know, I don't even know where you want to go next. But when you said... Oh, man. So, I can feel this pull that you're... And not in a bad way pull. But like you're bringing kids into this. We're thinking big candy bar. Then I'm helping you zoom in on a fifth. Then we're diving deeper into zooming on a fourth of a fifth. And then we're zooming you back out. And that is the reunitizing that, of course, we would expect his kids to be like, "Wait, what?" That shift in thinking is shortcutted... Shortcutted? Is that a word? We short cut that shift in thinking by saying, "Oh, just multiply across," It makes no sense to them because they haven't experienced what you just did with them. And changing what they're focusing on is a necessary part of really understanding where are those 4 pieces within a fifth that then make it be 20 total pieces?

Pam  17:24

Nicely said. Yeah. And so AVR people, I think would call that coordinating units. It's like how many different levels of units? You said reunitizing. So, we sort of have a candy bar. That's a unit. And we cut it into 5 equal pieces, so now we've got 5 units, right? Well, first, we have 5 units. And then we can now zoom in, like you said, on one of those, and that's 1 out of those 5 equal share, so then now 1 of those 5 gets its own name. One-fifth. And now, we zoom in again, and we cut that one-fifth, just that 1 piece, into 4 equal piece, so now we've got 4. But we only drew 3 lines. And we cut it into 4 equal pieces. And so, now we've got 4 equal pieces. And now we have a new name for those. Now, a new unit of each of those is one-fourth. That's a new unit. And that brilliantly said, we have to zoom back out to talk about how many total of those little guys would be if we've cut it all up that same way. And now we've got the number 20 of those little pieces. But then it's only 1 out of those 20, so now those little pieces have a new name. There's 20 of them. There's one-twentieth of that whole candy bar.

Kim  18:33

And can we also acknowledge that we haven't even talked about one thing? That when we say one-fourth of a fifth, kids are used to drawing. Like, if we have a 3 by 5 array, we're used to telling them that means 3 groups of 5. Think about the 3 first. When we say one-fourth of a fifth, we're suggesting that you focus on the fifth first, and then operate on that fifth.

Pam  19:00

Oh, that's very nice. Yeah. Yeah, let me just do that with an example. Like, if we had 3 times 4. We're thinking about 3 groups. And then you can think about of 4 things. Right? 3 groups of those 4 things. So, you're thinking about the 3, and then you can picture in each group those 4 things. 

Kim  19:19

if you're representing on an area model, you draw (unclear). And

Pam  19:25

(unclear).

Kim  19:25

3 down.

Pam  19:26

Yep. 3 rows by 3 columns. Yeah. So, you're totally thinking about the 3 rows first by the 4 columns. Nice. And in this case, we're saying hey, to think about a fourth of a fifth, you got to find that fifth, so that you can think about one-fourth of it. And that's, yeah. And then... ...let me add one more layer of unitizing on that. So, it's one-fourth of one-fifth of the whole unit. Like, it's one-fourth of one-fifth of the candy bar. So, lots of units. Lots of reunitizing. Kim, when you said, "There's so much to unpack here," you know what was screaming in my head? Was the teacher, the honest well-meaning teacher right now, who... Thank you for still listening. Who's like, "Oh, my gosh. You guys are making this so complicated. Why don't you just tell them multiply the numerators, multiply the denominators, and you're done. Why are you making it so hard for kids? You're being mean. Like, this is... And you're wasting your time. And you're causing unnecessary struggle. When they talk about productive struggle, this is exactly the wrong thing that you want to do in your class because now the kids are all..." If you are thinking that, if you have colleagues who think that, I respect the fact that your instinct right now is to go, "Whoa, that is really confusing, and it doesn't need to be. For these kids to get answers, all they have to do really is just multiply across." Correct. If all we want is for kids to just get answers to this problem, all they have to do is multiply straight across. But if we want to build their brains to think and reason proportionally and multiplicatively about fractions, then I'm going to suggest that we cannot just tell them to multiply straight across, or what we'll get is what those groups were doing when I threw out the question. When I threw out this question, and they were honestly like sure that it was one of three rules, and none of them were multiply straight across, that's what we get when all we say is, "Let's just do the easy way. Let's not make kids struggle." Ya'll, we don't ever want kids to unproductively struggle. We don't ever. We're not asking for struggle just for struggle sake. What we're asking for is for kids to honestly grapple with the math, the mathematics, the relationships at hand, making sense of it, so that they can continue to reason and make sense of math. 

Kim  19:46

It's tricky.  Yeah.

Pam  20:13

Okay. No, go ahead. 

Kim  21:04

Well, I was just going to say, you know I think I think many of us acknowledge that when we do what we've traditionally done, and say, "Here are some rules. Memorize those things," there are some kids who can hang on to it a little bit, and then there are some who have not traditionally been able to do that. And if we keep doing that, we keep saying "just memorize these rules," we're automatically discounting certain kids. Were saying, "This is too hard." Or, "You can't hang, and so sorry for you.

Pam  22:31

But we really mean, "You can't rote memorize and mimic." 

Kim  22:33

Yeah, and that just like feels to me like we're automatically saying, "Well, I know some kids are going to have a difficult time with this because they have for years, and years, and years, and generations. But oh, well."

Pam  22:43

But it's so much easier for the other kids that let's just go with that." And I get it. Like, if math is about rote memorizing and mimicking, then yeah. You would want to do that the easiest way possible. We're asking everybody to consider what if it's not? And we're suggesting that it's actually not. Alright, so, Kim. I asked you when I said one-fourth of the fifth that I was going to kind of... Thank you for letting me walk through kind of what I did with these kids. How would you? If I just had said, "Hey, Kim, what's fourth times one-fifth?" What would you have done mentally? 

Kim  23:21

You know, I think I've had enough experience that I don't necessarily picture a candy bar. But I can lean back on that and say I know a fifth. I know that I'm focusing on a fifth first. And I know that if I... I don't know. I guess I just know if I have a fourth of it, then I'm going to have a twentieth. I'm not sure how I know to be honest with you. 

Pam  23:43

Okay, you've just done it so many times. It's just right there.

Both Pam and Kim  23:46

Yeah. 

Pam  23:46

And so, one of the things that we would want for kids, I would submit, is that we would want them to be able to look at one-fourth of a fifth and be able to think about, "Hey, like I can think about a fifth. That's like 5 pieces. One-fourth of those would give me 20 pieces. Bam, it's 1 out of those 20 pieces." 

Kim  24:00

Yeah. 

Pam  24:01

Now, not that they... I don't want that to be like a long, laborious. I don't think it is. But we would want them to lean back on that in any case. Like they can (unclear).

Kim  24:10

You know what we don't want? We don't want them to say multiply the 4 by the 5, and that gets you 20." Like, I actually think about a fifth and a fourth of it.

Pam  24:21

And that gets you the one-twentieth. You actually think about it. 

Kim  24:23

Okay. 

Both Pam and Kim  24:23

Yeah. 

Pam  24:23

Nice, nice. Cool. So, if I were to ask you one... Oh, actually. Let me do one other quick thing before I ask you that. There is a different way that we could visualize the one-fourth times one-fifth. And I think this goes back to kind of what the two teachers were looking at me strangely. I could draw a rectangle. And I'm going to go ahead and draw kind of... I'll go and draw the same Hershey bar. And I could say that the height of the Hershey bar... Let's say I was in the top left hand corner, and I'm an ant on that Hershey bar. And I'm going to walk from the top left hand corner down to the bottom of the Hershey bar. I can say that I was going to walk the whole height of that candy bar, right?  And if I were to start in that same top left hand corner, I could have walked to the right, and walk all the way across the width of that candy bar. Are you good? You're good? Yeah?

Kim  25:16

Yeah.

Pam  25:16

So, let's say I'm that ant. And I could have said, "Hey, instead of walking down..." I'm starting in that top left hand corner. Instead of walking down that candy bar, I'm only going to walk down one-fourth of the height of the candy bar, and I'm going to stop. So, I'm just going to walk that length. Could you like? Like, what would you do to show where the ant had sort of walked? They like kind of stopped there? 

Kim  25:37

Yeah, so I'm starting in the top left corner.

Pam  25:39

Mmhm. 

Kim  25:39

And instead of going the whole length, the whole unit, I stop a fourth of the way down, and I put my ant there. Yep. 

Pam  25:46

And you put your ant there. And so, if I were going to represent that on on the board and for the class, I might actually say like, "How do I know where this is? And the kids might say, "Well, you could, you know, cut it in half, and then cut it that half into half." And so, I've actually drawn kind of where I've cut the whole candy bar horizontally in half, and then I've cut that half in half. And then I've cut the bottom half in half. So, you can kind of see fourths. But the ant really has just crawled down to that sort of spot. And then I could say like, "If that was kind of how I was describing that dimension of the candy bar, that that's kind of one-fourth of the height of that candy bar. And I could kind of label that distance the ant has walked as one-fourth. Does that track? Okay.  If I was that same ant up in the upper left hand corner, and I was walking across the candy bar to the right. This time I only want to walk one-fifth across the candy bar. And so then I would say like, "How can I find that fifth?" And then the kids are like, "Well, divide that into fifths." And so, then that's kind of hard to do, but I would sort of eyeball dividing that into fifths. And I went ahead and kind of drew vertical lines, where the fifths are all the way across. But if that ant was walking, it would only walk like a fifth of the way. And so, then I've kind of labeled that over to that first fifth, where I've kind of labeled that one-firth does that track? Okay, can you tell me what your rectangle looks like now?

Kim  27:02

Yep. I actually didn't put all the cut lines.

Pam  27:10

Okay.

Kim  27:10

But I can picture what you did. So, if the ant started in the top left corner and was walking horizontally, it only went a fifth of the way of that whole unit, the whole length. So, do you want to know what I have left on my paper? Like I kind of have... 

Pam  27:27

Yeah.

Kim  27:28

Go ahead. 

Pam  27:29

Well, no. Tell me what your whole thing looks like.

Kim  27:32

So, I have this giant rectangle.

Pam  27:34

Mmhm.

Kim  27:34

But I am only focused on this tiny corner here, where the ant walked down a fourth of the unit and across a fifth of the unit. So, I'm kind of left with this little. It's not quite a square. But there's this area here on the left corner.

Pam  27:51

That's like a fourth of the height by a fifth of the width. 

Kim  27:55

Mmhm. 

Pam  27:55

And it's this kind of rectangle right there. 

Kim  27:57

Yep. 

Pam  27:57

Cool. Excellent. And I would want students to be able to picture that as well. Which I think is kind of what the teachers were going for when they were like, "Why are you cutting the fifth vertically into fourths?" They thought I was going to cut the whole rectangle horizontally into fourths, and then end up with sort of this double shaded thing. When I say sort of this double shaded thing, the reason I say that is because I think a lot of teachers out there as we've tried to bring meaning to multiplication, and multiplication of fractions specifically is they've taken a multiplication problem like a fourth of the fifth, and they've cut the rectangle into fourths, and they've drawn the three lines to cut into fourths horizontally. And then they've cut it into fifths vertically, and they've drawn the five lines to cut into fifths vertically. And then they've shaded 1 out of the four-fourths, and they've shaded 1 out of the five-fifths. And where they double shaded, then that was like their answer. And then they would sort of read off the answer. They would be like, "Okay, the double shaded part. I've got 1 double shaded here. And so 1." And then they would count the total number of things, and they would say it's 1 out of those 20 total things. So, teachers, I would invite you to consider if you have taught this as a series of things to do. Here's the rectangle. Cut it into these pieces. Cut it into those pieces. Shade this. Shade that. Count the double shaded. Count the total. Then, what you've done is you've turned reasoning about fractions into a procedure to mimic and count. And if you go back to the development of mathematical reasoning graphic, counting strategies is that least sophisticated of all of the kinds of reasonings that we're trying to develop. And so, yeah, kids can do it, but they're actually... We've sort of stuck them down into counting strategies because they just draw the lines, shade, count, count. So, I'm trying to lean away from that procedural counting strategy, and instead really asking, "Hey, if the ant walked, where would this one-fourth be? If the ant walked that way, where would the one-fifth be? Cool." And then what would it look like to talk about the area of one-fourth of the candy bar by one-fifth of the candy bar? And, oh. Well, it would be this little guy here. And then we would also have to talk about how can you tell how many other pieces there are? And it might even make sense to not draw all the cuts because now at this point, you've just got that little rectangle you were focused on in the upper left hand corner, and now they have to visualize how many. Now, I'll go ahead and let them draw them in there. But I want them to really be thinking about that they kind of have a 4 by 5 rectangle if they've cut it into fourths and they cut it into fifths. So, how many total pieces are there? And I want them to think about that in a multiplicative way. Not just counting the total number. Or just doing, "Oh, just multiply those, and you'll have the total number." Does that makes sense?

Kim  30:54

Yeah, and it might be hard for some of our listeners to hear because for the first time ever, they might think, "Oh, my gosh, my kids totally made sense of multiplying fractions by fractions because now they have a visual."

Pam  31:07

"They were using the model. And that's what we're supposed to do. We're supposed to use manipulatives or models. And they've got it. We've done the CRA thing. And so, you know, magic has happened." When in reality, they were actually just like mimicking and counting. 

Kim  31:20

Yeah.

Pam  31:20

So, we'll just invite you to consider, everybody, that maybe a first thing to do would be to think about cutting that candy bar just vertically and really focus on what's actually happening and helping the kids reunitize and giving them some space to have their brains strengthened with all of those different levels of units that they're coordinating, so that their brain can actually strengthen to do that kind of work before you even move on.

Kim  31:46

Hey, so you you did not intend to go into this eighth grade class and spend... I don't know how long you spent.

Pam  31:52

I know! 

Kim  31:52

I'm assuming you didn't spend, you know, a little bit of time on a fourth of a fifth. 

Pam  31:58

that conversation about one-fourth times one-fifth took the entire time we were supposed to do a whole Problem String.  Yeah. Well, so

Kim  32:04

Yeah. 

Pam  32:05

And so, that's about as far as we got. Yeah. So, Kim, we're totally over on this podcast, so I would just ask... We're not going to do it. But this is where I would finish. Could you think of...so, listeners...one-third times one-fourth in both ways. Could you actually think about one-third of a fourth by splitting that candy bar up, and then splitting the piece up? Could you think about it using area but not in a rote memorizing, mimicking, counting way. But in like where's the ant crawling? And so, what does that piece look like? And then out of total how many pieces would that area be of something like one-third times one-fourth? And then could you generalize if that's true, then what is if I gave you a fraction that was 1/a? Now, I'm going to use a positional description  because just so we're all. 1 divided by a. 1 over a. That fraction 1/a times 1/b. So, 1 divided by b or 1 over b. If I have 1 divided by some number times 1 divided by another number, could we generalize what that resulting fraction will be? And I don't know, Kim, do you want to? Do you have some words for how you would generalize with that?

Kim  33:19

Yeah, but I want to say that again because when you said 1/a, I thought you were saying the number 8. So, 1 divided by a times 1 divided by b.

Pam  33:27

Correct. Correct. Yeah, letters, letters. I'm trying to be general. 1 divided by x times 1 divided by y. I'm trying to be general. Yeah.

Kim  33:35

You're want me to say what I think that is?

Pam  33:36

Yeah. What would be? What might? I mean, I could tell you what I'm hoping for. But what would you hope like? Or maybe just tell us. If I say that, 1 divided by a times 1 divided by b, what do you think? Because here's what I think you don't do. I don't think you go, "Mmm, which rule is that?"

Kim  33:53

No.

Pam  33:54

What do you do? 

Kim  33:55

So, in the context of your ant, I'm thinking that the a is one dimension that it's walking, and b is the other dimension, so it's going to be 1 divided by a times b.

Pam  34:09

Okay. I'm sure you've got like more happening in your head to make that happen. But you got 1 divided... Lke, sort of dimension of 1 divided by a by a dimension of 1 divided by b. And the area (unclear). 

Kim  34:24

Yes, so the... 

Pam  34:25

Go ahead.

Kim  34:26

Yeah, the two dimensions tell me how many of the total pieces that are in... It's 1 of the total number of pieces that it could have covered.

Pam  34:40

Ah, okay. So, you're sort of thinking about an a by b rectangle. 

Kim  34:45

Mmhm.

Pam  34:46

And so you're like, "Bam, then then I know it's a times b is the total number of pieces."

Kim  34:50

Yeah.

Pam  34:50

Just like 4 times 5 was the total number and pieces. Or if we had done a third by fourth, then it would have been 3 times 4 would give you the total number of pieces.

Kim  34:58

Yeah. 

Pam  34:58

And then how do you know it's 1 divided by?

Kim  35:01

Because it only went 1 unit fraction each way, so it's one of the total number of pieces.

Pam  35:12

Yeah, when that ant crawled down and it crawled over, it only went. And it was 1 of those total number of pieces. So, a generalization I would want kids to come up with is if I've got a unit fraction... And we're saying a unit fraction, which means the numerator is 1. If I've got a unit fraction times another unit fraction, then I can really think about that's going to create a total number of pieces that is that a times that b. That's a total number of pieces. But I only want one of them. So, 1 out of the a times b pieces. 1  divided by a times b. That's a generalization, actually, I want to make with understanding that kids are really thinking and reasoning about that. And then in our next episode, we'll build on that. So, I think a lot of teachers are like, "Hey, multiplication of fractions. Multiply straight across, Bam, we got it! Boom! The only other thing you do with it is the cross cancel thing." Or whatever you were going to to call that. I don't like that verbiage. But where you sort of are dividing out common factors. Those are usually... When I see a typical fraction instruction in textbooks, that those are usually the two things that happened with fraction multiplication. I'm actually suggesting we need to start with a unit fraction times a unit fraction, and we're going to build from there. You can't wait to hear next week, eh?

Kim  36:22

Yeah. 

Pam  36:23

Alright. Ya'll, 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 Let's keep spreading the word that Math is Figure-Out-Able!