[00:00:00.350] - Pam
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris, a former mimicker turned math-er.

[00:00:10.870] - Kim
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.

[00:00:19.190] - Pam
Y'all, 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.

[00:00:33.060] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:39.940] - Pam
And we invite you to join us to make math more figure-out-able. Alright, Kim, it's been a minute since we've recorded.

[00:00:48.040] - Kim
Yeah.

[00:00:48.700] - Pam
We put podcasts out every week, but we batched a few lately. And today, man, I feel like it's like slogging through, I don't know what. Just like. Okay. What do we do here?

[00:01:00.260] - Kim
Reminder, reminder.

[00:01:01.280] - Pam
Who are we? That might have been the most boring intro we've ever recorded. I don't know. I felt like it was just...

[00:01:06.120] - Kim
Well, I had to tell you that you... I could hear you move your stand-up desk when I was talking.

[00:01:11.750] - Pam
Totally. I waited though, right? I waited until you were talking. Yep. Yep.

[00:01:15.500] - Kim
It distracted me for a second, so I might have mumbled more than usual. It's alright. It's all good.

[00:01:21.530] - Pam
Hey, when I just present standing up, it's at one height. Like today, I want the microphone closer. And anyway, it's a different height when we do podcasts. I forgot that. So, it was like, "Oh, up, up." Sorry, Craig. Yeah, Craig, our podcast editor just messed with that. Nobody else could hear it. They're like, "What? I didn't hear it."

[00:01:43.110] - Kim
I did.

[00:01:44.110] - Pam
Okay, let's dive in. Let's pretend we know what we're doing.

[00:01:47.290] - Kim
Alright, alright. Okay, so a while back, we did a Problem String on the podcast that...

[00:01:53.160] - Pam
Like, one? Like one, right?

[00:01:54.450] - Kim
One.

[00:01:55.550] - Pam
We've never done...

[00:01:55.650] - Kim
We did a specific Problem String.

[00:01:57.480] - Pam
Alright, alright. Okay.

[00:01:58.280] - Kim
That got a lot of people talking. And it was a great string, and it made people think. It made like the listeners think, and I heard from several people who said, "I cannot wait to do this in my classroom.", But also, it was because you posed some questions for which there is no traditional algorithm that you can use to solve it. So, it's just different.

[00:02:18.350] - Pam
Yeah.

[00:02:18.410] - Kim
And so...

[00:02:19.460] - Pam
Yeah.

[00:02:19.710] - Kim
Let's do some more.

[00:02:21.330] - Pam
So, these are fun, right? Because especially as a high school teacher, boy, I remember so strongly. Or I have a strong memory, I should say, that if my kids saw... Not all, but many of the students would see a fraction, and a fraction, and then something in between, an equal sign, a multiplication sign, addition, subtraction, division. Like, it didn't matter what was in between them. They would like just flip a coin and start applying one of those rote-memorized rules. They'd find common denominators, or invert and multiply, or cross cancel, or whatever, all the things. Rather than like try to reason through the problem. And we've just done so much work about how to reason that when I found these and realized this could be a string that really there's no like set procedure to do, that you kind of have to like reason through them, I was kind of on fire. And yeah. So, let's do another one because people asked for it, and what you asked for you might get, so keep asking. Okay. Kim, you eat... Well.

[00:03:17.500] - Kim
Everyday.

[00:03:17.590] - Pam
Do you eat?

[00:03:19.760] - Kim
I love to eat.

[00:03:22.160] - Pam
Do you ever eat protein bars? Is that a thing? Like, as a snack or as a breakfast meal replacement? No?

[00:03:28.070] - Kim
No, not so much. But like a granola bar or something. We'll go with it.

[00:03:32.660] - Pam
A granola bar.

[00:03:33.550] - Kim
I do eat bars sometimes.

[00:03:35.930] - Pam
Alright. Well, a bar. We'll just talk about a bar. It could be a candy bar, could be a granola bar, could be a protein bar, could be something like that. I remember when you and I were in a store once and we were comparing protein amounts. Yeah. I remember that distinctly.

[00:03:47.760] - Kim
I mean... Anyway, okay. I think some are better than others, but... So, if I'm going to have one, I'm going to like. Yeah.

[00:03:53.910] - Pam
Well, well done, well done. Okay, if I have this bar. And we're like, we're looking because we're curious.

[00:04:01.340] - Kim
Yeah.

[00:04:01.810] - Pam
And for whatever reason, what we know is that two-fifths of that bar weighs 8 grams.

[00:04:08.800] - Kim
Okay.

[00:04:09.510] - Pam
Can you picture that?

[00:04:10.250] - Kim
I'm writing that down.

[00:04:11.430] - Pam
Yeah, yeah.

[00:04:12.060] - Kim
I'm assuming you're probably going to ask me questions about that. Two-fifths of the bar weighs 8 grams. Okay.

[00:04:16.580] - Pam
Yes. Two-fifths of the bar weighs 8 grams. If two-fifths of this bar weighs 8 grams, what would one-fifth weigh?

[00:04:26.100] - Kim
One-fifth is going to be 4 grams.

[00:04:28.550] - Pam
Because?

[00:04:29.520] - Kim
Because it's half as much, half of the bar. If I know two-fifths is 8 grams, then half of the fraction, half of the amount of bar is going to weigh half as much.

[00:04:43.760] - Pam
Because half of two-fifths is one-fifth.

[00:04:45.930] - Kim
Mmhm.

[00:04:46.890] - Pam
So, if I have two 1/5s, half of that's one 1/5, and half of 8 is 4. Cool.

[00:04:51.600] - Kim
Yeah.

[00:04:52.110] - Pam
So, if I was drawing this on the board, I would have two-fifths weighs 8 grams, and then I would put kind of a scaling arrow divided by 2. Two-fifths divided by 2 is one-fifth, and 8 divided by 2 is 4 grams. Cool. What would one-tenth of the bar weigh? Same bar, same bar.

[00:05:08.220] - Kim
I'm going to half it again because...

[00:05:10.770] - Pam
Okay.

[00:05:10.820] - Kim
So, I'm going to get 2 grams.

[00:05:12.260] - Pam
You're just in a habit? You're just like, I might as well just keep...

[00:05:14.680] - Kim
Half things, yep. Because two-tenths makes a fifth.

[00:05:19.200] - Pam
Two-tenths makes a fifth, so half of a fifth is a tenth. So, if you half the fifth, that's a tenth. And half the 4 grams, that's?

[00:05:26.540] - Kim
2 grams.

[00:05:27.190] - Pam
2 grams. Okay. So, so far we have two-fifths of the candy bar, weighs 8 grams. Or candy, protein, whatever. Two-fifths of the bar weighs... I can't say "weighs". Weighs 8 grams. One-fifth weighs 4 grams. One-tenth weighs 2 grams. Might be some sort of pattern happening here. What about one-half of the bar? Love to hear about that.

[00:05:48.130] - Kim
Well, the first thing I'm thinking about is I know a half is five-tenths.

[00:05:53.590] - Pam
Mmm, nice.

[00:05:54.630] - Kim
So, I'm going to scale times 5.

[00:05:57.210] - Pam
Next to the one-half, I just wrote five-tenths. I'm sorry for interrupting.

[00:06:00.800] - Kim
No, it's okay. Mmhm, yeah. So, then I'm going to scale from the tenth of 2 grams times 5 to get 10 grams.

[00:06:08.270] - Pam
Nice. So, if a half is 5 times bigger than a tenth, then 5 times more than 2 grams is the 10 grams.

[00:06:17.310] - Kim
Mmhm.

[00:06:17.330] - Pam
I wonder if you just wanted to play a little bit. Like, I love your strategy. It's very nice. If you wanted to play a little bit, could you go from the one-fifth to one-half?

[00:06:26.070] - Kim
Yeah, I could think about two and half 1/5s is the same thing as a half.

[00:06:34.800] - Pam
Okay.

[00:06:36.090] - Kim
So then...

[00:06:38.290] - Pam
How do you know that?

[00:06:41.030] - Kim
Because half of 5 is 2 and a 1/2. So, (unclear).

[00:06:46.750] - Pam
Nice, okay.

[00:06:47.630] - Kim
Okay.

[00:06:48.510] - Pam
So, 2 and a 1/2 is half of 5. So, two and a half 1/5s is equivalent to 1/2. Okay.

[00:06:53.880] - Kim
Yeah. Sorry, I'm distracted because you asked me about a fifth, but I wanted to talk about two-fifths. So, let me get focused.

[00:07:00.290] - Pam
Oh, okay. That's interesting.

[00:07:02.450] - Kim
Okay, so one-fifth. You asked me could I use the one-fifth? So, yeah, I could do two and a half 1/5s. Two and a half 1/5s. Which would be the 4 grams twice. And then half it again would be 2 more grams. So, 4, 4, and 2. It's kind of additive, but...

[00:07:21.120] - Pam
4, 4, and 2.

[00:07:22.248] - Kim
A fifth, a fifth, and a half of a fifth.

[00:07:22.320] - Pam
And that's how you got your 10 grams?

[00:07:25.350] - Kim
Yeah.

[00:07:26.110] - Pam
Okay, cool. So, we could either 5 times the tenth, which is 2 grams times 5 is 10 grams. Or you could 2.5 times the fifth, which is 4 grams times 2.5 is 10 grams. And did you say you wanted to go from the two-fifths to get to one-half?

[00:07:40.770] - Kim
Yeah, because...

[00:07:41.250] - Pam
That's close to two and half 1/5s.

[00:07:43.100] - Kim
Yeah, but I... The two-fifths to me... Like, it's one of those percents that just screams like I see two-fifths as 40%. And so, when you asked about a half, then I think 50%. So, if 8 grams is 40%, then what would 50% be?

[00:08:01.390] - Pam
Nice.

[00:08:01.900] - Kim
So, yeah. Then if I know 40% is 8, then 50% would be 10.

[00:08:08.000] - Pam
Very nice. I thought what you were going to do was say that since you knew the two-fifths was 8 grams and we already had the tenth.

[00:08:16.240] - Kim
Mmhm.

[00:08:16.390] - Pam
The tenth is a half of a fifth.

[00:08:18.240] - Kim
Mmhm.

[00:08:18.450] - Pam
Right? You said that already.

[00:08:19.820] - Kim
Mmhm.

[00:08:19.920] - Pam
So, the two-fifths and a tenth is a half. Therefore, the corresponding 8 grams and 2 grams would be the 10, the corresponding 10 grams.

[00:08:28.390] - Kim
Mmhm, mmhm.

[00:08:28.510] - Pam
But you're just so, percent oriented. That's interesting. You can just probably see that 40. Or that... How did I say that? That 8. 40% is 8, so 50% is 10.

[00:08:40.210] - Kim
I mean, certainly easier than I can picture the two-fifths and a tenth together. Yeah.

[00:08:47.190] - Pam
Okay, okay. Well, especially if you had to come up with it. It's sitting right in front of me.

[00:08:50.720] - Kim
Yeah, yeah.

[00:08:51.680] - Pam
Yeah, nice. Okay, cool. What if we wanted only to eat a fourth of the bar? I wonder how many grams we would eat.

[00:08:59.720] - Kim
Mmhm. Well, I'm glad we just did a half because now I can half that. Okay. So, a fourth would be 5 grams.

[00:09:08.260] - Pam
Nice. Cause half of the 10 grams. Could you go from anything else just for fun? Not as easily, for sure.

[00:09:17.950] - Kim
Yeah. I mean, I actually find myself looking at the grams, so it's probably cheating a little bit.

[00:09:25.080] - Pam
Oh, like how could you get from something to 5?

[00:09:27.860] - Kim
Yeah. Yeah.

[00:09:28.960] - Pam
Yeah, that's cheating.

[00:09:32.020] - Kim
Fourth. A fourth, a fourth, a fourth.

[00:09:35.120] - Pam
I mean, you just said a fourth was...

[00:09:36.360] - Kim
I'm trying to think about how many tenths would be a fourth. Two and a half 1/10s?

[00:09:39.800] - Pam
That's kind of what I was... Yeah, I was kind of wondering. Two and a half 1/10s because two and a half 2 grams.

[00:09:46.800] - Kim
Mmhm.

[00:09:47.440] - Pam
That's also 5.

[00:09:47.740] - Kim
Yeah. So, I feel like it's the same relationship from... We have two and a half 1/5s was a half, so two and a half 1/10s would be a fourth. It's like we're doubling and we're halving from a fifth to a tenth and we're halving from a half to a fourth. So, there's this kind of a nice relationship between those.

[00:10:07.200] - Pam
Yeah, that's... Yeah. And then we got kind of some practice multiplying something times 2.5.

[00:10:12.560] - Kim
Yeah.

[00:10:13.370] - Pam
Yeah. Like, the 4 grams times 2.5 or the 2 grams times 2.5. Nice. Okay, cool. Alright, last problem. How about nine-tenths of the bar? How much would nine-tenths of the bar weigh?

[00:10:24.360] - Kim
Nine-tenths of the bar.

[00:10:26.090] - Pam
In at least 2 ways.

[00:10:29.050] - Kim
Mm-hmm. Well, I haven't figured out what the whole bar is yet.

[00:10:32.890] - Pam
Yeah. Nice.

[00:10:34.670] - Kim
So, I could. I could figure out the whole bar.

[00:10:36.920] - Pam
Okay.

[00:10:37.930] - Kim
If I know half a bar is 10 grams, then the whole bar, I'm going to put a 1. The whole bar is 20 grams.

[00:10:45.280] - Pam
Nice.

[00:10:46.100] - Kim
And then I can back up the tenth of a bar.

[00:10:48.670] - Pam
Because we know a tenth of a bar is 2 grams. Mmhm.

[00:10:51.650] - Kim
Yeah, so that would give me 18 grams.

[00:10:53.690] - Pam
Nice. You can find ten 1/10s of the bar. Get rid of one-tenth.

[00:10:57.260] - Kim
Mmhm.

[00:10:58.300] - Pam
20 grams minus that two-tenths.

[00:10:59.490] - Kim
Ooh, you know what I just saw?

[00:11:01.270] - Pam
Hmm?

[00:11:01.770] - Kim
Remember, I was talking about percents? So, I had two-fifths is 40% and a half is 50%.

[00:11:06.950] - Pam
Nice.

[00:11:07.070] - Kim
So, now you're asking me about 90%.

[00:11:09.990] - Pam
Very nice. I like that a lot. So, two-fifths and a half. But you were thinking about that as percents, which gets you to.. I don't know that if you would have said two-fifths and a half, if I would have said 90%. But if you say 40% and 50%, bam, that's obviously 90%. Nice.

[00:11:27.000] - Kim
Nice. Man, I like these problems.

[00:11:28.840] - Pam
They're fun, aren't they?

[00:11:30.710] - Kim
Yeah.

[00:11:30.870] - Pam
And part of what's interesting about them, I think, is that so many multiple relationships and really like thinking about how a fifth was related to two-fifths and a tenth is related to a fifth and how a half is related to two-fifths, a fifth, or a tenth. Just all those relationships. I don't know. I think everyone I've done this with kind of gets freed up to play a little bit.

[00:11:53.950] - Kim
Yeah, for sure.

[00:11:55.100] - Pam
Maybe even more than other Problem Strings that we do. They, start to just feel playful. It was interesting that you did not. Let's talk about the whole bar for a second. Every once in a while, I'll work with somebody who, like, from when we know the very first one. Two-fifth weighs 8 grams. Well, it's one-fifth. Sometimes they'll say, "Well..." They'll figure out the one-fifth there, and then they'll like check it. They'll say, "Well, if one-fifth is 4 grams," which you got, "then times 5 to get five-fifths." So, 4 times 5. And instantly they kind of have the 20 grams. If they do that, I try really hard to not have them share that yet. Because once that 20 grams comes out and then I say," "Well, how about one-tenth?" Then they tend to just start doing stuff like finding a tenth of 20. And then I say a half, and they find a half of 20. That's not bad. You know, we can definitely do it. But boy, I try to really focus the conversation around the relationships between everything else, and then ask people, "Well, no matter how you did it, you know, is there another relationship up here that we could have used?"

[00:12:54.360] - Kim
Yeah.

[00:12:54.920] - Pam
Notice that that question is a little bit different. "Is there another relationship up here that's true or that we could use?" Rather than, "Okay, you have to solve it two ways to get a grade." Like, I don't know. There's something about being more playful instead of like commanding that somebody solve something two ways. Here, we're looking for relationships and playing rather than commanding multiple ways.

[00:13:16.840] - Kim
Yeah.

[00:13:17.040] - Pam
Yeah.

[00:13:17.240] - Kim
Well, and we're getting a lot of different relationships out. So, like you said, we're doing the messing with times, two and a half of something.

[00:13:25.420] - Pam
Mmm, mmhm. Especially in this one. Yeah.

[00:13:25.610] - Kim
And, you know, we're thinking about the fractions, like how they relate to each other. And, you know, this one we got to pul pull in some percents conversation. So there's a lot of different things happening. And once somebody finds the whole bar, then it could very quickly become just using the operator meaning of fractions...

[00:13:43.520] - Pam
Nice.

[00:13:43.970] - Kim
...when you say two-fifths of whatever 20. And one-tenth of 20.

[00:13:46.630] - Pam
Yeah, so just find a tenth of 20. Yeah, just find a fourth of 20.

[00:13:49.440] - Kim
It becomes a little more repetitive of the way you're going about it is the same.

[00:13:53.600] - Pam
Yeah, that's a nice point. So, one other thing that I'll just mention briefly. When I do this Problem String with kids, which I have, I think the very first time I did it, I drew a candy bar right away, and I said if two-fifths weighs 8 grams, what do we know? And so even before I said, what's 1/5? I just said, you know like, let's... Here's the candy bar. I drew a really long candy bar. And I was like, two-fifths weighs 8 grams. What do we know? And they kind of together, we sort of cut it into fifths. And then I said, but we don't know one-fifth weighs 8 grams. We know two 1/5s. So, then I kind of put a bracket above two of the fifths to kind of say that's the two-fifths, and how much does that weigh? And then below that, I kind of put another bracket. So, you could kind of see I've sort of bracketed these two 1/5s. And that's two-fifths, and it's also the 8 grams. Then when I said, "How much does one-fifth weigh?" a lot of kids were able to go, "Oh. Well, one-fifth." You know, we've already sort of described that when we cut the bar into fifths, we created fifths, right? So, they were already kind of... I don't know. I felt like I'd kind of done some of the thinking for them. Not maybe, maybe we did it kind of together, but it was almost too obvious if I did it right off the bat. So, what I've started doing is asking the question, and then circulating and hear what kids are doing. If kids are like, "I don't even know what you're asking," man, we start drawing candy bars. But if kids are like, "Okay, if two-fifths..." and, you know, they can do some reasoning at all, then I will tend to get those kids to start sharing and then I'll draw the candy bar.

[00:15:27.440] - Kim
Yeah.

[00:15:28.060] - Pam
Like, as they're sharing, so that I can make sense of what they're sharing. But if they're like you, I don't necessarily ever draw the candy bar.

[00:15:36.750] - Kim
You know, it's funny that you say that though because when you posed the problem, I drew a rectangle because... I mean, I didn't do anything with it after that. But I drew the rectangle, I cut it into fifths, I didn't write any other numbers or anything, but it was like, "Okay, here's the context."

[00:15:54.820] - Pam
Kind of anchored you.

[00:15:55.790] - Kim
It was like, that's the context. I never put any numbers, I didn't split it up any further later on. But it just, for whatever reason for me, like, established the context. So.

[00:16:06.840] - Pam
And to be clear, when I said I wouldn't draw the candy bar, I meant as the teacher. I'm only going to draw it when I feel like it's going to be helpful for most of the kids. Or enough of the kids, or the teachers that I'm working with. Then I'm going to draw it and I'm going to draw it at that point where I'm like, "This would help us." And then I'm only going to draw just as much as we need.

[00:16:29.220] - Kim
Right, right.

[00:16:29.440] - Pam
So, I think this is delicate balance between doing too much work up front, doing... What do we say? Doing it just in case they need it rather than just in time when they need it. So, I would just encouraged...

[00:16:42.060] - Kim
I would rather just in time.

[00:16:42.930] - Pam
Yeah, I would just encourage everybody. I think with this Problem String, if you know your content, know your kids, you kind of play with how much you write down when. This might be a good example that I might not write very much on the board, but as I'm working with individual kids, I might sketch out what they're saying to kind of help. Anyway. Okay, cool. So, then somebody brought up in our teacher coaching group that we have" one of the questions that they asked in one of our monthly coaching sessions was they saw a video of me doing this in our Building Powerful Fractions 2 workshop. And they said, "Hey, why didn't you put it on a ratio table?" And that's an interesting question. And Kim, do you want to, do you want to talk about that a little bit? What do you think?
Like, could we?

[00:17:24.290] - Kim
Could we describe what you normally do? So, what I did was write the sentence, two-fifths of the bar weighs 8 grams.

[00:17:31.690] - Pam
So do I.

[00:17:31.980] - Kim
And so then.. Okay, and then as you asked more, I put the fractions underneath the fractions. And then over, several words away, under 8 grams, then I just wrote the numbers that I was coming up with.

[00:17:45.490] - Pam
Mmhm.

[00:17:45.650] - Kim
So, I remember when you were filming this for the workshop. And even in that moment, you know, there was you did some sort of erasing at one point and I heard you say, and I think we have it on film. You said, "I wish these numbers weren't so far apart." And so, you kind of scootched them a little closer.

[00:18:00.600] - Pam
So many words in the middle. Yeah, mmhm.

[00:18:02.401] - Kim
Yeah.

[00:18:02.680] - Pam
Yeah.

[00:18:02.770] - Kim
So, you scootched them a little bit closer, but you continue to just record the fraction and the number.

[00:18:08.800] - Pam
Mmhm.

[00:18:09.010] - Kim
So, that's why the question came up about why not a ratio table.

[00:18:12.720] - Pam
So, what could this look like in a ratio table? I think it could look like on the left-hand side, we could put the sort of the fraction of the bar or the two-fifths. And on the right-hand side, we could put the 8 grams. So, how would I label that? Like, fraction of the bar on the top left and then grams or weight on the right? So, it could be two-fifths to 8. And then underneath the two-fifths, I could have written 1/5. And one of the questions was, you know... Well, I think you and I discussed would we do that the first time we did this Problem String? And I think I'm leaning towards no. I'm still a little unclear why I'm leaning towards no. But I think you agreed with me. Do you have any words around why you might not do that the first time?

[00:18:56.400] - Kim
I think one of the things we talked about was that when it's set up in a ratio table, if kids have already seen a ratio table, if people have already seen a ratio table, then it kind of suggests to them a thing to do is go back and look for other relationships. And we do want that to happen here.

[00:19:15.590] - Pam
Mmhm.

[00:19:15.700] - Kim
But it's the same reason why I sketched the bar first is that once you're establishing the context, this is so different and so unusual for so many people that...

[00:19:26.700] - Pam
This context, this way of asking this question.

[00:19:29.220] - Kim
Yeah, this type of problem. That their brain could be going in lots of different ways about the way that they want to solve it, and so I think not putting in a ratio table gives a little bit more freedom to how they might want to model. And the ratio table is just a little bit more narrow of a nudge. But that's not to say we would never do it on a ratio table.

[00:19:50.200] - Pam
Right. And I think what I kind of thought was fun was that the people that were asking, "Could you put on a ratio table?" were some of our more advanced people in our coaching group.

[00:20:01.020] - Kim
More seasoned. Yeah.

[00:20:01.570] - Pam
Yeah, more seasoned. Well, not just seasoned, but also I would say advanced, like sophisticated. Like, they know a lot. They've been learning a lot. They've been doing a lot. And so, for them to recognize this is a proportional relationship, that's brilliant. Like, I'm just... That made me smile as kind of like a, I don't know, parent moment of like, "Oh, look at the little kids growing up." I don't know, I hope that doesn't sound... I don't mean that to sound arrogant. It was just kind of that mentor moment. Oh, look at the mentees like being so cool. It was nice. Good moment. Alright, kind of fun. Why would we want to pose these kinds of problems to students?

[00:20:35.130] - Kim
Yeah, yeah. These are the kinds of problems that don't feel super obvious, like you said earlier, where you would pluck two numbers, put some sort of operation to work. This is really can feel to a lot of people like a problem to solve, to engage in, to wonder about, to go, "Wait, if I know this, then do I know this?" And you're asking the next question. But like when you asked one-half, I couldn't just go straight to the fraction one-half. I had to think of it in terms of what could I think about that's equivalent to a half to get to your answer? And so, there's a little bit more playfulness.

[00:21:18.990] - Pam
And so many relationships involved, right?

[00:21:21.480] - Pam and Kim
Yeah.

[00:21:21.960] - Pam
Connections and actually thinking about the fractions, not just doing something.

[00:21:26.540] - Kim
Yeah.

[00:21:27.460] - Pam
Sorry for interrupting you.

[00:21:28.640] - Kim
No, it's okay. I mean, just these are the kinds of things that we want to engage students in that really make them think hard about how they want to go about problems.

[00:21:35.850] - Pam
And this is a particular one that it just kind of naturally does that. Which is nice. We don't have to kind of snooker kids into it. Yeah. Cool.

[00:21:42.520] - Kim
Yeah.

[00:21:42.920] - Pam
Alright, well that was fun. We might do it again someday. 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.