Pam  0:01  
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris, a former mimicker turned math-er. 

Kim  0:10  
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  0:17  
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  0:34  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:41  
We're so glad you're here to join us to make math more figure-out-able. Okay, so I have a funny story to start with. Funny? Is it funny? Maybe it's strange. I don't know what you'll think. Okay, so I had this radioactive iodine treatment after I had my thyroid cancer. It's a whole thing. And it killed a bunch of my salivary glands. So, you know that I don't produce enough saliva, which is delightful because it's a whole dental thing. All the stuff. But when I talk, I suck on stuff in order to produce more saliva. 

Kim  1:09  
Mmhm.

Pam  1:10  
And what I have found works the best are these manuka honey lozenges. But they're huge, so I cut them up into little, teeny pieces, and then I just suck on a little teeny piece. If you've ever been in a live presentation with me, you'll notice that I'll go about 10 or 15 minutes, and then I'll just quietly stick something. And so, as we are recording podcasts today, I have a bunch of... Well, anyway. I cut up a bunch of them. Kim, they come in different batches. This latest batch that I bought, I cut them all up. I put them in this mug that sits right next to my desk, so I can just like pull one out. They're stuck back together. Like, I can't get them apart. So, I'm like, digging in here with this pair of scissors trying to get the little tiny pieces.

Kim  1:49  
Wait, did they like stick back together?

Pam  1:51  
They stuck back together. Like, it's like this glob now. It looks like a hockey puck. 

Kim  1:55  
Was it like maybe humid in your house? 

Pam  1:58  
Probably.

Kim  1:58  
I bet. The weather's changing.

Pam  1:59  
Well, I think it's both. I think it's super humid. And I think that it's this batch, because I had a batch before that honestly, I didn't like as much because they disintegrated. So, then it was all these like dust. Like, come back together! Anyway, but these are like too together. Yeah, it's a whole thing. So, I think I finally have enough broken up. But my hand hurts now because it's like, yeah, it's like trying to break this hockey puck up of like, sharp. Like, they're sharp. You don't care. Okay. There, see. Little personal. Next time you get to share something stupid personal.

Kim  2:32  
Alright, I'll think of...

Pam  2:34  
Or you don't have to. 

Kim  2:35  
I don't care. 

Pam  2:36  
Ah, okay. Hey, I was thinking that it would be fun to do a Problem String that I've been doing lately. In fact, I did it with a group of fifth grade kids not too long ago, and I'm just having a ball with this particular Problem String.

Kim  2:49  
Yeah.

Pam  2:50  
And I'm going to make a point about it at the end.

Kim  2:53  
Okay.

Pam  2:53  
So,  if you don't mind, Kim. We're going to dive in. We're going to do a Problem String, and then we'll make sort of a point about what I love about this particular Problem String.

Kim  3:01  
Okay.

Pam  3:01  
So, here we go. Okay. So, Kim, picture a candy bar.

Kim  3:05  
Yep. 

Pam  3:05  
Okay. It's kind of a long, skinny candy bar. It's not like a fat, squatty candy bar. It's a long, skinny one. It's really long. And what we know. It's kind of weird what we know. But what we know is that three-tenths of that candy bar weighs 12 ounces. That's what we know. 

Kim  3:21  
Okay.

Pam  3:23  
Okay, so for whatever reason, we know three-tenths weighs 12 ounces. I'm super curious. Well, should I ask the question first? I'll tell you. With the younger the student that I work with, before I move on, I will say, "So, tell me about that. Like, where?" And I'll draw a big, long, skinny candy bar. 

Kim  3:41  
Yeah.

Pam  3:41  
And then I'll say like, "Help me think about three-tenths of this candy bar weighing 12 ounces. Can we find some tenths?" And then I might even ask kids like, "How would you cut this into tenths?" Kim, what do you think kids say when I say, you know, "How would I cut this into tenths?"

Kim  3:56  
They'd probably cut it in half first.

Pam  3:58  
Often. Often. There will at least be some kids that will cut in half. There are definitely a whole group of kids that will just start kind of estimating tenths, and they are relieved when one kid goes, "Well, can't we cut it in half, and then cut halves into...

Kim  3:58  
5 pieces.

Pam  4:00  
5 pieces. So, then that's not all that easy, right? You try to cut something into 5 pieces equally, but it's easier than cutting something into 10 pieces equally. And so, we'll kind of have that discussion a little bit where I cut the whole thing in half. And often I'll make that half line a little bit outside the candy bar, so you can kind of see that half line. And then I'll cut 5 pieces on the left, 5 pieces on the right. And I'll label 1 of those pieces as one-tenth. So, 1 of the pieces has one-tenth. But we know that three-tenths weighs 12 ounces. So, Kim, what could I put on this candy bar to sort of see those three-tenths? I mean, I'm kind of asking to read my mind here. I don't know if...

Kim  4:55  
I mean, like, what? You want me to label on the candy bar? I'd put 12 ounces over... Or maybe three-tenths above it and 12 ounces below where the three-tenths would be. 

Pam  5:06  
So, just kind of like just 3 of those pieces. And I kind of put a little brace below it to kind of encompass those three-tenths. And like, that's 12 ounces. So, I like it. Like, we got three-tenths kind of above those three-tenths, and then 12 ounces kind of below them. Cool. So, we know that. Now, with adults, I don't often do that much. I will just say, "If three-tenths of a candy bar weighs 12 ounces, then how much does one-tenth weigh?" And then I'll just ask that. And then if I need to, maybe I'll draw the model. I'm still kind of grappling with if I should have drawn it first or... Anyway. If three-tenths of the candy bar weighs 12 ounces, how much does one-tenth weigh?

Kim  5:39  
If I know three-tenths, so I'm trying to get one-tenth then I can divide three-tenths by 3 to get one-tenth. And I can divide 12 ounces by 3 to get 4 ounces.

Pam  5:50  
And so, what I have written right now on the board is I have three-tenths weighs 12 ounces. And underneath the three-tenths, I wrote one-tenth. So, when you said, "Divide by 3," I literally wrote like the scaling arrow from the three-tenths to the one-tenth, divide by 3. And then a scaling arrow divided by 3 from the 12 ounces. And I'm like, oh, what is 12 divided by 3? Sure enough, it's the 4 ounces. Kim just said. Cool.

Kim  6:12  
Are you on a ratio table right now? Is that what you just said? 

Pam  6:11  
I'm not. I could be. That's interesting. Kim, I have never done this on a ratio table. 

Kim  6:14  
Well, when you said scaling marks, I wasn't sure if you were. I wrote three-tenths is 12 ounces and one-twelfth is... 

Pam  6:27  
Yeah. 

Kim  6:28  
But I'm wondering (unclear).

Pam  6:31  
I have usually written kind of a sentence. Sorry to interrupt.

Kim  6:34  
No, it's okay. I'm just wondering, when you said scaling marks, I know what you mean, and I did the same thing. But I...

Pam  6:34  
Well, now you have me wondering if I should put in a ratio table. Okay, I'll try that next time. 

Kim  6:39  
Okay.

Pam  6:44  
Next question. So, we know what three-tenths weighs. We know what one-tenth weighs. I'd like to know what one-fifth of the candy bar weighs. What do you think? 

Kim  6:52  
So, I know that a tenth is half of a fifth, so I wrote times 2.

Pam  6:59  
You just said a half, and then times 2. That's confusing.

Kim  7:04  
Yeah. So, maybe I should say in the same direction. I know a fifth is twice as much as a tenth. So, I wrote times 2 from one-tenth to one-fifth. And I wrote times 2 from four ounces to 8 ounces. So, I have 8 ounces.

Pam  7:20  
How do you know a fifth is twice as much as a tenth?

Kim  7:24  
Because if I can't cut a candy bar into 10 pieces.

Pam  7:28  
Yeah.

Kim  7:29  
I know that my portion is one-tenth. But if I'm thinking about cutting something into fifths, then 2 of those tenths is going to fit into the size of one-fifth piece.

Pam  7:42  
So, this is often when I will draw a candy bar underneath the first one. So, I have a candy bar up there where we cut it in half, and we cut each of those pieces into 5. So, now I have the ten-tenths. But now, in that same size candy bar that's underneath the first one, I'll say, "Let's cut this guy into fifths." And as we cut it into fifths, I'm looking up and I'm trying to like make those fifth pieces line up with the two-tenths above them. And someone will say, "Huh, we could have actually done those tenths by cutting the candy bar into fifths, and then cutting each of them in half. 

Kim  8:16  
Mmhm.

Pam  8:17  
Sometimes somebody will say that at the beginning, though I'm finding not very often.

Kim  8:21  
Yeah.

Pam  8:21  
Not very often do people cut it into fifths, and then cut those in half. But now we can sort of see that relationship that you were just talking about. We can see those two-tenths in that one-fifth. Nice. Okay, so we got three-tenths weighs 12. A tenth weighs 4. A fifth weighs 8 ounces. What about three 1/5s of the candy bar? How much does that weigh? Three 1/5s?

Kim  8:40  
Well, we just talked about one-fifth, and I know that three-fifths is 3 times as much as one-fifth.

Pam  8:47  
Yeah. 

Kim  8:47  
So, I'm doing 8 ounces times 3 is 24 ounces.

Pam  8:53  
Brilliant. I wonder if you could also use the three-tenths to get to the three-fifths.

Kim  9:00  
Yeah, that's nice. So, three-fifths is going to be twice as much as three-tenths.

Pam  9:07  
And sure enough, there's that 12 ounces. Oh, sorry,

Kim  9:10  
No, 12 ounces doubled is 24 ounces.

Pam  9:13  
Nice. So, two different ways to get that 24 ounces. And also a nice way to say that three-fifths is 3 times 1/5 and three-fifths is double three-tenths. So, sort of really hammering. This is an example, Kim, of when we talk about, "Don't make equivalent fractions into an algorithm." Don't make it be, "Put the denominator down, this goes into that times that." Don't do that. Like, help kids really experience tenths and fifths and the relationship between them. And here's a way to do that. Cool. 

Kim  9:43  
Yeah.

Pam  9:43  
We're almost done. How about one-half of the candy bar, half of the candy bar, half of the candy bar.

Kim  9:49  
Mmhm. Well, my initial thought... I'm going to give you two. My initial thought was that I know that a half is equivalent to five-tenths. 

Pam  9:58  
Oh, nice. Mmhm.

Kim  9:59  
And so, I want to go from one-tenth to one-half and do times 5.

Pam  10:06  
Because it's like one-tenth to five-tenths.

Kim  10:09  
Mmhm. 

Pam  10:16  
Yeah, times 5, nice.

Kim  10:17  
So, 4 times 5 is 20. 

Pam  10:18  
So, 20 ounces. Okay, great. 4 ounces times 5.

Kim  10:20  
And I just lost what my other one's going to be.

Pam  10:22  
Oh.

Kim  10:22  
Shoot.

Pam  10:22  
I'm going to let you think. Take your time. Take your time.

Kim  10:24  
I can't remember now. It's going to bug me.

Pam  10:27  
Can I just mention? These are ones that kids have told me.

Kim  10:30  
Oh yeah, yeah.

Pam  10:31  
It's kind of fun. So, one kid said, "Well, we know three-tenths, and we're trying to get to five-tenths. And we know a fifth, you just said was two tenths. So, that three-tenths plus a fifth, the two-tenths. We could add the 12 ounces and 8 ounces.

Kim  10:46  
Ah, so they're thinking additively about that. That's nice. Yeah.

Pam  10:49  
Yeah, they were thinking additively. Yeah.

Kim  10:50  
That's perfect. 

Pam  10:51  
Another kid. And this might be my favorite. Oh, do you have another one?

Kim  10:54  
Did somebody do three-fifths minus one-tenth?

Pam  10:58  
No, but they use three-fifths in a different way. Can you be more multiplicative? Wait, wait, wait. Three-fifths? Is that what I want? No, hang on, that's not what I want. What do I want?

Kim  11:08  
We're trying to get to one-half.

Pam  11:09  
Oh, what was I thinking about? That's not... Yeah, yeah, yeah. Oh, it was what you said. Sorry. Yes, it was three-fifths minus a tenth.

Kim  11:17  
So, somebody was adding three-tenths and one-fifth.

Pam  11:23  
Mmhm.

Kim  11:23  
And then somebody was subtracting three-fifths minus one-tenth. Oh, that's nice. Good on those people. Yeah.

Pam  11:30  
Yeah. Yeah, very nice. Nice, nice. And again, you're really supporting and strengthening the relationship between fifths and tenths is very nice. Cool. Okay, last question. Nine-tenths of the candy bar. We almost got all of it, almost all of it. Nine 1/10s of the candy bar. How much does that sucker weigh? 

Kim  11:50  
I kind of cheated a little bit earlier. When we said one-tenth...

Pam  11:54  
Yeah? 

Kim  11:54  
...was 4 ounces. I couldn't help myself to say like what is the whole bar?

Pam  11:59  
Thank you for not saying it then. And I work really hard to not have anybody say that. 

Kim  12:05  
Yeah.

Pam  12:06  
Yet, yet. Until we actually get to this last one.

Pam and Kim  12:09  
Yeah. 

Kim  12:09  
So, I knew then that the whole bar was 40 ounces because a tenth was 4 ounces. So, in this moment, you know, because I'm such an Over person, maybe. I don't know. I wanted to do the whole bar was 40 ounces minus a tenth, which is 4 ounces. And I got 36 ounces. 

Pam  12:28  
Nice, nice. Yeah, so just a couple other things that kids will often do is that they will say... Actually, before I do that.

Kim  12:37  
Oh, man. 

Pam  12:37  
So once you have... 

Kim  12:38  
Yeah.

Pam  12:38  
What?

Kim  12:39  
Well, because you could go from one-tenth to 9.

Pam  12:42  
Times 9. 

Kim  12:43  
Yeah, but I also like...

Pam  12:44  
4 times 9 is not bad. Go ahead.

Kim  12:47  
But I also like the three-tenths to nine-tenths. I just saw that one. 

Pam  12:49  
And that one doesn't come up as... In fact, I think that was only come up once. And I don't usually think of that three-tenths to nine-tenths, so I love that one. Three-tenths times 3 and 12 ounces times 3.

Kim  13:05  
Ooh, I like this string. 

Pam  13:05  
It's brilliant because there's so many really nice connections. 

Kim  13:05  
Yeah. 

Pam  13:05  
And let me just... Once you find that whole candy bar is 40 ounces, then you can kind of go back and say, so if the whole candy bar is 40 ounces, does it make sense that a tenth of it is 4 ounces? Sure enough. And that 3 of those would be 12 ounces? Sure enough. And that a fifth... So, now I can think a fifth of 40 is 8 ounces. Sure enough. And so, 3 of those is 24. Half of 40 is 20. Bam, like all of it. It's just kind of a really nice series of problems. Kim, here's the point that I want to make with this string. Well, no. We've made several points. But here's a point I want to make. What algorithm might teachers have been tempted to or kids are like stuck in using to solve these problems? You know, we really have to work hard to get kids not to use that algorithm for these problems. Like, does it exist? And everyone shake your heads and say no. Like, there is no traditional algorithm to solve these problems. You have to actually reason your way through these. That's brilliant. And I'm going to suggest that as soon as you get out of the four operations, that is true. Like, it is true that there are going to be problems that there's just not this method that we want to teach kids. Well, I don't. There's not this method that we have historically taught kids. I'm trying to find problems like that to create strings like this to in part send the message it's not about one and only one way. It's not about memorizing what somebody has created before you. It's about thinking, and reasoning, and developing relationships. And think about all the relationships we just strengthened by doing this particular Problem String. And we did the string with a group of fifth grade kids. I did two of them in a row. First one was rough! Like, we were really getting out what it even means to have a unit fraction of a whole thing, and then have 4 of those or whatever. Like, we really did a lot of that work. But boy. Then we did a Rich Task with them, and then we did a second Problem String like this. And man, that second one, we had some great. Like, y'all, it's not that you do once, and if kids are instantly there or not there, oh, we failed. Or, you know like, they're done. No, neither of those. It is a trajectory. It's a landscape. It is a continual building and strengthening. That's how we're advocating that we make math more figure-out-able. We teach math. What do you think? 

Kim  15:17  
Love it. Love, love, love. 

Pam  15:19  
Sweet. 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!