# Ep 89: Playing With Fraction Multiplication

March 01, 2022 Pam Harris Episode 89
Math is Figure-Out-Able with Pam Harris
Ep 89: Playing With Fraction Multiplication

Kim has a story to tell that Pam hasn't heard yet! Listen in to get a feel of how math can be when students understand that Math is Figure-Out-Able.
Talking Points:

• Pajama Math and exploring relationships in multiplying fractions
• Can you really just multiply straight across with fractions?
• Two ways to reason about 3/5 times 3/9
• Pam and Kim reason multiple ways about 3/4 times 2/5
Pam Harris:

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

Kim Montague:

I'm Kim Montague.

Pam Harris:

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:

Yes.

Pam Harris:

Go.

Kim Montague:

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

Pam Harris:

Fun.

Kim Montague:

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:

I mean.

Kim Montague:

But he gets me every time.

Pam Harris:

And you both enjoy it. So that works.

Kim Montague:

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:

I'm gonna interrupt you really quick.

Kim Montague:

Yeah.

Pam Harris:

So when y'all are on social media, and you're looking at MathStratChat Check out when Kim will post his strategy. Yeah, he's the one that asked for it. He's like,

Kim Montague:

Oh, yeah. Cuz 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. And if it's Thursday or Friday, and I

Pam Harris:

He'll get to you, huh? haven't shown him Oh, woo.

Kim Montague:

Yeah. So um,

Pam Harris:

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

Kim Montague:

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:

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:

Yes.

Pam Harris:

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:

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:

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

Kim Montague:

Um, it was actually a little random.

Pam Harris:

A little?

Kim Montague:

It was a little random.

Pam Harris:

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

Kim Montague:

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:

Okay.

Kim Montague:

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:

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:

Yep.

Pam Harris:

"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:

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

Pam Harris:

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

Kim Montague:

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:

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

Kim Montague:

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:

Brilliant reasoning.

Kim Montague:

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

Pam Harris:

Because 1/3 of three anything's, 1/3 of three things like. 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:

Yeah, yeah. Yeah.

Pam Harris:

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

Kim Montague:

They're a mess.

Pam Harris:

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:

Yeah.

Pam Harris:

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:

That's it.

Pam Harris:

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:

Well, can I have a second? Because,

Pam Harris:

Before you pause, wait.

Kim Montague:

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:

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:

You can go. How many are we gonna share?

Pam Harris:

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

Kim Montague:

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

Pam Harris:

You start then.

Kim Montague:

Okay.

Pam Harris:

Okay.

Kim Montague:

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:

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

Kim Montague:

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:

How do you know that?

Kim Montague:

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:

I mean like where you're going.

Kim Montague:

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

Pam Harris:

Yeah, cuz each half of that fifth is a tenth. You have three of those halves of fifths. Half of

Kim Montague:

Yeah. it is 3/10. Yeah.

Pam Harris:

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:

Yeah, one and a half fifths.

Pam Harris:

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:

Uh-hum.

Pam Harris:

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:

Oh, sorry.

Pam Harris:

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:

A symptom?

Pam Harris:

What?

Kim Montague:

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:

Well, so why don't I let you share that since you're

Kim Montague:

Pam Harris:

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

Kim Montague:

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:

We better say that again.

Kim Montague:

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

Pam Harris:

it's equivalent to 75% of 40 cents.

Kim Montague:

Yeah.

Pam Harris:

Because 2/5 is 40 cents.

Kim Montague:

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:

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:

That's great.

Pam Harris:

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:

Wait, wait.

Pam Harris:

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

Kim Montague:

Yes.

Pam Harris: