Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

And we're here to suggest that mathematizing is not about mimicking or rote memorizing. But it's about thinking and reasoning, about creating and using mental mathematical relationships. That mathematics class can be less like it has been for so many of us and more like mathematicians working together. We answer the question: if algorithms are not your end goal, then what is?

Kim Montague  00:35

Yeah, so we've seen a lot of different ideas out there about how to learn fractions. And two weeks ago, we started a series, and we talked about five interpretations of rational numbers. And last week, we talked about how, with young learners, we believe that the best way for kids to gain intuition or get a feel for fractions is to gain experience about them.

Pam Harris  00:59

Absolutely. So Kim, we're going to exemplify this idea today, by an experience that you had with your sons. You have two sons.

Kim Montague  01:09

Yep. 

Pam Harris  01:10

Tell us about that experience. Go. 

Kim Montague  01:11

Okay. So a couple years ago, it was probably five years now.

Pam Harris  01:16

Wait a couple is five? 

Kim Montague  01:17

Okay, five years. Um, that's probably the word that I say incorrectly all the time. Couple is probably what, two? Anywho. I just found this video. And it was precious because they were five years younger. And I just watched it again. But here's the story. So I had these little bundt cakes, these little kind of like bundtinies in our freezer, and there was a chocolate one and a vanilla one. And so I wanted to pull them out. And I wanted to share them among Luke, and Cooper and myself. And so I pulled them out. And I called Luke over and I said, "Hey, Luke, I've got these two bundt cakes. How much would we each get?"

Pam Harris  01:55

Only you would do this, by the way. But more of us should think about this. I don't know that I ever said how much like we would just eyeball it. Anyway, so super. Keep going.

Kim Montague  02:06

So he had just turned eight. It was the summer and he had just turned eight like the week before. And right away he said two thirds. And I said, "How do you know that we're each going to get two thirds?" And so we dove into his thinking. And he was thinking about splitting each of those cakes into thirds. So the chocolate went into 1/3, 1/3, 1/3 and the vanilla one into 1/3, 1/3, 1/3 and dealing out a third of the chocolate to each of us and a third of the vanila to us, right? That makes sense. And it was a fine splitting into unit fractions strategy.

Pam Harris  02:38

Okay, I have to pause for a second. I actually had to think about that. So they're in thirds, there's two of them, and they're in thirds, I get a third of chocolate and I get a third of vanilla. You get a third chocolate, you get a third of vanilla. The other kid gets a third. Okay. And then they're all gone. 

Kim Montague  02:50

Yeah, yep. 

Pam Harris  02:50

Okay, that works. Alright. 

Kim Montague  02:51

But then he said, "Mom, I only like the chocolate one."

Pam Harris  02:57

A kid after my own heart.

Kim Montague  02:59

"I only like the chocolate one, and you can't have the chocolate one. So that's going to be a problem." That's like problematic. And it was so fantastic. Because then it was a great problem for us to have, right? It made us like think a little bit further. And it led him to try to figure out how we could fair share the cakes so that we each get two thirds, but also take the flavors into consideration. And frankly, what he wasn't thinking about at that time was let's just trade like, it was a new scenario. He went back to the idea that there's these two whole bundt cakes, and I want to cut them so that we each get a fair share. And honestly, in this video that I watched, you know, when he was eight, he tried to give some away. He was like, let's just give some to dad. And he tried to say that he didn't need as much.

Pam Harris  03:47

Wait, wait, like if he could just give some to dad and then it was easier to share fairly? 

Kim Montague  03:51

Yeah. It was like a new problem for him. Right? 

Pam Harris  03:55

Interesting.

Kim Montague  03:55

It was like a new thing to solve. And he tried to say, "Well, I don't need as much. So like, if you give me a chocolate, maybe I'll take a smaller portion." And eventually, he did settle on him getting two thirds chocolate. Me getting two thirds vanilla and Cooper getting 1/3 of each. 

Pam Harris  04:13

Nice.

Kim Montague  04:14

So he kind of worked through that. And interestingly he said that wasn't equal for each of us, until he considered it again. At first he said, "Oh, I'm not really sure if that's equal." And I had him talk through what he had just said, "Two thirds chocolate for me, two thirds vanilla for you, Mom, and 1/3 of each flavor for Cooper." And it was such an interesting conversation.

Pam Harris  04:39

And I just want to point out part of what is so noteworthy about the story is that -listeners I don't know if you caught it, but Kim was like, I mean obviously he could have just traded, but for him it was a new problem and everything. So in the moment you were clear, dude just trade and we're good. But you let him do the considering. You let him do the grappling. This is what we mean by productive struggle. He's interested in it, she can go with it. Even if you didn't sort of know the outcome, it would be worth just kind of letting everybody sort of grapple with the question. Nicely done.

Kim Montague  05:12

And wanted bundt cake, right? He was motivated.

Pam Harris  05:15

Alright, you have two kids. So what was Cooper doing?

Kim Montague  05:17

Yeah. So Cooper, you know, Cooper was hanging out because he wants bundt cake and Luke said -

Pam Harris  05:24

And to be clear, Cooper's younger. 

Kim Montague  05:26

Well, yes. So Cooper had just turned five. It was like, you know, they have really close birthdays, he had just turned five like, two days before or something. And Luke walked away. And it was super cute and super fun for me, because he jumped in the conversation. And he said, "I have an idea." And when Luke walked away, Cooper jumped up on the counter. And he talked about how to split those two bundt cakes if their dad had been home, if my husband was there. And it was ideal, because we talked about two cakes, being four halfs, he said, "You can cut this one in half, and you can cut this one half, and we each get one half." And I said, "And how many halves will we have?" And he said, "Four halves." And we got to talk about four halfs being two wholes. And then we got to talk about whether or not he would get more bundt cake if there were the three of us there. Or if he would get more bundt cake if their dad was home. And there were four of us there.

Pam Harris  06:30

Oh, wonderful, wonderful. Yeah. All the vocabulary, all the things. That is so cool. But Kim, you in that instance, that time you had time with your kids, it was a real experience. You actually had two bundt cakes.

Kim Montague  06:44

Yeah. 

Pam Harris  06:45

To share with three people. We can't do that in class, right?

Kim Montague  06:49

Well, actually. One of our - and probably where I got some inspiration was - that one of our favorite rich tasks is one that we use and tweak based on Cathy Fosnot's Sub Sandwich problem. You and I have done this a lot with kids of varying ages. I mean, a lot. And what's super cool is that we get to have different conversations, different congresses with different outcomes based on the goal for that group. So let's tell a little bit. So this Sub Sandwich problem -

Pam Harris  07:18

Yeah, so maybe we'll just mention that this is from Cathy Fosnot's Field Trips and Fund-Raisers and we'll put the link to that in the show notes. Like you said, we've tweaked it a little bit. Go ahead and tell us the problem.

Kim Montague  07:28

Okay, so the problem is that this class of kids is going on a field trip. And the teacher has organized these parent volunteers. And she's going to send a group of students off to one location in the city, and a different group of kids with different numbers off to a different location in the city. And for lunch, rather than the kids bringing their own lunch, they've organized these long sub sandwiches to share among the group. So there's one group has eight kids in it. And they're sharing seven subs. And another group has five kids, and they're sharing four subs. Then there's another group with five kids, and they're sharing three subs. And then there's the last group that has four kids, and they're sharing three subs. And so the story, the context is you've got these kids, they're sharing the subs and they want to know, was it fair?

Pam Harris  08:19

To be clear 'subs' are sub sandwiches. They're sort of those big long sandwiches, but just because we're colloquial, just cutting it off and calling them 'subs.' So these big sandwiches are too big. 

Kim Montague  08:29

Right. 

Pam Harris  08:30

So y'all, you might want to write those numbers down. Because we're gonna actually dive in and do a lot of work. You might want to think about was the situation fair? Did each kid get the same amount of sandwich? And think about that. If you've done this problem before, just keep listening. If you haven't done it, you might want to like, think about how much of a sub did each kid get in each of those sort of different locations? 

Kim Montague  08:53

Yeah, we want them to stop, right? Stop and take some time. 

Pam Harris  08:57

Pause the podcast. Yeah, better if you have thought about it a little bit, because what we're gonna do is what we would never want to do in class, we're just gonna now kind of tell you a bunch of stuff. We're gonna share with you some of the results that happen after you've grappled with those relationships. So we want to give you guys a chance to grapple first because the learning is better that way. Absolutely. Yeah. So there's a ton of cool things that we could talk about with this rich task, and we'll see how many we could get to today in the podcast. Let's see if we can. So go, Kim. Let's go. 

Kim Montague  09:27

Okay. So we worked with this with some third graders, right? We're talking about a specific instance. We've done third graders through adults. So when I went into a third grade class, I launched the problem. I sent the kids off to work. And I expected to see a couple of the challenges that I saw. But there was this one thing that happened that I didn't anticipate with this particular group, and it's that they were struggling to name the pieces. You know, I thought that would be something that they could handle right away. But it was interesting to me because when they cut -

Pam Harris  09:58

Well and in fact, because the teacher whose class you went into - this is not your group of kids -

Kim Montague  10:01

Right. 

Pam Harris  10:02

You went into it and said, "Tell me what the kids have done with fractions." And she said, "Oh, we've done everything we need to this year." 

Kim Montague  10:06

Yeah. 

Pam Harris  10:07

So you're really clear that if you cut something equally into three pieces, kids should be able to call that 1/3. Or if you cut it into nine pieces, kids should have been able to call it one ninth. That should be a beginning third grade task. So you're really clear. So you were a little surprised when there were some students who struggled to name the pieces, right?

Kim Montague  10:24

Yeah, yeah. And you alluded to this, in an earlier episode, I think maybe it was last week, two weeks ago, sorry. But when they cut a sandwich into halves, they could easily call it one half and one half. They could label those two pieces. But when they got one of those halves, and they had to cut it in half, again, to make two 1/4. So they had one half, and then they had the other half cut into half to make two one fourths, they now wanted to call those pieces 1/3 and 1/3. Because they had three pieces. So in reality, they had half a sandwich, and a fourth and a fourth. But they were calling those three pieces equally, one thirds. They were calling it half of a sandwich, and a third of a sandwich and a third of a sandwich.

Pam Harris  11:13

Yeah, because they sort of saw three pieces, and their kind of limited understanding was if I see three pieces, right, regardless of the fact they're not the same size, then I'm going to call them 1/3. 

Kim Montague  11:23

Right. 

Pam Harris  11:24

Yeah, cool. So notice the numbers in the problem that we gave you a minute ago, and how nicely they're chosen. When I was watching the video of you doing this, with these third grade kids, you said, "What are you guys thinking? Was it fair?" And a couple kids were like, "I don't think it's fair, I think it's fair." They didn't really have a lot of reasons. And you called on one, and I'll never forget, he goes, "Well, it's not fair, because there's a five and a five and a four to four and a three and a five and a four." And I was like, and you even said, "Whoa," like your eyes got kind of big. I knew that you kind of followed what he did. But you were also really clear that the rest of the class did not. Right? The rest of class was like, "What just happened here?" because he literally was sort of pulling some of those numbers out. So let me slow down a little bit. There's a five and a five and a four and a four and a three and a five and a four and a five. What was he talking about? So one of the things that this wonderful young man was talking about, was he was comparing the same number of kids getting different amounts of or different numbers of sub sandwiches. 

Kim Montague  12:26

Yeah. 

Pam Harris  12:26

So for example, in the problem, five kids sharing four sub sandwiches went one place, and five kids sharing three sandwiches went to a different place. And so he was really clear: five and a five and a four and three, what he means is the five kids are sharing four sandwiches, they're gonna get more than five kids sharing three sandwiches. Like it's the same number of kids sharing more sandwiches, they get more. The same number of kids sharing less sandwiches, they get less. So he was clear on that right away that it wasn't fair. So that's an example of same denominator because we can think about four subs shared among five kids, that's like four subs to five kids. So four fifths of a sub per kid. And three subs shared among five kids, that's like three fifths of a sub per kid. So just sort of comparing four fifths to three fifths, that's like same denominator, but totally thinking about it in context about how I could reason about which one was greater. And context helps. Great example.

Kim Montague  13:33

Yeah. And so you can also compare, and I think one of the other things he pulled out was when the number of the subs is the same. So there was a group that had three subs shared by four kids, or there were three subs shared by five kids. And comparing three fourths of a sub per kid and three fifths of a sub per kid. Since the numerator was the same, you can compare the denominators. And really the context makes it so nice, because more kids sharing means that you're going to get less per kid. 

Pam Harris  14:02

More kids sharing the same number of sandwiches means that you get less Yeah, totally cool. So then in the scenario, the kids get to work, and they're splitting up the sandwiches, because you actually say to them, to determine if it's fair figure out how much of a sandwich each kid gets in each group. How much  of a sandwich does each kid get in each group, right? 

Kim Montague  14:24

Yep. 

Pam Harris  14:24

So one of the strategies that we consistently see whether we do this with third grade kids or fourth or fifth or adults, we see this strategy that I think is kind of the smart volunteer strategy, like the volunteer that went with the kids is cutting up the sandwiches, and it's to start by cutting sandwiches in half. So they create halves, and they give everybody a half of a sandwich. And then they're left with some pieces and they cut those up. So for example, if I'm in the scenario where I've got three, so I got to think for a second, where I've got three subs shared with four kids. Then I'm going to take those three subs, and I'm going to cut the first one in half, and you get a half and you get a half. And now the second one in half, and you get a half, you get a half. Now, each of the four kids have a half, right? And I've gotten rid of two of the subs out of the three subs. 

Kim Montague  15:18

Yep. 

Pam Harris  15:18

But now I have a sub sandwich left, what do I do to split those up?. And so this is kind of the general idea where they get rid of the halves, and then they deal out the leftovers. So one of the strategies to compare who got more, because they've already determined it's not fair - and so then you're deciding which group got more - is to just sort of kind of do what they did as they were dealing out. Well, if everybody got a half of a sandwich, and then we divvied up the leftovers, each group that happened, then we can just like ignore the halves and just compare the leftovers. So for example, one of the groups gets a half of a sub sandwich, plus a fourth of a sub sandwich, that's a group actually, I was just talking about the three subs shared among four kids. Because after you split out the halves, you're left with one sub sandwich left to split among four kids, then they each get a fourth. And to save time, I'll just say in one other group, each of the kids gets a half plus a fifth of the sandwich. So you have these two groups where in the one group, they got a half plus a fourth of the sandwich. And then the other group, they got a half plus a fifth of the sandwich. Well, now you can end up comparing a fourth of a sandwich to a fifth of the sandwich. So important where now we're comparing unit fractions. And so since all of these are unit fractions, where we're giving each kid a fair share of the next piece to cut up, kids are really sort of ingrained in this idea of what unit fractions are and how to compare those unit fractions. Great outcome for that strategy.

Kim Montague  16:51

Yeah, and so in another scenario, you just talked about splitting, dealing out half first, there's another strategy that a lot of kids do. The students would cut each sandwich into all of the pieces. So in the group where four kids are sharing three subs, three subs per four kids, each sub gets cut into four pieces, and they deal them out. So they cut the first sub into fourths, and you get a fourth, you get a fourth, you get a fourth, and the last kid gets a fourth. And they keep doing that with each sandwich. And all the kids have these tiny pieces. And then they count the number of pieces that each kid got. Since they cut up three sandwiches, each kid got three, one fourths, three of those 1/4 pieces. So like 1/4 from each sandwich, or three fourths of a sub per kid, because when you put those three one fourths together, like smoosh them into a sandwich, that's three fourths of a sandwich.

Pam Harris  17:47

Totally cool. And so this is the the strategy that I think is not maybe what our real volunteer would do is cut up, right? Because now you have all these little tiny pieces. Can you picture that if I cut up every sandwich into those fourths, and again, like you said, each kid gets a fourth somebody is gonna get the end every time. So you got one kid that gets three 1/4s, but they get the heel every time. That's not very nice. I don't know. But it is actually a very typical strategy, because it's sort of easier for kids to go, "Well, if I got four kids, I'm just gonna take each sandwich and divide it evenly. And now the next sandwich divide it evenly." That's kind of a way to think about that. So interestingly, the kids in my scenario cut unit fractions, but they're all different sizes unit fractions, because they were looking at different wholes, different units. They started with a whole sandwich and cut as many as they could into equal halves. But then they cut those halves into pieces, which means the unit shifted. Now as I cut that half into halves, like I divide a half into two pieces. Now I'm thinking about what's half of a half. And so that's the unit shifting, it's what's the whole? That's really interesting. We're re-unitizing. So one of the things that makes fractions so hard is that we have to be really careful about what the whole is. But Kim in your scenario, kids were cutting each sandwich into the same size pieces always from one whole sub. Their unit never shifted, right? So it was always, "I'm going to take this whole sub and cut it into all these little unit fractions." 

Kim Montague  19:13

Yeah.

Pam Harris  19:14

Either way, a super great beginning way to mess with fractions. Notice that both are all about unit fractions where the numerator is one.

Kim Montague  19:23

Yeah. So how do kids compare in that scenario? Since they've been cutting those tiny pieces, fourths in the group that shared among four kids, and fifths in the group that had five kids, and eighths in the group that had eight kids, they're really clear on the size of those tiny unit fractions, tiny pieces of sandwiches. So now when they compare sometimes a really cool strategy comes out. What's really cool is that they can compare the amount of subs by thinking about how much a student didn't get.

Pam Harris  19:51

I love this strategy. So cool.

Kim Montague  19:52

Yeah if a kid got four fifths of a sub, they didn't get 1/5 of a sub and if a kid got seven eighths of a sub they didn't get an eighth of a sub. And they can still use unit fractions to compare. They can consider how much to the next whole sub.

Pam Harris  20:08

Oh, that's so important. So, I can think about how three fourths of a sub compares to four fifths of a sub compares to that seven eighths of a sub. And those are numbers from the problem, because often kids will look at it and go, "I don't know, like they're all close to the whole sub." But how close are they? Oh, they're just a unit fraction away. So they could use what they know about the size of those unit fractions to help them think about the whole fraction. So cool. So many important things happen when we focus on a well written, purposely planned rich task. In this case, sharing fairly, students get a chance to develop unit fractions and get a great sense for their size. And also what happens when you combine them, that three one fourths is three fourths. Students get a chance to re unitize. What is a fifth of a half of a sandwich? Students get to start developing the strategy of comparing fractions by looking at how far the fraction is from the whole, how far that unit, or how far the fraction is from the whole by looking at the unit fraction. So many cool and important notions about fractions. Then y'all after I do a rich task like this with students, I'd want to do a problem string to help solidify some little part of that. So cool. Stay tuned because we're going to continue to talk about how to build fraction sense in our students and even get to some fraction operations coming up. I'm so excited. So y'all remember to join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday eves as we explore problems with the world.

Kim Montague  21:38

Hey, if you find the podcast helpful, would you please rate it and give us a review? That way more people can find it wherever they get podcasts. 

Pam Harris  21:45

Yeah, don't forget that we're collecting your questions that you want answered. Send those to Kim@mathisFigureOutAble.com. Kim, do you like how we're sending it to you? 

Kim Montague  21:52

I love it.

Pam Harris  21:53

Because Kim will keep them organized. And we'll tackle those questions in an upcoming episode coming soon. So send those in soon so we can make that happen. So if you're interested to learn more mathematics and you want to help yourself and your students develop as mathematicians, don't miss the Math is Figure-Out-Able Podcast because Math is Figure-Out-Able!