Ep 308: Fraction Multiplication, What?!

Can your students confidently reason about fraction multiplication? In this episode, Pam and Kim facilitate a Problem String that develops two strategies for fraction multiplication and discuss the thoughts that went into designing the string.

Talking Points:

• #TryThisTuesday and #TryThisThursday
• Scaling with fractions
• Building fraction relationships
• Goal of math is to build mathematical reasoning, not just get answers.
• Reasoning is the reason!
• Designing Problem Strings for multiple outcomes

Math is FigureOutAble Blog: https://www.mathisfigureoutable.com/blog

Check out our social media
Twitter: @PWHarris
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC 

Transcript

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Pam  0:00  
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned math-er.

Kim  0:09  
And I'm Kim, 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  0:17  
Y'all, 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  0:30  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships. We invite you to join us to make math more figure-out-able. Hello, hello.

Pam  0:43  
Hey, hey, hey. What's up today? 

Kim  0:44  
Well, I'm super excited because I get to go see classrooms today. We got to record first.

Pam  0:51  
Hey, that is your fave. Alright, so let's get these recorded, so then you can. Yeah, you're on today. That's a lot to do in a day. Okay, cool. So, you might know... Not you, Kim. Everyone else. That...

Kim  1:02  
I might know too. 

Pam  1:03  
Well, I think you do know; there's no might in there, But on social media, on Tuesdays and Thursdays, I put out a thing that I've just kind of made up my own hashtag. Because why not. Hashtag TryThisTuesday or hashtag TryThisThursday. And we just started putting out things that are like Problem Strings, or things you can try in your classroom. Relational Thinking, As Close As It Gets, Factor Puzzles. Like some of our favorite kind of... Some of them are quicker. In fact, most of them are quicker, but the Problem Strings are, you know, definitely longer. And then you could just give those a go. You know, you're like, "What should I do today?" You could just go look up the hashtag TryThisTuesday or TryThisThursday. Or on a Tuesday or Thursday, you could see what I posted, and you could try those things. So, I posted one not too long ago, and I thought that we would do that one on the podcast today. It's kind of become a favorite. And I think... Was this your idea? Kourtney's idea? I don't remember whose idea this was to begin with. I don't know. Somebody cool. Somebody cool came up with this idea. And it's not in Lessons & Activities for Building Powerful Numeracy. So, middle school teachers, if you've got that book, and you're using it to... Even grade four. Grade four. Look, I was in Korea lately. Fourth or fifth grade. Then you could totally add this to your repertoire of fraction Problem Strings. It's pretty cool. Alright. So, ready, Kim? Yep. What is one-fourth of 8?

Kim  2:32  
Oh, 2.

Pam  2:33  
And how do you know?

Kim  2:36  
If you split 8 into 4 chunks, then there's going to be 2 in each one.

Pam  2:42  
Nice. So, fourth of 8 is 2. Cool.

Kim  2:44  
Yeah. 

Pam  2:44  
What about three 1/4s of 8? 

Kim  2:48  
Okay, that's 6. 

Pam  2:49  
Because? 

Kim  2:50  
Because if one-fourth is 2, then you have three of those 2s because three 1/4s, and so that's 6. 

Pam  2:57  
I wrote the second problem directly underneath the first problem. And I scaled from the one-fourth to the three-fourths. I just have this kind of arrow, and I wrote times 3. And then from the 2 to the 6, I wrote times 3. It almost feels kind of ratio table-ish a little bit.

Kim  3:11  
Okay. 

Pam  3:11  
Okay, cool. In fact, I could have written this in a ratio table. That's interesting. I might go back and think about that in a minute. Next problem. What if I were to ask you instead for one-fourth. So, back to one-fourth. One-fourth. But this time of eight-ninths. Wonder how you're thinking about that? 

Kim  3:28  
Mmmm, I'm going to call that two-ninths.

Pam  3:34  
Why?

Kim  3:35  
Because I know that two-ninths, and two-ninths, and two-ninths, and two-ninths is eight-ninths. It's kind of like the four 2s make 8, then the 4 two-ninths make eight-ninths.

Pam  3:51  
So, I actually kind of wrote two-ninths plus two-ninths plus two-ninths plus two-ninths. I'm actually writing it right now. And so, that's like four 2/9s. And you're saying that's eight-ninths. And so... Go ahead. 

Kim  4:04  
It's also related to the first problem because instead of one-fourth of 8, this was eight-ninths. So, like if I... You said something about scaling. If I divided the 8 by 9, then I could divide the 2 by 9. So.

Pam  4:25  
Very nice. Yeah. One other way I might say that back to you is, so if you know one-fourth of 8 things is 2 things, you could kind of say it that way in the first problem. Like, if I said what's a fourth of 8 pizzas, you could say it's 2 pizzas. And in the last problem what's one-fourth of eight 1/9s, you could say well it's going to be 2 of those 1/9s, two-ninths. Couple different ways of thinking about that. Nice, cool. Next problem. What about three-fourths? Or three 1/4s of eight-ninths? I really wish our listeners right now could see all the problems. Y'all, you might pause right now and just write down one-fourth of 8 is 2. Three-fourths of 8 is 6. One-fourth of eight-ninths is two-ninths. Three-fourths of eight-ninths. Sorry to interrupt, Kim.

Kim  5:14  
I could do one-fourth of eight-ninths was two-ninths. So, I could scale times 3 from the one-fourth to three-fourths. Scale times 3 from two-ninths to six-ninths. 

Pam  5:26  
Nice. 

Kim  5:28  
Do you want another way?

Pam  5:31  
Do you have another way? 

Kim  5:31  
Yeah.

Pam  5:34  
You're like, "Of, course."

Kim  5:39  
From the second problem, we have three-fourths of 8 is 6. So, if I don't want 8, I want eight-ninths, then I can divide the 8 by 9 and divide the 6 by 9. So, now I have six-ninths. 

Pam  5:53  
Nice. And another way I could say that is if you know three-fourths of 8 things is 6 things, you can think of about three-fourths of eight 1/9s as 6 of those one-ninths. Whoo! Okay, cool. Next problem. Nicely done. Great connections. I like how you sort of use both of the kind of problems before in different ways to think about that last problem. What about changing tacks completely? What about one-fifth of 10? 

Kim  6:22  
Also 2.

Pam  6:25  
Oh, I wonder if I did that on purpose. I might have to think about that. I don't think I meant it to be the same as the first problem. One-fourth of 8 is 2. One-fifth of 10 is 2. I have to think about that. Alright, so one-fifth of 10 is 2. How about one-fifth of ten-elevenths?

Kim  6:42  
That would be two-elevenths.

Pam  6:45  
Pretty sure?

Kim  6:47  
Yep.

Pam  6:48  
Because?

Kim  6:50  
Because if I cut up ten-elevenths into 5 pieces, that's two-elevenths for each of the 5 chunks. 

Pam  6:57  
Ooh, nice. 

Kim  6:58  
Or. Or I could say the previous problem just said one-fifth of 10 was 2. But now I have 10 somethings, which would give me 2 somethings. And the somethings are elevenths.

Pam  7:11  
Nice. So, one-fifth of ten-elevenths is two-elevenths. Cool. I like it. Next problem. How about two 1/5s of 10?

Kim  7:20  
That's going to be 4. 

Pam  7:25  
Okay.

Kim  7:27  
If one-fifth of 10 is 2, then two 1/5s is going to be worth 4. 

Pam  7:32  
Like, double that. So, I'm doing this doubling, the scaling arrows again from the one-fifth to the two-fifths is times 2. And the 2 to the 4 is times 2. Teachers, the reason that I narrate that is so that you can kind of think about... You know, especially if you're writing these problems as we're doing them. You could say, "Oh, okay. So, when a student says that, Pam's suggesting that you would do the same thing. That you would scale from the one-fifth to the two-fifths times 2. Scale from the 2 to the 4 times 2. Cool. Last problem. I might actually, at this point, say, "Anybody want to guess the next problem?" It is two-fifths of ten-elevenths. I wonder if there's anything up there that could help you. I don't know.

Kim  8:16  
I'm going to go from the second problem in this chunk of 4. And I know that one-fifth of ten-elevenths was two-elevenths. So, I'm going to double that because now you're asking me about two 1/5s of ten-elevenths, and I'm going to get four-elevenths. 

Pam  8:32  
Nice. I like it. That's probably it, right?

Kim  8:37  
The third problem has two-fifths of 10 was 4. So, now if you're asking me about two-fifths of 10 somethings, then I'm at 4 somethings, and the somethings are elevenths, so, four-elevenths.

Pam  8:52  
Nicely done. Kim, are you suggesting that if somebody walking down the street has to find two-fifths of ten-elevenths that they could actually say to themselves, "Well, I know what one-fifth of ten-elevenths is, so I can scale that, double it to get two 1/5s of ten-elevenths." Or they could say, "Well, I know what two-fifths of 10 is." So, then I could just find two-fifths of ten-elevenths, and that'd be four-elevenths. Like, you're suggesting that somebody could actually approach three-fourths of eight-ninths by reasoning. They can actually think about that problem. 

Kim  9:28  
Yeah, yeah. And they have choices.

Pam  9:32  
Yeah, and they have choices. Kind of a couple different ways to go. 

Kim  9:35  
Yeah. Yeah.

Pam  9:35  
I find it noteworthy that... Well, I'm going to guess here. Tell me if you think I... Maybe. Maybe. Maybe I'm crazy. But I wonder if there might be a teacher or two out there that's like, "Yeah, but that worked for these problems. Like."

Kim  9:49  
Sure.

Pam  9:49  
"That's not generalizable. Like, you're not going to be able to do that for crazy numbers. The fact that you could find fifths of 10 or fourths of 8 had everything to do with the fact that you could reason through these problems. How do you... What do we say to that, Kim?

Kim  10:08  
I think the more experience that we give students, the more that they can start looking for relationships that they own. And I think that if we give them varieties of opportunities thinking about... So, one of the things that we did here was scaling.

Pam  10:25  
Mmhm.

Kim  10:26  
So, if they know how to scale, that's a fantastic strategy. We also talked about the two-fifths of 10. So, if they're thinking about a unit fraction, and then they can scale, that they can think about what does a fraction even mean? So, ten-elevenths was the one that we landed on.

Pam  10:43  
Yeah.

Kim  10:43  
Yeah, so I think it's really easy to say that kids can't until we give them the opportunities to try. 

Pam  10:51  
Yeah, and I would add to that... Nicely said. That if we are coming at teaching math from a perspective of my job here is to help students get answers, then I don't blame you for going, "Pam, this is only going to get answers to some problems. It's not going to get answers to like every general, crazy fraction problem." But what if our goal in teaching math is actually to build mathematical reasoning and mathematical reasoners? Then the things you just mentioned are not just strategies or techniques or methods that we sure hope kids memorize. Nope. They are actual relationships that kids are owning in their heads, which means they can now reason through them for enough... I was going to say for enough problems. But not just that. Like, it's not just that they got answers to these problems. And maybe they won't use these same strategies to get answers to other fraction multiplication questions. But in the process of reasoning through these they're building a whole lot more than just can I get an answer to a fraction multiplication question? Y'all, if your goal is to help kids just get answers to fraction multiplication questions, hand them AI. They're done. Like, give them a calculator. They're done. That's not our goal. Our goal is to actually build mathematical reasoning, and so that's another reason why we pulled out both of the strategies that we're pulling out in these strings. Anyway, thanks for letting me just kind of mention that.

Kim  12:22  
Yeah, so we talked about...

Pam  12:23  
Yeah.

Kim  12:24  
We did the particular string the way we did it. There was a chunk of four, a chunk of four. 

Pam  12:27  
Mmhm.

Kim  12:29  
And originally, you and I had talked about... I think when you first wrote this string, it was a chunk of three and a chunk of three. 

Pam  12:36  
Mmhm, mmhm.

Kim  12:37  
And we thought, "Oh, man. This would be even better." Because we talk about these things, right? We talk about strings. And I think maybe in social media, you may even posted a set of three problems, a set of three problems. 

Pam  12:50  
I exactly did. If you go look at the TryThisTuesday that I posted this one, it'll be missing a problem out of each of those sets. 

Kim  12:57  
Yeah. So, we were like, "Hey, let's add to this Problem String, so that both of those nice strategies come out." And so, when we were tinkering with it, it was so funny because we both started talking about a way that you could make a change, but we made a change in a different way. So, can we talk about that for a second? When you  wanted to add to the Problem String, we did it the way that we just did. We did one-fourth of 8, three-fourths of 8. So, you could scale. And then we went back and did one-fourth of eight-ninths. So, you then added the eight-ninths portion into it, and then you scaled up again to three-fourths of eight-ninths. So, that was the 4 chunk. And then in the second 4 chunk, you said, "Now, let's switch that. Let's do one-fifth of 10, one-fifth of ten-elevenths. So, it was kind of you took it to fractions first, but stayed with the one-fifth. Then you moved on to two-fifths of 10, two-fifths of ten-elevenths. 

Pam  13:53  
Then the scaling, mmhm.

Kim  13:55  
Mmhm. And the suggestion that I made was to say let's not have 4 problems in each of those chunks. Let's do three problems, but let's do one-fourth of 8, three-fourths of 8, three-fourths of eight-ninths. And then in the second chunk, let's do one-fifth of 10, one-fifth of ten-elevenths, two-fifth of ten-elevenths. So, a little bit of a variety where you don't have all four problems in each of those. And we decided that both of those would work. And it's just we had different preferences. You liked yours better.

Pam  14:31  
Well, if I can describe yours, yours was a little bit... One of the pair... The pairs. One of the chunks of 3 problems sort of focused on one strategy. 

Kim  14:41  
Mmhm.

Pam  14:41  
And the second pair. Pair. The second chunk of problems focused on the other strategy. And then, Kim, I didn't actually even finish the string.

Kim  14:50  
Yeah. 

Pam  14:51  
Because after what we got down to the two-fifths of ten-elevenths, I forgot that there was a last problem, which is kind of just in a clunker. Now, we don't give you any helpers.

Kim  14:59  
Yeah.

Pam  14:59  
So, for my string where I had the four problems that did both strategies, four problems that did both strategies. Now, when I give you the last problem, two-thirds of six-sevenths, now you could kind of look up and go, "Huh. Let's see, we had two strategies happening in each of these in a different order. What can I use?" In your Problem String, you kind of did a chunk of problems where it was one strategy, then a chunk of problems where it was another strategy, and then you gave the same clunker problem and said, "Which of those two?"

Kim  15:33  
Yeah.

Pam  15:33  
And I think what we're saying... And maybe I'm just repeating, sorry. That you could do both, and probably should. Like, one day do one form, and one day do the other, and give kids... Don't be too predictable. 

Kim  15:46  
Yeah.

Pam  15:48  
Right? Like, too predictable. No. Everybody gets bored when it's too predictable. I'll tell you why I like mine a little bit. Not that I don't like yours, and I do want us to use both. I like mine because there's this sense of... Oh, how do I even describe it? Because using either of them for the fourth problem in the first chunk, and then switching it up, just, I don't know, has the opportunity to kind of keep kids on their toes a little bit. 

Kim  16:14  
I like yours for the first way, the first time, the first experience because they get a higher dose. So, I do like that you switch the order. But they get one opportunity to see it in two strategies in the first set, a second opportunity to see it both strategies in the second set, and then they get to use it in the clunker. And then the next day or two days later, I'd come back with my version, where they still are reminded of the two strategies, but maybe a little bit less often. 

Pam  16:45  
Interesting. Okay. I like it. 

Kim  16:48  
Are we solving the clunker or no? Maybe we should leave it. We should leave it for our listeners, and then they can reply with how they... Which strategy they like better.

Pam  16:57  
Ooh, I love it. yeah.  So, go find this one on social media and reply to that post. Or just reply. Just send out a TryThisTuesday. Use the hashtag TryThisTuesday and say, "Hey, here's how I did two-thirds of six-sevenths. And we'll check it out. We'll see what you were thinking. How are you thinking about two 1/3s of six 1/7s? Like, which of those two strategies that we were kind of throwing out today are you thinking about? Or tell us your own strategy.

Kim  17:25  
Yeah.

Pam  17:25  
Alright, y'all, so check out social media, TryThisTuesday, TryThisThursday for great use right away problems for your class. Can I point other one other thing, Kim? Point out one other thing? I hope today we also modeled that we're all about continuing to improve our craft. Like you said, "Pam, but you've already put this out one way. How dare you put out another?" You know like, "How dare you fix it?" So, just... You didn't say it that way. But just so we're really clear. If we figure out that something we've been doing, we could do even a little better, we then do that. We're not... Sometimes... So, academics can be a little bit like, "Ooh, I've already written it that way. I don't want to look stupid by putting out or inconsistent by, you know, putting out something different. We'd rather put out the better as we go. I've actually just found something that I had published. And I'm like, "Huh, I've changed my tune on that. Like, I've learned more. I have a better way to kind of help everybody understand that." So, we're all about working together to improve our craft. Notice that we're sort of sharing what you're thinking, what I'm thinking, and when we do that, we're really helping each other improve our math content for teaching. That is the field that is not filled out enough. We have a lot of math, but we don't have enough math content for teaching. Alright, 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. Let's keep spreading the word that Math is Figure-Out-Able!