Pam  00:01

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

Kim  00:09

And I'm Kim Montague, a reasoner, who nows 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

Did you say nows knows?

Kim  00:19

Probably. Just move on, Pam.

Pam  00:21

We know that algorithms.... We know algorithms are amazing historic achievements. But ya'll 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  00:37

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

Pam  00:45

We invite you to join us to make math more Figure-Out-Able. Bam!

Kim  00:50

This is a no mocking zone.

Pam  00:52

I'm sorry. I wasn't trying to. That wasn't mocking. That was me just... I was noticing. I see you, Kim. 

Kim  01:00

Oh. 

Pam  01:00

I got your back. That's what I meant. 

Kim  01:04

Hey, last week, we talked a lot. It was a big episode. 

Pam  01:07

Oh, my gosh, we both looked up. It was like, Whoa, that was long. 

Kim  01:10

Yeah. 

Pam  01:10

Alright, (unclear).

Kim  01:11

It's good stuff though, right? Alright, so last week. We're just going to dive in because we have so much to say about factions. 

Pam  01:16

Go, go go. 

Kim  01:17

So, last week, we talked about making meaning of a unit fraction by unit fraction, and all the different things that come into play with that, where kids are re-unitizing. And it's really important that we spend that time and we spend that work, rather than just tell them to multiply across because then we can build on it. So, let's build on it. 

Pam  01:36

Yeah, and you just said a unit fraction by a unit fraction. And I'll add two other ways to say that. So, a unit fraction times a unit fraction. And a unit fraction of. A unit fraction of a unit fraction. Yeah, so three different ways that we want to build all of those. And it's more than just telling kids "of" means times or times means "of". Yeah, either way. It's really like, let's get those three meanings down. So, ya'll, if you're just tuning in to this episode, and did not listen to last, you are going to want to listen to the last one. By the end of it, we were like, "Kids now have some feel, some sense of a unit fraction." 1 in the numerator. Times another unit fraction. 1 in the numerator. So, like one-fifth times one-seventh or one-third times one-fourth. They're not just multiplying straight across. They are really reasoning that if it's one-fourth times one-fifth... Oh, let's do another example. If it's a third times a seventh, then they've got 3 times 7 total number of pieces in the area model. And they could think about that. And they could think of if they've just got one of them. Or they can really think about one-seventh and that there's like 7 of these chunks. They can cut one of those sevenths into... What did I even say? A fifth of a seventh. Into 5 pieces. And so, they could really think about then they would have 35 total pieces. And they got 1 out of those total pieces. So, anyway, we kind of get that down. Kim? 

Kim  02:59

Yeah?

Pam  02:59

One of the things that we would really want to have before we dive completely into multiplication of fractions would be to have kids think about five-sevenths, a fraction like five-sevenths, not only as 5 out of 7 equal pieces. But we would also want them to think about five-sevenths as five 1/7s. 

Kim  03:19

Yeah.

Pam  03:19

In fact, in our challenge that we just did a few weeks ago, we did a fraction Problem String, and in that fraction Problem String, when I would talk about fractions like three-fourths or two-thirds, I would often say three-fourths and I would say three 1/4s. Or two-thirds, and I would say or two 1/3s. And we had several challenge participants ask, you know like, "Why are you naming fractions that way?" And so, I would suggest that thinking about five-sevenths as five 1/7s really helps kids get a more multiplicative view of five-sevenths. if it's just five-sevenths, then I could just count 5 and count 7. And then, yeah, that's 5 out of 7. And I could really be only in counting strategies. But if I can think about five 1/7s. Five-sevenths as five 1/7s, I could think about it as one-seventh plus one-seventh plus one-seventh plus... Like, five of those one-sevenths. That's more of an additive thinking. It's kind of like skip counting. And then I can move from that to thinking about well, I've got five of those 1/7s. And that's a bit more multiplicative. I can really think about five 1/7s. That's like a scaling view. I'm scaling one-seven times 5.

Kim  04:38

Yeah, and I think it helps kids recognize that you have five of those 1/7s. It's not that five-sevenths is this random, weird amount that isn't connected to any other amount.

Pam  04:51

Mmm. So, like 5 pieces, 5 pizza pieces out of 7 pieces, pizza pieces. It's 5.

Kim  04:58

Of those 1/7s.

Pam  04:59

Of those one-sevenths. Yeah, (unclear).

Kim  05:00

Yeah. And, you know, fractions becomes this time or this place where kids may start to think it's all about rules, right? Because some of the things that we say. But this scaling idea that you're talking about really is and can be connected to some work that they've done in multiplication. 

Pam  05:17

Work we would hope they have done (unclear).

Kim  05:19

Oh, yeah. (unclear). So, when we talk about 25 times something, we can scale that to get 75 times something. Or for smaller numbers, if we're finding 20 times something, we can find 10 times something and double it, scale up times 2. Or find 2 of something and scale up times 10. So, you know, if they're experiencing scaling in multiplication of whole numbers, then it's not such a reach when we're talking about scaling with fractions.

Pam  05:49

Yeah, nice, nice. So, if that went really fast right there, then check out some of our multiplication episodes. Or maybe we'll just do one on scaling and multiplication one time. Anyway. So, today, let's build some multiplicative reasoning with fraction multiplication and think about this idea of how we can bring scaling into it. 

Kim  06:07

Okie doke. 

Pam  06:08

Alright, Kim. So, the very first problem I'm going to give you is one-fourth of one-fifth.

Kim  06:12

Oh, okay. So, that's going to be one-twentieth. 

Pam  06:15

And if... Yeah, you can say more if you want. 

Kim  06:18

Do, I need to talk about it?  I mean, we...

Pam  06:19

I mean.

Kim  06:20

We did last episode.

Pam  06:21

Yeah. Listen to last episode. Okay, so if one-fourth of one-fifth is one-twentieth, then what might be... Or maybe I'll just say. Next problem. What's three 1/4s of 1/5.

Kim  06:40

So, one-fourth of one-fifth was one-twentieth. But I don't have just one-fourth. I have three 1/4s, so I'm going to scale from the one-fourth to the three-fourths times 3. 3 times as many. So, then instead of only one-twentieth, I have three-twentieths. 

Pam  07:02

Nice. Nicely done. And I wonder. When I said the problem, I might should have said I was going to write down three-fourths times one-fifth. Because I said if one-fourth of one-fifth is one-twentieth, what's three-fourths of one-fifth? And I wrote down three-fourths, 3 divided by 4 times 1 divided by 5. And I wonder if based on what you just said, I might write right under that. I might write 3 times with parentheses. So 3 "parentheses" one-fourth times one-fifth.

Kim  07:34

Mmhm.

Pam  07:35

And I wonder if that gives everybody kind of that scaling feel just a little bit.

Kim  07:39

Yeah.

Pam  07:40

It's one-fourth times one-fifth, but 3 times that.

Kim  07:45

Mmhm.

Pam  07:45

And you're saying that we can think of that as three-twentieths. Cool. So, another way that I could represent that three-fourths times one-fifth is I could have represented... Last time when we talked about one-fourth of the fifth, I kind of drew a candy bar, and we talked about an ant walking down one-fourth of the height, and then over one-fifth of the width. And we kind of had one-fourth of one-fifth was one sort of shaded rectangle inside the candy bar. And so, one-fourth of one-fifth is that shaded rectangle in top left hand corner, and now I have three-fourths of one-fifth, then I can kind of think about that one-fifth of the candy bar. But I don't only need one-fourth of it. Now, I need three 1/4s. And so, I could sort of picture 3 rectangles in that first column. 

Kim  08:39

Mmhm. Those little slivers. 

Pam  08:41

Yeah, can you tell me what you? Did you draw what I was?

Kim  08:45

I didn't, but I'm picturing it. So, that long candy bar we talked about.

Pam  08:49

Mmhm.

Kim  08:50

That we cut into fifths.

Pam  08:51

Mmhm.

Kim  08:52

That first fifth on the left side. Last week, we talked about just the one little sliver that was one-fourth of one-fifth.

Pam  09:00

Mmhm.

Kim  09:00

And now you have 3 of those slivers. 3 of the fourths inside the one-fifth. Three of 1/4s of the one-fifth of the candy bar.

Pam  09:11

And so, are your 3 slivers going down? 

Kim  09:14

They are.

Pam  09:15

Yeah. Okay, cool. Cool. Cool. Cool. So, we might have listeners going, "Yeah, that's kind of like if you cut the candy bar into fifths vertically, and you cut the candy bar into fourths horizontally, and you shaded 3." You could do that really procedurally, but you can also actually really think about if I know what one-fourth of one-fifth is. It's that little rectangle in the top left hand corner. But I need 3 of them. Then you really can just like scale and see those 3 rectangles now down that first column. Okay, cool. So, the next problem that I'm going to ask you is what is... Let's see. You just said three-fourths of one-fifth is three-twentieths. Now, I'm going to ask you for two-fifths... Actually, let me restate that problem. Sorry, three-fourths times one-fifth, you said was three-twentieths. I'm just going to restate that same problem that we just did. But I'm going to restate it as one-fifth of three-fourths. Is it okay if I use the commutative property a little bit there? 

Kim  10:15

Sure. 

Pam  10:16

So, you're saying that one-fifth of three-fourths. And I just wrote that down. One-fifth times three-fourths is three-twentieths. We just established that. The next problem is what is two 1/5s times three-fourths.

Kim  10:31

So, this is just I'm scaling again. Except instead of scaling the three-fourths, this time I'm scaling the one-fifth to two-fifths. So, three-fourths of one-fifths is three-twentieths. But now I need twice as many because I'm scaling times 2. So, it's going to be six-twentieths.

Pam  10:53

So, you scale the three-twentieths times 2, and that's six-twentieths. 

Kim  10:57

Yeah. 

Pam  10:57

Cool. And if you were to say that in class, I might go up to that same rectangle that I have on the board, where I had the one-fourth by one-fifth kind of shaded in. And then we we tripled that. We scaled it times 3, and so I had 3 of those rectangles going down that first fifth in the left hand side. But now you just said we need twice as many of that. So, I've sort of taken those three rectangles in that left hand column, and I've doubled it. I've scaled it. If you see my hand, I've kind of flipped my hand over like it's like I'm taking those 3, and it's like. And now, I have those 3 in the second column as well.

Kim  11:36

Mmhm.

Pam  11:36

So, now I have 6 total pieces out of the... Or, I have 6 pieces out of the 20 total pieces. 

Kim  11:44

Mmhm.

Pam  11:44

Which is the same six-twentieths that you just said.

Kim  11:46

Yep. 

Pam  11:47

But visually, here's what I didn't do. I didn't do what I've seen. And what I was shown to do is like cut it into fifth, cut it into fourths, shade this. shade this number of fourths, shade that number fifths, and then count 6 out of the total count 20. Instead, I really thought about that one-twentieth, and I scaled that times 3. And then I took those three-twentieths, and I doubled them because it was two 1/5s now, and so I just doubled those. Oh 3 doubled is 6.

Kim  12:16

Mmhm.

Pam  12:16

Bam. And so, two-fifths times three-fourths is six-twentieths. What do you think? 

Kim  12:23

I like it. I am sitting here thinking about if I'm not in the midst of a Problem String, how do I think about scaling? And I don't know if it's a good time to mention that. But I'll wait till we're done.

Pam  12:34

Are you sure? 

Kim  12:35

Yeah. Yeah, it's fine. 

Pam  12:36

Okay, cool. So, then let me ask you the next question in the string. What is one-third of one-fourth.

Kim  12:43

One-third of one-fourth. That would be a twelfth.

Pam  12:46

And can you give me some? Why?

Kim  12:50

If I have a fourth of a candy bar, and I cut that one... So, I have a whole candy bar. I have a fourth of the candy bar. Picturing that left side again. If I cut that one-fourth into 3 pieces, then I just have 1 of the 12 total pieces of the candy bar. 

Pam  13:05

Cool. And so, I could draw that. And I could kind of picture of this candy bar, I've got a one-third on the left hand side and a one-fourth across the top. And I've got, that's 1 piece out of the total 12 pieces. Cool. And so, what if the next question was two-thirds of one-fourth?

Kim  13:25

So, I double. From one-third to two-thirds, I have twice as many. So, then I'm going from one-twelfth to two-twelfths, which is actually a sixth.

Pam  13:37

And I wonder when you're at that two-twelfths if you could actually... Well, so if you were to say that in class, I would say, "Oh, so now you have twice as many." So,  have that candy bar drawn up there. I have that one rectangle in the top left hand corner. That's a third by a fourth. And then you doubled that. So, now I've shaded in the one right underneath it. So, that's two 1/3s by one-fourth. And I can clearly see that's 2 out of those 12 pieces. How can you see that that's 1 out of 6 pieces? Where are your 6 pieces that that chunk of 2 is 1 out of those. Does that make sense? 

Kim  14:16

I think I... I don't know that I understood your words, but I know what you're getting at. So, if 2 of those... So, I have them side by side. So, if 2 of my slivers now becomes 1 of the units, 1 of the amounts.

Pam  14:34

You're going to see it as a unit (unclear).

Kim  14:35

Yeah. 

Pam  14:36

(unclear) as a chunk.

Kim  14:37

There's only 6 of those now instead of 12. 

Pam  14:40

So, there's six 2 pieces. 

Kim  14:42

Mmhm. 

Pam  14:43

So, if you look at that candy bar, you can go circle two pieces, and there's 6 of them. The way my candy bar looks like I have to circle some pieces along the bottom to make them two pieces. But I can find a bunch of two chunkers, and I can find 6 of them. And you're saying we got 1 of those. Nicely done. Cool. So, you're saying that two-thirds of one-fourth is two-twelfths, which is equivalent to one-sixth.

Kim  15:10

Mmhm.

Pam  15:10

I'm going to say that slightly differently with using the commutative property. So, one-fourth of two-thirds, you're saying is two-twelfths, which is equal to one-sixth. Cool. 

Kim  15:21

Mmhm.

Pam  15:21

The next problem is, what is three-fourths of two-thirds? Three 1/4s of two-thirds. Because you just told me what one-fourth of two-thirds is. What's three 1/4s of two-thirds?

Kim  15:34

So, I'm just going to scale again from what I had before.

Pam  15:37

Mmhm.

Kim  15:38

I'm going to scale times 3. And so, I have one-sixth times 3, which is three-sixths. And so, that's a half.

Pam  15:49

And you know three-sixths is a half. Can you see three-sixth?

Kim  15:54

Oh, I totally abandoned my picture. Sorry.

Pam  15:55

That's okay. Because you're just scaling. 

Kim  15:58

Yeah, yeah. 

Pam  16:00

So, we totally could. We could go on that picture where we had the one-fourth of two-thirds being that one-sixth.

Kim  16:07

Mmhm.

Pam  16:08

And we could take that one-sixth, that two chunker, and we had 1 of the 6. And we could literally do 3 of them. And when we do 3 of them. On my picture, I've got 6 out of the total 12. Which is one-half. I've also got three 2 chunkers out of the total six two chunkers. Which is also one-half. And you can actually see that it's one-half of the candy bar because I've got one-sixth chunker out of the two-sixth chunkers that make up that 12 candy bar, or the 12 pieces of the candy bar. Cool. So, in a way, Kim, what I just heard you say is if you can think about one-third of one-fourth being one-twelfth, then when you think about two-thirds of a fourth, you can just scale that up times 2. That's just two-twelfths. 

Kim  16:56

Yeah. 

Pam  16:56

And if you know that that two-thirds... Or that one-fourth of two-thirds is that two-twelfths. Then you could also think about three 1/4s of that two-thirds is just scaling that up times 3. And so, you could take that two-twelfths and scale that up times 3. And 2 times 3 is 6, so that's six-twelfths. 

Kim  17:13

Yeah. 

Pam  17:14

And then you could, if you have to, simplify at the end. 

Kim  17:16

Yeah. 

Pam  17:17

So... Go ahead. 

Kim  17:18

Well, I was going to say. What I was going to mention earlier is that, you know, sometimes people will suggest that in a Problem String, "Well, yeah, Kim can follow what you're saying. Or a student can follow what you're saying because you're giving them the problems that that you know are the helper problems." 

Pam  17:33

You're leading them by the nose. You're like, Do this, do this, do this. Bam." Mmhm, mmhm. 

Kim  17:37

And I think what I'd like to suggest is that because I have built understanding, or if we would help kids build understanding of a unit fraction times the unit fraction. Like that's so within their knowledge base and their understanding. Then a student could look at a fraction multiplication problem like two-thirds times three-fourths, and they could say themselves, "What is a third time's a fourth? That's a twelfth. And I could scale up by 2. Scale up by 3. Or scale it by 6." And so, they can lean on the fraction understanding and scale for themselves. Double scale or scale all at once. But it still has a meaning to it. 

Pam  18:20

Yeah, you're actually suggesting that instead of looking at a fraction multiplication question and reaching for a rule. "Gal, which rule is this? Is that the one we did on Tuesday or is that the one we did on Thursday?" Yeah. Instead of reaching for a rule, they can say themselves, "What do I know? Well, I know what a unit fraction times unit fraction is, so I can find the total number of pieces. And then I'm just going to scale times the first numerator and scale times a second numerator." If they have if they've experience doing that, they literally think about fraction multiplication just like that. 

Kim  18:49

Yeah.

Pam  18:51

"Unit fraction times unit fraction? Bam, I know the total number of pieces."

Kim  18:54

Yep.

Pam  18:54

"Scale times the first numerator. Scale times the second numerator. Bam." And what they literally just thought through was multiplying the denominators. Isn't that interesting they multiply the denominators first to find the total number of pieces? And then they scaled times the numerator, scaled times the other numerator. They multiplied the numerators together. And they end up with multiplying the numerators divided by multiplying the denominators. But they've reasoned through it in such a way that now everything else is going to follow with more meaning from there.

Kim  19:25

Mmhm.

Pam  19:27

Yeah, it's so interesting. I wish I had learned to reason about fractions that way. Kim, I remember the day when I was working on writing Lessons & Activities for Building Powerful Numeracy, and I was doing some stuff with the area model for fractions. The kind of more memorized, rote county way, where I would divide the rectangle, divide the rectangle. Maybe I should say, divide the rectangle and shade. Divide the rectangle the other way and shade. And then I would look at the double shaded piece and count really. And I remember I was checking some work that I was doing, and I asked my kid. I was like, "Hey, how do you think about this fraction?" And he did kind of what we just talked about. And I was like, "Wait, what?" And then I called you. And I was like, "Hey, how would you think about this?" And you did something different. And I was like, "Mmm! Maybe I need to rethink a lot of..." I just was aware in that moment how little I had ever thought about, you know like, actually reasoning with fractions. The more I dove into reasoning about fractions, the more it influenced so much other work that I was doing. So many other things now I look at differently. Even the slope of a line I look at differently now that I can reason about fractions differently. 

Kim  20:48

Yeah.

Pam  20:49

It just impacts scaling everywhere in really, really interesting ways. 

Kim  20:55

Yeah. Alright, but we're not done. So, we've talked to unit fraction by unit fraction. Spent some time with scaling this week. And there's more. And this is when it gets really fun, so I hope you join us next week.

Pam  21:07

Yeah, I wonder if anybody's thinking right now, "But, Pam, what about simplifying those fractions? What about..." Well, you just join us next week. Let's have some fun with that. Alright, ya'll, thank you 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!