Ep 208: Fraction Multiplication Part 3

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  00:17

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  00:32

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

Pam  00:39

We invite you to join us to make math more Figure-Out-Able!l Alright, that's getting down. I'm getting that down. That sentence is so long. 

Kim  00:47

Well, you said that. You can do something about it. 

Pam  00:50

No, but it's such a good one. Like, I've had several people tell me that they really like it. 

Kim  00:53

Yeah.

Pam  00:54

I just take a deep breath, and... It's like my days of playing the trombone. It's like...

Kim  01:01

You played the trombone?

Pam  01:02

I did. Do you want to know why? Because I wanted to be... She's like, "No," but I'm going to tell you anyway. I wanted to be in the band. And I told my parents, I wanted to play the drums. I was like, all over. My dad played the drums. I was like, "I want to play the drums." They were like, "Okay, you could be in the band." And my dad came home one day, and he had this beat up old trombone. And he goes, "Hey, check it out. I got an instrument for you." And I was like, "I'm sorry, what? I want to play the drums." And he's like, "Yeah, but I was able to get this free trombone." I was like, "Um..."

Kim  01:29

That is not the drums. 

Pam  01:31

So not. They said, "Play it for a year, and then you can play the drums and whatever." Anyway, I played it for like four years. It was fine.

Kim  01:38

That's amazing. How do I not know? It's always fun when after 20, what, 2 years (unclear). Okay.

Pam  01:43

The good part about playing the trombone is I was the only girl always. 

Kim  01:47

Yeah, of course. 

Pam  01:47

It was like me and the guys. It was awesome. 

Kim  01:48

I shouldn't say "Of course." Lots of trombone players are women (unclear).

Pam  01:51

I'm sure there are, but not in mine, and it was great. The ratio was in my favor. I had a blast being a girl (unclear). Alright, Kim, let's do math. (unclear).

Kim  02:02

But first. 

Pam  02:03

Oh, okay, okay. 

Kim  02:03

We have this lovely, lovely woman named Kim. 

Pam  02:06

Oh. 

Kim  02:06

Hi, Kim.

Pam  02:07

We like people named Kim.

Kim  02:08

Kim Bright. She left us a review. It's been a while now. And I've been saving it because...

Pam  02:14

It was that good? 

Kim  02:15

Well, it's fun. She says, "My favorite math podcast."

Pam  02:19

Aww, thanks. 

Kim  02:20

"I love listening to Pam and Kim share different ways to think about teaching math. I've gained so much math from them..." Sorry, "I've gained so much from them and implement a ton of what they're sharing into my teaching practice. They're fun to listen to, and break down math in real fun and practical ways. Thank you for sending the message that the use of reasoning and understanding of numeracy is the way to understand math. I'm here for it, and so are my students."

Pam  02:45

Aww, that's awesome

Kim  02:46

We're here for it too, Kim. Thanks.

Pam  02:48

Thanks for the five stars, Kim. 

Kim  02:49

Yeah. 

Pam  02:50

And, listeners, thank you so much for giving us reviews, and... What do you call that? The rating, right? Rating? Rating reviews. The algorithms for podcast, the more ratings and reviews we get, then the more listeners can find the podcast, and the more we can help revolutionize the way we teach math, so we appreciate you guys helping us get the word out.

Kim  03:09

Okay, so the last couple of weeks, we've been talking about fractions, fraction multiplication, and we did an episode on unit fractions and unit fraction by a unit fraction. And then last week, we talked about taking that understanding and using scaling to think about non-unit fraction multiplication. 

Pam  03:28

Yes. 

Kim  03:28

So, today, let's go some more. 

Pam  03:31

Yeah. So if you have not listened to those, we are really not going to recap a lot of that. So, we're going to build off of it, so you might want to listen. I mean, you're welcome to listen to today. But as soon as you hit something you're like, "What?" then it might be because maybe we're building off of... We are. Not maybe. But yeah, check out the last two if you haven't heard them yet. Alright, cool. So, Kim, I'm going to throw out a question, and I just want to hear your thinking. Don't try to give me what you think I want. Just tell me actually how you think about it. 

Kim  03:59

Okay. Do I ever? 

Pam  04:02

I mean, pretty much. So, Kim, if I were to say two-fifths times three-fourths. Two-fifths times three-fourths? What do you? 

Kim  04:09

Yeah. So, I think I mentioned this last week that when I think about a fraction times fraction multiplication, then I think about... I'm actually going to write down what I think in my head. So, I actually think 2 times one-fifth times 3 times one-fourth. So, I think about the fifth and the fourth as the twentieth. But then I scale up, like we talked about last week, times 2, and then scale that up again, times 3. And so, for that, I get six-twentieths. 

Pam  04:41

Cool. And then you would probably also think about simplifying that six-twentieths just... 

Kim  04:47

I mean, if I had to. Somebody (unclear).

Pam  04:48

If you have to. 

Kim  04:49

Yeah.

Pam  04:49

But you wouldn't have to. You can think about that as six-twentieths. 

Kim  04:52

Yeah. 

Pam  04:52

Cool. So, that would definitely be a way that you could think about that. If you weren't thinking about the last two weeks. I wonder if there's any other way you might think about two-fifths times three-fourths. 

Kim  05:02

Do I get to play?

Pam  05:03

Yeah, you get to play. Play a little bit. 

Kim  05:06

Okay, so if I was thinking about two-fifths times three-fourths, then I would actually think about three-fifths times two-fourths. And that would be half of three-fifths. 

Pam  05:17

Because two-fourths is one-half. 

Kim  05:19

Yeah. 

Pam  05:20

Okay. 

Kim  05:20

So, half of three-fifths is three-tenths.

Pam  05:25

How do you know half of three-fifths is three-tenths?

Kim  05:28

Because I think about one-fifth, and I think about half of that. Half of one-fifth is one-tenth, so half of three-fifths is three-tenths. 

Pam  05:39

Nice. Nice. Nice. Nice. Any percents coming to mind just while we're playing a little?

Kim  05:47

Well, yeah. I mean, I think 50% of three-fifths. You could also think about 40% of 75. Or 75% of 40. Like, (unclear).

Pam  05:58

(unclear) Yeah, if you go back to the original problem, two-fifths times three-fourths, you're saying you could think two-fifths is 40%?

Kim  06:05

Even more fun would be 75% of 0.4.

Pam  06:11

Slow that down just a little bit.

Kim  06:12

Okay.

Pam  06:12

Because the problem is two-fifths times three-fourths. But you're thinking you can think about three-fourths times two-fifths...

Kim  06:18

Yes.

Pam  06:19

...as 75% of two-fifths, which is 0.4. 

Kim  06:24

Yeah. 

Pam  06:24

Why would you think about 75% of 0.4? I love it. I love it. I love it.

Kim  06:31

Because 75% of... 

Pam  06:36

Like, how do I verbalize this? Three-fourths? How about

Both Pam and Kim  06:39

three-fourths of 40

Kim  06:40

cents is $0.30.

Pam  06:42

$0.30. Bam. And there's your three-tenths again?

Kim  06:44

Yeah.

Pam  06:44

Yeah.

Kim  06:44

This would be a great MathStratChat problem once you do a lot of work with your kids (unclear) so many fun things on this problem. I love this problem. 

Pam  06:52

So many things you could do with it. Yeah.

Kim  06:54

Yeah.

Pam  06:54

Cool. Okay, so now that we've established that two-fifths times three-fourths is six-twentieths or three-tenths. Which is equivalent to three-tenths. Another way that we can also model that would be kind of based on a little bit about what you said earlier. I'm going to draw a candy bar, and so I've got a rectangle kind of a horizontally. A little bit longer rectangle. And I'm going to think about two-fifths of three-fourths by thinking about a fifth of a fourth first. So, I'm thinking about fourths. I'm going to cut the whole candy bar vertically into fourths. And then I'm going to cut the whole candy bar horizontally into fifths. Okay, so now I've got sort of a 5 by 4 to represent fifths by fourths. And if I think about a fifth of a fourth, that's kind of that upper left hand rectangle, but we need to two 1/5s, so I double that, so now I've got two of those fifths in that left hand column. And then we need three 1/4s, so I'm going to triple that across. And so, my rectangle now looks like kind of a 5 by 4. Do I hear your pencil? I think I hear your pencil. 

Kim  08:06

Sorry. 

Pam  08:07

No, that's awesome.

Kim  08:08

I'm sketching your picture.

Pam  08:09

You pencil person, you. And then... So, where did I even say? I've got like a 5 by 4 where I've got the 2... How do I say this? A 2 by 3 shaded in. In the middle of it. Does that track for you? 

Kim  08:24

Yeah, so your big rectangle, your outside big is a 5 by 4.

Pam  08:28

Mmhm. 

Kim  08:29

And then inside of that you have a 2 by 3 shaded.

Pam  08:32

Yes, yes. 

Kim  08:33

Okay, cool. 

Pam  08:33

And so, that's kind of those six-twentieths. And like we talked about last time, I could count each of them. I could look at those 6 as three 2 chunkers, and that would be three 2 chunkers out of the ten 2 chunkers. That would be another way to see the three-tenths. But we're not going to go there today. So, focusing on  the six-twentieths, I've got sort of 6 out of the 20 shaded. Okay, cool. Next problem. What is three-fifths times two-fourths? And I think you even did that when I asked you to think about the other one

Kim  09:05

(unclear) You let me play with (unclear).

Pam  09:06

I did let you play. What is three-fifths to two-fourths? Say it again.

Kim  09:11

That is also six-twentieths or three-tenths. 

Pam  09:15

And that's because you thought about two-fourths as one-half. And that's when you did the whole half of 3. Yeah, anyway. So, okay, so you're saying they have the same answer, right?

Kim  09:24

I do. Mmhm. (unclear).

Pam  09:25

Okay, so three-tenths or six-twentieths. But I'm going to go ahead and represent a rectangle, that is three-fifths by two-fourths. Okay. So, underneath my other, the rectangle for the first problem, I'm drawing the same rectangle. But this time, I'm going... So, like the outside is the same. But now I'm going to draw... I can think here. I'm going to still do fourths vertically and still do fifths horizontally. That takes me a second. Okay. And now I'm going to do 3... So, I've got a fourth of a fifth is that top left hand rectangle, and I've got 3... I'm going to scale that times 3. That little guy times 3. So, now it's down there times 3. So, in that left hand column, I've got 3 rectangles shaded in. But now it's two-fourths, so I'm going to double that. And so, now my rectangle looks like a 5 by 4, but the inner shaded rectangle is a 3 by 2. Does that track? 

Kim  10:33

Yep. 

Pam  10:33

Okay. So, if I was doing this with real kids because you're not a real kid.

Kim  10:38

Sometimes.

Pam  10:39

Then I would have the problem two-fifths times three-fourths, with the one rectangle, and the problem three-fifths times two-fourths with the other rectangle that we've just described. And then I would step back, and I would say, "What do you see?" So, Kim, if you were a kid, or yourself, what are some things you might see? 

Kim  11:00

So, in the first problem... I have numbers all over the place. I think the first problem was two-fifths times three-fourths.

Pam  11:07

Mmhm.

Kim  11:08

Okay. So, in in the original first problem, I still am thinking about fifth time's fourths. But I only had two of the 1/5s, and I had three of the 1/4s. 

Pam  11:20

Mmhm.

Kim  11:21

But this time, I'm still thinking about fifth time's fourths, but I have three of the 1/5s and two of the 1/4s. So, you can see that in the representation that you sketched because the 2 and the 3 are reversed, their swapped, and it looks like a rotation in your model.

Pam  11:44

The shaded part, right? Because the outside rectangle looks the same. It's still a 5 by 4. And the inside shaded guy went from a 2 by 3, rotate to a 3 by 2. But like you said, both of them are thinking about a fifth times a fourth, and then scaling either times 2 times 3 or in the second one times 3 times 2. So, we end up with 6 shaded pieces out of the 20 shaded pieces. It's just they're sort of rotated a little bit. And that can be a spatial way to help kids sort of think about the commutative property in fraction multiplication. That they can literally reason through a problem like two-fifths times three-fourths. But they could also say to themselves, "Yeah, but I could swap. I could rotate that inner array." Thank you, Cathy Fosnot, by the way. So, I learned this rotating the inner away from Cathy Fosnot. I probably should have mentioned at the beginning, maybe. So, golly is that in Parks and Playgrounds? So she has a (unclear).

Kim  12:46

Yeah.

Pam  12:46

I think it's called Parks and Playgrounds. Measuring Parks and Playgrounds? Something about parks and playgrounds, where it's this idea that you can look at the shaded inner array, and you can watch it rotate, and that kids can go, "Oh, yeah, multiplication is commutative." So, I think there's two things happening here. One, we have a spatial justification for the idea that the numerators can kind of switch places or that you can use the commutative property. But also there's kind of, Kim, you're scaling idea going on. When you talked about two-fifths times three-fourths, you said that you were thinking about that as two 1/5s times three 1/4s. And so, once you can kind of think about that one-fifth times one-fourth. You know like, the way I have it written down on my paper is 2 "parentheses" 1/5 times 3 "parentheses" 1/4. I can use the commutative property right there. 

Kim  13:36

Yep. 

Pam  13:36

And think about one-fifth times one-fourth. That's the one-twentieth. And I think about that 2 times 3 as the 6. 

Kim  13:42

Yeah. 

Pam  13:43

So, what that means is, we can have other problems where once we realize that we can make those fractions into the numerator times that unit fraction, and the other numerator times its unit fraction, then we can kind of go wild with the commutative property, and out comes this idea of simplifying either first or at the end because, really, we kind of use the commutative property to do it. So, for example, I could give you a problem like two-fifths times five-sixths, and you could, at that point, decide... Well, let me back up. The way that I've seen a lot of teachers handle this is they go "Okay, either multiply straight across." Now, kids are rote memorizing a rule and they're mimicking.

Kim  14:37

Yep. 

Pam  14:37

And then they have to remember which rule to do when.

Kim  14:39

Yep. 

Pam  14:39

Or the teacher says, "No, no, no. We're going to get some modeling out of this," and so they say, "Draw a rectangle. Cut one one side into fifths, shade 2 of them. Cut the other side into sixths, shade 5 of them. Count the double shaded pieces. Count the total. And then represent the shaded pieces out of the total." We're submitting that that's another procedure that kids are rote memorizing and mimicking. And then they're stuck in counting strategies because they're counting what's coming out of it. So, we're not recommending either of those when a kid hits two-fifths times five-sixths, but we're also not recommending that teachers start to slash and burn at this point.

Kim  15:21

Mmhm.

Pam  15:21

Where they're like, "Okay, so I can see two-fifths and five-sixths. Hey, those fives are the same. I'm just going to like cross them out (unclear)."

Kim  15:27

The same. 

Pam  15:28

Yeah. 

Kim  15:29

The same.

Pam  15:31

And so, now we've got this sort of slash and burn thing happening, and they're cross canceling or whatever. All these things that then kids start to cross cancel no matter what is happening in between the fractions.

Kim  15:42

Right.

Pam  15:43

So, we're also not suggesting that that is the first go to. How could kids look at a problem like two-fifths times five-sixth and use reasoning to go from there? Kim, I'm not going to say much more than that.

Kim  15:57

Yeah.

Pam  15:57

What are you thinking about?

Kim  15:58

So, I thought two-fifths means 2 times one-fifth. And I wrote two times one-fifth. And then I also wrote times 5 times one-sixth. So, on my paper, I have 2 times one-fifth times 5 times one-sixth. That's what that fraction multiplication means. So, within that, I have one-fifth times 5. Which means like it's equivalent to 1.

Pam  16:25

And we would want kids to know that, right?

Kim  16:27

Right. 

Pam  16:28

Like, that's not a, "Ooh, look, there's a 5 in the numerator and a 5 in the denominator. We can slash and burn them." No, we like want to actually think about what five 1/5s means. 

Kim  16:37

Yeah. 

Pam  16:38

That's an important understanding. 

Kim  16:40

Yeah.

Pam  16:40

Five 1/5s. Cool. (unclear).

Kim  16:42

So, then I have 2 times one-sixth (unclear).

Pam  16:44

Leftover. 

Kim  16:45

Yeah.  Two-sixth is a third.

Pam  16:48

And two- sixths is equivalent to one-third. Cool. And how are you thinking about two-sixths being equivalent to one-third? And I know you're probably not thinking about that very much, but if you think about it?

Kim  17:04

I don't know.

Pam  17:05

You've got that one pretty ingrained, right?

Kim  17:07

I mean, yeah, yeah. 

Pam  17:08

Yeah. I wonder if you were to think about your two 1/6s. Does that... I don't know. Does that do anything?

Kim  17:19

I mean, if I thought something was, you know, cut into 6 pieces, and I had 2 of them, then it would be equivalent to if I had that same thing cut into thirds. 

Pam  17:27

Yeah. 

Kim  17:28

Yeah.

Pam  17:29

That's one way to think of it, sure. Yeah, nice. So, Kim.

Kim  17:35

Yeah?

Pam  17:35

We actually want kids to think about fraction problems, where it's not just slashing and burning, but they're actually thinking about unit fractions, the numerator times unit fractions, scaling times unit fractions, so that they can make sense of the fact that they're using the commutative property, and that they're going to end up with things like 5 times one-fifth and 2 times one-sixth, and that they can make sense of what those things mean. 

Kim  18:01

Yeah.

Pam  18:02

Instead of just mimicking an area model. Go ahead. 

Kim  18:04

Yeah. Well, and I was going to say when you give kids the opportunities to have some real understanding, that is one way that they can think about that problem. But there's several other ways. And I think that's what we're looking for in a classroom, right? We wouldn't want an environment where every single kid in the classroom does what I just said to solve two-fifth time's five-sixths because it's a rich problem where we should have a variety of strategies. And we only talked about one. I don't know if we want to talk about some other ways to solve that problem. But if we built on unit fraction. We talked about scaling. We just did this kind of commutative property swap the numerators thing. But also, we could have kids that say, "Well, I know one-fifth of five,-sixth, so I can double it to get two-fifths of five-sixths. 

Pam  18:54

What is one-fifth of five-sixths?

Kim  18:56

One-fifth of five-sixths. So, one-fifth of 5 is 1. So, one-fifth of five-sixths is one-sixth.

Pam  19:06

Nice. But we need two-fifths. 

Kim  19:09

So, it's two-sixths.

Pam  19:10

So, you're going to double that to get two-sixths.

Kim  19:12

Mmhm.

Pam  19:12

I love it. Do you have any more? Put you on the spot.

Kim  19:18

So, two-fifths of 5 is 2, so two-fifths of five-sixths is two-sixths.

Pam  19:26

Nice. Well done. Yes?

Kim  19:31

Well, I was going to say, and that's, you know, I think that's what we're hoping for in the classroom. That we don't just say let's, have meaning and all still do one way to solve a problem. You know, it's about building kids brains, so they can look at the numbers and see what they're playing with in that moment. Oh, this reminds me we had somebody in one of our Facebook groups. You know, we have we have a couple of different groups that we run. And one of the people said, "Hey, I have this question about my son, and he used to think about this multiplication fact this way, but lately I've noticed he's been thinking about it this way. Is that okay?" And I said that's absolutely brilliant that he used to... The last time she asked him how he thought about... I don't know. Maybe it was 8 times 4, he was thinking about it like 4 times 4 doubled. But now he's thinking about it like 10 times 4, back 2 times 4. And she's like, "Is that Is that okay?" I was like, "Yes, that's desirable! Yay, go him!" 

Pam  20:24

Yeah, we actually want multiple connections in kids heads, right? We want multiple ways, multiple paths, multiple representations. We want... It's not about... Oh, this is a high school thing. A lot of high school teachers will say multiple representations. And I'm like, ah, it's not about multiple representations. It's about the connections between multiple representations. We want to have a web-like series of connections in kids heads, so that they can use the one today that flows the easiest but also the one tomorrow that's going to connect to a different problem. Kim, I was just having a conversation yesterday with a mathematician at church. This guy, he makes me think. Oh, my gosh, every time we talk, I walk away going, "Let me think about that for a half and hour." 

Kim  21:08

That's awesome. 

Pam  21:09

So amazing. And his new project is he's working on a new encryption algorithm. Not encryption. He used to do encryption. Now, he's doing... Oh, golly, what do you call it when you make files smaller? Not condensing, but... Oh, my gosh, my brain. When you...

Kim  21:27

No idea.

Pam  21:29

Yeah, you know when you like take something, and you make it into a zip file, so that it... Compress! When you compress files. Okay, people are yelling at their. Yes, compressed files. I know. Anyway, so he's working on compression algorithms, and he was saying, "Hey, a lot of compression algorithms out there only work for certain kinds of files, and when you compress, it's pretty good if you can get like a 70% compression ratio." So, you know, if it was this big, now it's only 30% as big as it was. Then that's really good if you can compress it that much. But he's working on a compression algorithm that you can iteratively compress. So, you compress it, and then you can compress that one, and then you can compress that one. And we were starting to talk about. I kind of get a lot of what he was talking about. But I don't get how you uncompressed. Like, how do you not lose data when... Anyway, blah, blah, blah. What we were talking about was he said, "Yeah it's super important, that people..." He can't find people to work with, because he said people don't understand topology and number theory enough." And I said, "Hey, Al, I'm super curious. When you say that, would you agree with me that, it's a big reason, in a big way, people don't understand number theory because when they were actually doing numbers, they were taught to mimic and memorize, memorize and mimic algorithms." And he goes, "Yes! All they did was mimic and memorize algorithms, and so now that when they get to me, they never reasoned through the numbers, and so they're number theory is poor." And then add on to that they haven't had enough topology, and they can't put those together. The reason I'm mentioning this whole story is his brilliance in mathematics...like he has published a lot...is that he brings together these sort of disparate areas of mathematics. Like one of the reasons he could do what he does is because he brings together topology with things like set theory with things like number theory. And his unique way of bringing those connections together is where he adds to mathematics. I know I just got all excited about it. But that comes to mind because when you're mentioning this kid solving a problem in two different ways, it's about connecting. It's about connecting different ways of thinking. That's mathing. That's what mathematicians do. Now, Al, my friend does that with higher math. But we can do that very kind of thing with today. Today, we did it with fraction multiplication. Bam!

Kim  23:49

Yeah. And it's why it's so important that we give kids so many experiences and bring all this stuff together because when they understand what a unit fraction is, and how to operate on it, and they own the commutative property, and the relationships between decimals, and fractions, and percents, they have choice, and they have really cool strategies at their fingertips.

Pam  24:06

They have options. More importantly, they've built reasoning, and they are mathing.

Kim  24:12

Yeah. Thanks for listening everyone, and for leaving us a review, and for sharing the Math is Figure-Out-Able movement alongside of us. We love doing this, and we want as many people to know that we are so grateful for you and this movement. Keep spreading the word. 

Pam  24:26

Bam! 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. And keep spreading the word that Math is Figure-Out-Able.