Ep 89: Playing With Fraction Multiplication

Pam Harris  00:00

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

Kim Montague  00:08

I'm Kim Montague. 

Pam Harris  00:09

And we make the case that mathematizing is not about mimicking steps, or rote memorizing facts. But it's about thinking and reasoning - about creating and using mental relationships. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what? Alright, Kim, we were talking and you were telling me the story about your boys. And I was like, "Stop, stop. Don't tell me any more. We're gonna put this on the podcast." So y'all, this is a little raw. I don't even know the end of the story. But I'm excited to hear it. Gonna be a lot of fun. Kim, you have a few boys? 

Kim Montague  00:58

Yes. 

Pam Harris  00:59

Go.

Kim Montague  00:59

Alright. So Cooper, who's my younger in fifth grade, has been working on multiplication and division of fractions. 

Pam Harris  01:07

Fun. 

Kim Montague  01:08

And yeah. And so I've gotten to listen to him a little bit. And he absolutely loves talking math with me. He's not super a fan of some math situations. But he will get a shower. I'm usually like yelling, "Get out of the shower." And he will jump into my bed afterwards and say, "Let's talk math." And I, you know, soak up every second of that. I think it might actually be a ploy to stay up late.

Pam Harris  01:34

I mean. 

Kim Montague  01:36

But he gets me every time.

Pam Harris  01:39

And you both enjoy it. So that works. 

Kim Montague  01:40

Oh, yeah, it's fine. So he is the one who will say, "Hey, is it MathStratChat?" And loves, like just thinking about things, especially if he hasn't had a chance to think of them before. So anyway,

Pam Harris  01:50

I'm gonna interrupt you really quick. 

Kim Montague  01:52

Yeah. 

Pam Harris  01:52

So when y'all are on social media, and you're looking at MathStratChat Check out when Kim will post his strategy. 

Kim Montague  01:59

Oh, yeah. Cuz like, 

Pam Harris  02:00

Yeah, he's the one that asked for it. He's like, "Hey, is tonight? It's Wednesday. Let's get on MathStratChat." Whatever. And then they'll post his strategy. So you can totally check it out.

Kim Montague  02:07

And if it's Thursday or Friday, and I haven't shown him Oh, woo. 

Pam Harris  02:12

He'll get to you, huh? 

Kim Montague  02:13

Yeah. So um, 

Pam Harris  02:15

Come on Kim, do MathStratChat with you kid. What's wrong with you? 

Kim Montague  02:18

I know. Um, so the other night, we were talking about multiplication of fractions. And I said, and so sometimes, you know, I will give him like a mini problem string just to see, you know, what kind of relationships he can see. So I was giving him like, 1/5 times 1/8,  and 2/5 times 1/8, and 4/5 times 1/8. And kind of cementing some of the language of 1/5 of 1/8, and just changing it up a little bit to see what he was messing with. But I realized that I hadn't been giving him problems, like, 3/5 times 3/9. And he was at a place where he was kind of discovering some things.

Pam Harris  03:03

Can I pause for a second, I'm just gonna little mathy here. We're assuming a lot of knowledge. So you were doing some things with unit fractions? Yes. 1/5 of 1/8. And so that the unit fractions, you were kind of like, what do you know, what are you thinking about with multiplication of unit fractions? And you realized that you just hadn't had fun yet with him doing something with non unit fractions. 

Kim Montague  03:23

Yes. 

Pam Harris  03:24

Where there's not a one in the numerator. But some other numbers in the numerator. Okay. Sorry to interrupt. Don't lose your turn.

Kim Montague  03:34

So he was also at a place where he was starting to say, "Wait a minute, I'm noticing something, you can just multiply the numerators and then multiply the denominators." And I was like, "Oh, you think that's true?" And he went back and looked at a couple of the problem string that we had done. And he said, "Yeah, I think that's true." And I said, "Okay, so let me give you this other problem." And the problem that he gave him was 3/5 times 3/9. And I said, "What are you thinking?"

Pam Harris  04:00

Random. Was that on purpose? Or was it like random? 

Kim Montague  04:03

Um, it was actually a little random.

Pam Harris  04:06

A little?

Kim Montague  04:07

It was a little random. 

Pam Harris  04:09

You just chose random numbers, you're like 3/5 times 3/9? 

Kim Montague  04:11

Well, 3/5 had been in the previous problem. So the last problem I had given him was 3/5 times 1/5. And so then I was like, "Oh, wait, let me give you one that is not, doesn't have a unit fraction." And so I literally use 3/5 again, and then was like, "Oh, let me go 3/9." 

Pam Harris  04:35

Okay. 

Kim Montague  04:35

And so he says to me, "I think that it's going to be 9/45, because if you multiply the numerators and multiply the denominators, you're going to get 9/45." And I said, "Okay, so like, let's talk about what you know, and see if that's true." And he said, "Well, not really sure. But I know what 1/5 times 1/9 is." Because he's had a lot of experience with two unit fractions. "So I know 1/5 of 1/9 is 1/45." And I said, "Okay." And he said,

Pam Harris  05:16

Do you mind if I pause for just a second? Sorry, because that experience he's had, he could think about a ninth of something, and then say to himself, "If I'm going to cut that 1/9 into five chunks," 

Kim Montague  05:27

Yep. 

Pam Harris  05:27

"It's as if I'm cutting the whole into those five chunks as well, that were cut into ninths. And now I have 45 total pieces. And but I only need one of them. That's 1/45." Sorry.

Kim Montague  05:39

He literally pictures, a ninth of something, and then a fifth of that piece.

Pam Harris  05:44

Yeah. Nice. And so he knows that a fifth of the night is 1/45. Sorry, keep going.

Kim Montague  05:48

And so I'm recording on a piece of paper as he's talking out loud. And so I wrote a fifth times a ninth equals 1/45. And then he said, "Oh, wait." And at this point, I'll pause, because I have another son, who's three years older, and he hopped out of the shower, and like, came and jumped on the bed. And he's like, "Oh, I see what you're doing." And I said, "Do not say a word. You may not speak. You may listen."

Pam Harris  06:12

We should start calling this pajama math. Sorry, keep going.

Kim Montague  06:18

And I said, "Do not you can listen, and you can, um, but don't say a word. Let him think." And he said, "Okay, got it." And so Cooper had just figured 1/5 times 1/9. And then he said, "Wait, wait wait, I can do 3/5 of a ninth. And that's 3/45. And so I wrote times three on the, to get him from 1/5 times 3/5. And then from 1/5, I'm sorry, 1/45 to 3/45. I'd written times three. And he said, "Yeah, exactly that." And so then I said, "Okay, so now you know, 3/5 times 1/9 is 3/45. So how does that help you solve 3/5 times 3/9?" And he said, "Oh, I can just scale again." And he said, 'scale', because we've used that language before when he's done stuff on a ratio table. So then he said, "So that means that 3/5 of 3/9 is 9/45," which is what he had predicted. And I said, "Okay." 

Pam Harris  07:28

Brilliant reasoning. 

Kim Montague  07:29

Yeah, absolutely. And I love that he was willing to think through, like, what made sense to him. Because when he said, "I think I can just multiply across." He's like, "I'm not really sure. And that doesn't make any sense. And I can't picture that." And so I love that he's making some understanding about scaling up and like thinking about the unit fractions. So Luke, of course, says, "Can I talk now?" "Yes, what?" And he said, "Well, the problem was 3/5 times 3/9, and that's really just 3/5 times a third." And I immediately was like, "Okay." And he said, "And that's the same as 1/3 times 3/5, or a third of 3/5. And that's just a fifth." And I was like.

Pam Harris  08:26

Because 1/3 of three anything's, 1/3 of three things is one of those things, and we're dealing with fifths. So 1/3 of 3/5 is 1/5, which is equivalent to 9/45.

Kim Montague  08:37

Yeah, yeah. Yeah. 

Pam Harris  08:38

Ah, I love it. I love it. That is so amazing. I love it. 

Kim Montague  08:42

They're a mess. 

Pam Harris  08:43

You know, you could have said that the older kid came in and said, "Dude, just will frustrate across its rule." Like, you could have said that. But you have this atmosphere in your house where your kids are like, no, like, we're reasoning about stuff. And he looked at that problem was able to see 3/9 as 1/3. You can sort of think about a third of three things. Nice. 

Kim Montague  09:02

Yeah. 

Pam Harris  09:02

Hey, so we actually maybe should have, at the beginning of this podcast, we might should have said, 3/5 times 3/9, think about that, solve the problem, use relationships, audience like listeners, and then told you what her boys did. That would be actually more in line with what we suggest you do in your classrooms. So what we don't believe is that we just say, "Hey, here's what kids did or here's what someone else did. Isn't that cool?" We actually like to have you solve the problem first, before we sort of tell you other strategies. So a little miss on that. Sorry. Okay, we'll teach you non example today. However, now, well actually Kim, did you want to say anything else about your kids? That was cool. 

Kim Montague  09:42

That's it. 

Pam Harris  09:43

So then, let's do it right this time. So let's give you another problem that we're gonna invite y'all to play with. Like be Luke and Cooper and play with these numbers. And how can you reason through. Ready? Here's a problem. A problem for you to think about. What is 3/4 times 2/5? 3/4 of 2/5? Pause the podcast, work on that. How can you think about 3/4 of 2/5? And then come on back and we'll share a couple strategies. 

Kim Montague  10:16

Well, can I have a second? Because, 

Pam Harris  10:18

Before you pause, wait.

Kim Montague  10:20

We have some listeners who know some rules, right? Who know some things to do. And so you're asking like, what are the relationships you can figure out? Not just what's the answer?

Pam Harris  10:30

I mean, Luke new rules, but he reasoned through it. So yeah, we're not interested in an answer. It's not about to answer getting and if you're confused on that, listen to last week's podcast. What episode would that be? 87, 88? I don't remember. Listen to the last podcast episode where we talk about is it about answer getting or about building relationships. And what we're, yeah, Kim's right? Build relationships, or use relationships to solve that problem. Alright, then come back.  Okay, so hopefully you paused the podcast. And now we're going to share a couple of strategies of 3/4 of 2/5 or 3/4 times 2/5. Kim, who's going first?

Kim Montague  11:10

You can go. How many are we gonna share? 

Pam Harris  11:14

I don't know. How much time do we have? 

Kim Montague  11:16

I'll share my favorite one. You share your favorite one. Maybe? We'll hope they're not the same. 

Pam Harris  11:20

You start then. 

Kim Montague  11:21

Okay. 

Pam Harris  11:22

Okay. 

Kim Montague  11:22

So I love, love, love, love this problem. It works fantastically for this idea of swapping that I love so much. Because this problem is multiplicative in nature, right? So I love to think about 3/4 times 2/5, and actually I like to think about that as 2/4 times 3/5. So because you can think about three one fourths and two one fifths. And I like to swap the numerators.

Pam Harris  11:56

Using the associative property. You're re associating in multiplication.

Kim Montague  11:59

Yep. So I think about 2/4 times 3/5, which is the same as a half times 3/5 or a half of 3/5. And so then a half of 3/5 is 3/10. 

Pam Harris  12:17

How do you know that? 

Kim Montague  12:18

It's a great question. So 3/5, then if I cut those fifths in half. How do I know 3/5? Half of those are three tenths?

Pam Harris  12:35

I mean like where you're going.

Kim Montague  12:37

Yes. If I'm picturing three, 1/5, and I cut those fifths in half. That I only have 3/10.

Pam Harris  12:46

Yeah, cuz each half of that fifth is a tenth. 

Kim Montague  12:50

Yeah. 

Pam Harris  12:50

You have three of those halves of fifths. Half of it is 3/10. 

Kim Montague  12:54

Yeah.

Pam Harris  12:55

Yeah, that's really nice. Another way to think of one half of 3/5 getting 3/10 is that you can think about a half of three is one and a half. Right? So you can have one and a half tenths. And one and a half tenths is equivalent to, sorry, we're gonna have fifths, my bad. One and a half fifths. Because I'm thinking about a half. 

Kim Montague  13:17

Yeah, one and a half fifths.

Pam Harris  13:19

A half of three anything's is one and a half. So half of three fifths is one and a half fifths. And one and a half fifths is equivalent to three tenths. That would be another way to do that.

Kim Montague  13:27

Uh-hum.

Pam Harris  13:28

I love that. I was not thinking you were gonna use that strategy. Y'all, I have given Kim this problem before and that is not the strategy I thought she's gonna use.

Kim Montague  13:35

 Oh, sorry.

Pam Harris  13:36

That's okay. So I'm going to do my favorite strategy, and then I'm going to tell you what I thought Kim was gonna say, because I've heard her say it. But that is a symptom, a sign of a mathematician. They use the relationships,

Kim Montague  13:47

 A symptom?

Pam Harris  13:48

What? 

Kim Montague  13:48

A symptom.  A symptom. Is that bad? They use relationships that are on the top of their head that day, like what what they're thinking about that day, that sort of, "I'm going to use what pinks for me." Okay, so my favorite strategy today for this problem is to think about three fourths of two fifths as three fourths of two anything's. So if I'm just gonna think about three fourths of two fifths. I'm going to sort of ignore the fifths for a minute and just think about three fourths of two. I can think about a half of two. A half a two is one, and a half of that as a half. So 3/4 of two is one and a half. Is that right? One and a  half? And that's a different way of getting 3/4 of 2/5 is one and a half fifths, and I back to one and a half fifths, which is equivalent to 3/10. I like yours as well. Can I tell you that I got a little distracted for a second because I was like, "Man, I wonder what I've done before?" And then I realized what I probably did, and I got a little sad because it was what I didn't share, because I think I know what I would have said before, and if you're, it's true. And I like your strategy.

Pam Harris  14:56

Well, so why don't I let you share that since you're sad that I was about sure.

Kim Montague  15:01

I'm not sad you were going to say it, I'm sad I didn't say.

Pam Harris  15:05

Oh my word. Kim, you're on. What is your now favorite strategy? 

Kim Montague  15:10

Okay, my now favorite is to think about it in terms of money or percents. So 3/4 of 2/5, I would think about as 75% of 40 cents.

Pam Harris  15:26

We better say that again. 

Kim Montague  15:27

Yeah. 3/4 of 2/5 is the same as 75% of 40 cents.

Pam Harris  15:37

it's equivalent to 75% of 40 cents.

Kim Montague  15:40

Yeah. 

Pam Harris  15:40

Because 2/5 is 40 cents. 

Kim Montague  15:42

Yeah. So 75% of 40 cents is just 30 cents, which is three tenths. And you're right. I love that one well.

Pam Harris  15:51

I think that one is so cool. I think the first time I ever heard you say that, I was like, "Wait, where'd the point four come from?" And you're like, "two fifths." And I was like, "Oh, that's right, because 1/5 is point two. So to like 20 cents, and two of those 40." Now I own that, like now, but I think the first time I ever heard you say this 2/5 was not point four to me. It wasn't just, bam, 40 cents. So I'm kind of happy that now it is. It kind of makes like excited. 

Kim Montague  16:14

That's great.

Pam Harris  16:14

Yeah. And I think you could have thought about your strategy as 1/4 of 40 cents instead of 75% of 40 cents. And I thought about 1/4 a fourth. 

Kim Montague  16:24

Wait, wait.

Pam Harris  16:24

Well, sorry, but 1/4 of 40 cents, and then find three of those. 

Kim Montague  16:27

Yes.

Pam Harris  16:29

Like so you went straight to 75%. But you could have found 25% And then tripled that or found 1/4 of 40 cents. And then so three of those one fourths is 30 cents. That might be, if somebody ever thought about that might be sort of a helpful relationship to think about. Alright, y'all. So playing with fraction multiplication can be fun, whether you're in your pajamas or not, thanks for tuning in. If you want to have more problems like this, we do problems like this that are rich, at least most of them on MathStratChat. So we do MathStratChat on Wednesday evenings on social media here in the United States around 7:45. Central time. It's not always, you know, kinda depends on what I'm doing. But we throw out a question to the world. And I try to write math questions that are rich enough that Kim can have a favorite strategy. And I can have a favorite strategy. And there's multiple ways, really slick, cool ways of using relationships to figure those out. So if you're interested to hear how or see how people around the world are figuring these rich problems, join us and MathStratChat Wednesday evenings on social media. We do Instagram, Facebook and Twitter, and it's hashtag Math, Strat, Chat as math strategy chat. That's what it stands for. MathStratChat, we'd love to have you join in. My favorite way for you to join in, is for you to read the question, solve it with your favorite relationship, type that out. Now we have a couple people, it's super cool. Mark, I don't have to say your last name, Mark. Mark N will always do several strategies. Oh, I wasn't planning to say these names. And now it's not coming to me. Who's my friend (unclear)? What's her name? Oh, okay, I totally know your name. And I'm blowing it right now. Huh, well always, she will always put in at least three strategies. And often she writes them on paper and takes a picture. Brilliant. Like I love it, when we have your first sort of go at it. And then even better, the two of them and more, more of who will join in, will then look at other people's strategies. And then we'll say, "Oh, I love how you did this," or "Oh, that was so cool, how you were able to use this relationship." Or sometimes, "Whoa, that was even better than what I did." And so it's totally fun when people join in MathStratChat by solving it first your way, and throwing that out there and then giving us, or going into other people's strategies and commenting on those. And that just makes the conversation even more fun. However, if you want you could just lurk. You just go head on in there and see what other people are doing and that's totally fine and totally fun. So we would just invite you to join us on MathStratChat for fun strategy conversations. So if you want to learn more math and refine your mathematics teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able. And her name is Karen. It's Karen Lambert, it's Karen.