Pam Harris  00:01

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

Kim Montague  00:09

And I'm Kim.

Pam Harris  00:10

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 math class can be less like it has been for so many of us and more like mathematicians working together. We answer the question: if you're not teaching algorithms, then what? Hey, Kim, I just got an email from Melissa Freese, one of the two teachers in the podcast, Two Teachers in Texas. And she said, "I've been listening to your podcast, and I'm so excited. I've been teaching for 25 years and your podcast makes me think!!! It also makes me rethink so much of my teaching."

Kim Montague  01:00

That's great,

Pam Harris  01:01

Right, super cool. So I'm honored that what we were talking about makes such an experienced teacher rethink her practice. And thanks for having me on your podcast, Melissa. I had a blast talking to Melissa and Todd, the two teachers in Texas on the Two Teachers in Texas podcast. Y'all, it was the March 31, 2021 episode of that Two Teachers in Texas podcast. We invite you to go listen to it. We had some great conversations.

Kim Montague  01:27

That's so great. I knew you were going to be on a podcast recently, but I'm super happy to hear that she listens to ours too. That's great. I can't wait to listen. So Pam, this week, we're going to begin a series that we think everyone's gonna love, right? 

Pam Harris  01:40

Everyone.

Kim Montague  01:41

Yeah, we saw how well received the short podcast about fractions, decimals and percents went. And we already expanded a little bit on my favorite: percents. And we want to take some time now to dive a little deeper into fractions for everyone.

Pam Harris  01:55

So today, we're going to start with some important information that if we kind of have in our heads and our understanding of fractions, decimals, percents - especially fractions - then it will help us as we then start with students as we do things that are sort of building up for students. So we're going to talk about the five interpretations of rational numbers. So what are rational numbers? Rational numbers are things like fractions, ratios, decimals, percents. In fact, a rational number can be expressed as the ratio of two integers. So for example, 1/2 is the ratio of one to two, 4/5 is the ratio of four to five. And if I made one of those negative, then we could have the ratio of negative four to five, and so on. So those are rational numbers. Consider for a minute the simultaneity going on in a fraction - things happening simultaneously in a fraction. You have the numerator, a number. You have the denominator, a number, usually a different number. And then you're asked to consider the relationship between those numbers. That's a lot to consider simultaneously. And not only the relationship between those numbers, but then we can consider the relationship between those numbers as a number. Just lots of things going on simultaneously. Susan Lamon, in her book called Teaching Fractions and Ratios for Understanding helps us think about these five interpretations. In fact that's where we got them from.

Kim Montague  03:26

Yeah,  I remember it. Pam, I'm gonna interrupt you. I remember the first time you've ever talked about with me these five interpretations, and it was super important. So I hope everybody's going to really listen to these five interpretations.

Pam Harris  03:37

Excellent. Alright, so here they are. So not in any specific order. The five interpretations of rational numbers are: part-whole, measurement, operator, quotient, and ratio. Now, we didn't expect you to get all those down, because we're actually going to dive into each of them right now. And so we thought we'd start today by discussing these five interpretations of rational numbers. Kim, let's start with the most common interpretation in the US. That doesn't, by the way, mean it's the best one to start with. It just happens to be the most common one. And that is the part-whole way of looking at fractions.

Kim Montague  04:18

Yeah, it's definitely the most common, right? And in elementary school, it's likely the only way that fractions are taught.

Pam Harris  04:23

Yeah, not good.

Kim Montague  04:26

So we can picture a typical task in elementary school, right, young elementary school, where there's a pre-drawn object or set of objects, and students are asked to shade in a fractional amount, right. So like a rectangle's already cut for them into five pieces. 

Pam Harris  04:42

And five equal pieces, right? Already cut?

Kim Montague  04:45

Uh-hum. Everything's done for them. And students are supposed to shade 4/5. Can you picture that? They might be asked to do the shading when they're told the fractional amount. Or they might be given an image with something pre-cut and shaded in and might be required to tell the fractional amount not shaded, that's really the only two things that they're asked to do.

Pam Harris  05:07

So consider the development of mathematical reasoning, ya'll. Remember Counting Strategies come first. And then we build on that to get Additive Thinking. And then we build on that to get Multiplicative Thinking. And build on that to get Proportional Reasoning and then Functional Reasoning. If a student is given that kind of task - so let's review the task. They were given a pre-drawn object that is pre-cut into equal sections, and they're asked to shade 4/5. What kind of thinking would that student probably be doing? So I'm gonna act it out here. I've got this pre-drawn thing. It's cut into equal five pieces, and I'm supposed to shade 4/5. Y'all, I'm gonna count the total, Yep, there's five. It says 4/5. I'm gonna count four and shade those four. And I'm done. That is the only thing happening. Now maybe I've given them that pre-drawn figure with it cut into five equal pieces, and three of those pieces already shaded. And now the instruction is what fraction is represented. Again, what kind of thinking is happening? They're going to count the total, write that down in the denominator, count the total that are shaded, write that down in the numerator, and they're done. Like, that's it, they're literally only using counting strategies in order to do those fractions sort of tasks at those young grades. What we haven't done is create some sense of what 4/5 means. We haven't had students like grapple with the idea of what 4/5 are. Or even better, starting with 1/5 to help them think about four 1/5s or 4/5. They're just using counting strategies. That's a problem. 

Kim Montague  06:48

Right. So we need to actually help students create a sense of what a fraction means, right? So in this case, we're talking about 4/5. What does 4/5 actually mean?

Pam Harris  06:59

Yeah, so we do need the part-whole interpretation. But we need to do so much more, so it will be richer, and not only the part-whole interpretation. In fact, we don't think maybe that you're going to start with the part-whole. So we need to talk about how to develop this richer, deeper meaning. We're going to do that in an episode coming up. For now let's talk about one of the other interpretations. And we're going to talk about the measurement interpretation. So this is all about 4/5 - we're going to use 4/5 over and over again - this is all about 4/5 being thought of as four 1/5s. So in other words, it's all based on my share. If I'm sharing with four other people, so there's five people total, there's five of us, then how much do I get for sharing fairly? I get 1/5. Right. That's my share. So then what does 4/5 represent? Well, that's four of them. That's four of those 1/5s. It's almost like I said, "Hey, guys, let's share this fairly. But you guys, you don't want yours? Okay, I'll take them, cool. I'll take four of them." So then I get four of those 1/5s. Right? Hey, Kim. It's almost like we're sharing a chocolate bar. Because you would always give me your piece, right? You'd want me to give you your skittles? Because I eat chocolate and Kim's the sweet and salty.

Kim Montague  08:21

Hey, I like the way that you actually said that just now, right? You said four of the 1/5s. Interestingly, that's how fractional amounts will be named in a lot of other countries: four-1/5s. So here's a teacher tip for you guys. If you're only saying the words four-fifths, to describe that fractional amount, you can bring more meaning by also saying four one-fifths. Try hard to alternate your use of those two ways, because four fifths really is four 1/5s. 

Pam Harris  08:51

Absolutely. So now let's talk about your favorite interpretation of rational numbers, Kim. The operator meaning.

Kim Montague  09:00

Yeah, why is that my favorite? I don't even know. I love the operator meaning.

Pam Harris  09:04

Yeah, what's going on there? 

Kim Montague  09:06

You know, I think it's because it's actionable. It's in our daily lives. And you do something with the fraction, right? 

Pam Harris  09:13

Yeah, that makes sense. Because you're a doer. Alright, so give us an example of what we mean by the operator interpretation of rational numbers.

Kim Montague  09:19

Okay, I actually have a recent story. So my younger student is currently learning about the operator meaning of fractions and the representation of a fraction times a whole number, right? He's being asked to think about some quantity like 24, and then find 1/3 of 24. So he's literally thinking about 1/3 of something is going to be a part of that thing. Right? He knows that he needs to think about the 24 into three parts, which is pulling on his understanding of division. 

Pam Harris  09:50

Nice. 

Kim Montague  09:50

And what's even cooler about what he's doing right now is that it's a combination of the operator meaning and the measurement meaning. So he's being asked to think about, like, what's a fourth of 32? And he's thinking then about 3/4 of 32. So like if you know 1/4 of 32, that can help you with three 1/4s of 32. I love that.

Pam Harris  10:16

And that's the operator meeting. And the reason it's called operator meaning is because you're operating. It's an action thing. You're finding 3/4 of 32. You're like feeling like you're acting on 32 to find 3/4 of it. Yeah, super cool.

Kim Montague  10:31

Okay, so the next one we're going to talk about is the quotient interpretation. Tell us a little bit about that one.

Pam Harris  10:37

Yeah. So this is the one that I'm gonna admit that I rote memorized. Because I was sort of told it, like the teacher said, this is the thing. And I could use it in a very limited way. But now that I own it, wow. Like, it's so powerful in so many ways. This is all about the relationship between fractions and division, or between ratios and division. Here's an example. So let's say that I'm sharing those four candy bars among five kids. Okay, so I've got four candy bars, and I've got five kids. Each kid gets...? So we might want to pause right here. Just think about it. If I have four candy bars, and I'm sharing them among five kids. How much of a candy bar is each kid going to get? Maybe even pause the podcast right here and think about if four candy bars are being shared among five kids. Five kids are sharing four candy bars, how much does each kid get? In other words, we're thinking about sort of less than one candy bar, right? If it's five kids sharing four candy bars, they don't get a total of one bar, how much do they get? Yeah, they get 4/5 of a candy bar. So those four candy bars shared among five kids is equivalent to 4/5 of a candy bar per kid. Four divided by five is 4/5. And if you can see what I what I'm writing, as I say that, then I would write four division symbol five, is equivalent to, an equal sign, four fraction bar five. Or if I said that in an opposite way, 4/5 is equivalent to four divided by five. Now the limited way that I use that was to find the decimal equivalent for fractions. So that's the way that teachers sort of said, "Here's how do it." If I was given 4/5 and the teacher said, "Like find the decimal equivalent of 4/5." Then I would say, "Oh, I know how to do that one. Yeah, I remember I remember that. Let's see, that was the rule where yeah, yeah, we do division, right?" And then I would have done long division. I would have thought about - Kim's like, what? Yes, Kim, even for something like 4/5. I mean, it didn't matter the fraction except maybe a half, one half and fourths. Other than that - like with halves and fourths I think I could have thought about them in decimals. But other than that, literally all of them I did long division in order to get their decimal equivalent. But Kim, that's not how you would have reasoned about finding the decimal equivalent to 4/5. Tell us about that. 

Kim Montague  13:08

You know, so I'm such a person's person. I think about percents more than decimals really. And I know that 4/5 is 80%. So then I could go 80% is point eight or eight tenths. And honestly, I kind of own that one, right? It's fifths, so that one's not... 

Pam Harris  13:26

It's funny, though, but you're like, it's fifths. You know, you sort of say it, like everyone knows fifths. I mean, not everybody knows fifths, yet. But we all can. We can all think about fifths, and we can reason about them enough that then they sort of become ingrained. Kind of like Kim just went straight to 80%. Cool. So how about a number like, you just admitted, right, you own fifths? So what if I give you one that maybe you don't own as much? How about something like 3/8. Three eighths? How do you think about the decimal equivalent to 3/8? 

Kim Montague  13:53

So actually, I own that one too. That's 37.5%. 

Pam Harris  13:56

Ok, alright.

Kim Montague  13:57

But if I didn't own that one, I could use the fact that an eighth is 12 and a half percent, and then I could scale up times three. So I know 1/8, and that will help me with 3/8. 

Pam Harris  14:08

Okay, so you know 1/8, but what if you don't know 1/8?

Kim Montague  14:11

Well, so 1/8, man, so I would think about a fourth is 25%. So an eighth would be 12 and a half percent? 

Pam Harris  14:19

Because?

Kim Montague  14:20

I think a lot of us know a fourth.

Pam Harris  14:22

Yeah. Alright. So if a lot of us know a fourth is 25%, how did you get from a fourth to an eighth?

Kim Montague  14:27

So two eighths make a fourth. 

Pam Harris  14:29

Okay. 

Kim Montague  14:30

So then two 12 and a half percents make 25%.

Pam Harris  14:34

Cool. So kind of thinking about the relationship between a fourth and an eighth. If you know that an eighth is half of a fourth and a fourth is 25%. You can half 25% to get 12 and a half. Cool. So maybe you've done that enough that you just sort of own what 1/8 is. And then you can scale that 12 and a half times three to get 37 and a half. That's how you were finding the equivalent for 3/8? 

Kim Montague  14:55

Yeah.

Pam Harris  14:56

You know what I tend to do a little bit, I don't really scale 12 and a half times three. I think about a fourth being 25%. I know an eight is 12 and a half. So I think about that 25 plus 12 and a half. 

Kim Montague  15:08

That's nice. 

Pam Harris  15:08

That's how I know it's 37 and a half. Okay, so how about a fraction like 3/25? Go. 

Kim Montague  15:17

Oh, so this is also a kind of operator and measurement meaning together, right? So I would think about 1/25 is 4%. So then - 

Pam Harris  15:26

Wait, wait, how do you know that? How do you know 1/25 is 4%?

Kim Montague  15:29

Because I'm thinking about 25 times four is 100. 

Pam Harris  15:33

Okay.

Kim Montague  15:34

So 1/25 is 4%. And then 3/25, I would scale up times three, so that's 12%, or point 12.

Pam Harris  15:46

So you use the relationship between four and 25 being 100. To think about 1/25 is like 4%. And then if 1/25 is 4%, than three of those  twenty-fifths, is three times 4% or 12%. And then you got point one two.

Kim Montague  16:01

Yeah. 

Pam Harris  16:02

Or twelve hundredths. For those of you who are purists out there, that we can't use point one two. 

Kim Montague  16:06

Haha, sorry.

Pam Harris  16:08

Yeah, that's okay. Because if you remember our percent conversation, we said specifically that we want kids to be able to say, point one two and 12 hundreths. 

Kim Montague  16:17

Right. 

Pam Harris  16:18

I think that that's important. But they should also be able to say, by the way, 12 hundredths and 1 point 2 tenths. Anyway, we'll go on. Okay. Let's see, I think we're ready to talk about the last... Are we ready for the last one?

Kim Montague  16:31

Yeah. Okay, so last one is going to be the ratio interpretation.

Pam Harris  16:36

Alright, so the ratio interpretation actually has come into our conversation already today a little bit, when we were talking about four candy bars shared among five kids. That's a ratio of four candy bars to five kids. So that's kind of all we're going to talk about with ratio today. We're going to get into all of these a little bit more later. We've talked a little bit about each of the five interpretations. And what we want to emphasize today is that what's so interesting and important, is how we actually shift back and forth between all of these interpretations, as we saw problems with rational numbers. So Kim, let's dial in on that a little bit. Let's talk about the fact - we've already kind of mentioned a bit  that we were using one or two or more of the interpretations. But let's actually get really specific about how we're solving problems, and kind of shifting in and out of all of these interpretations with rational numbers as we're solving those problems. Alright, you ready? 

Kim Montague  17:35

Sure.

Pam Harris  17:35

So I'm gonna ask you a question. And as you solve it, I might have to help you pull out your reasoning. Like, right you guys after you've done something so often, it can be hard to kind of, like think about your thinking, right? So Kim and I are going to work together on this. She's going to solve a problem and then we're going to pull out the different thinking involved. Okay, so here we go. Eight kids are sharing seven large candy bars. Eight kid's sharing seven candy bars, how much of a candy bar does each kid get? Now, listeners, before Kim just goes, you might want to pause here and you think about seven candy bars for eight kids. How much of a candy bar does each kid get? You might pause here. But ready Kim, go. 

Kim Montague  18:19

Okay, so if there are seven bars for eight kids, I know they're not going to get a whole bar each. That's like my first thought. 

Pam Harris  18:27

Yeah, great beginning thinking.

Kim Montague  18:29

So I know it's gonna be a fraction of a bar for each kid. So a thing I could do is take each of those bars and slice them into eight pieces. And I have a whole bunch of eighth pieces. And each kid after - 

Pam Harris  18:46

Like one eight pieces. I'm going to interrupt you here.  You said 1/8 pieces? 

Kim Montague  18:50

Uh huh. So I could cut each candy bar into eighth pieces, eight pieces, which would then be called 1/8. And I can deal them out. So each kid gets seven 1/8 pieces. So they'd get seven eighths of a bar.

Pam Harris  19:03

Where did the seven come from?

Kim Montague  19:05

Because there were seven bars.

Pam Harris  19:07

Oh, right. So they get an eight from each of the seven bars. 

Kim Montague  19:11

Mmhmm. 

Pam Harris  19:11

So they get seven 1/8 pieces. Okay, good. Alright, I got it.

Kim Montague  19:17

Or honestly, that might be something that I could see myself doing, you know, with one of my kids, or I might just consider -

Pam Harris  19:24

Before you go on, before you do another kind of way to think about it - 

Kim Montague  19:27

Yeah. 

Pam Harris  19:28

Let's pause on that first way that you just said. And let's see if we can parse out all of the different interpretations. 

Kim Montague  19:33

Okay. 

Pam Harris  19:33

So we started with seven candy bars shared among eight kids. That's ratio, like we started with ratio. So the kids have to consider this ratio of seven candy bars to eight kids. And then you said I'm going to cut each candy bar into eight pieces, right? So you're sort of creating those unit fractions. We call those unit fractions, those one eighth pieces, kind of what's each kid share? And that's kind of has to do with that measurement meaning that we're thinking about one eighths a little bit. Well, at the end when you decided that you had seven 1/8 pieces, if we kept going and said, so those 7 1/8 - there's, again, the measurement meaning - turns into seven eighths, then that's  that sort of completing the measurement meaning, that I can think about seven one eighths as seven eighths. Does that make sense? 

Kim Montague  20:30

Yeah. 

Pam Harris  20:30

Okay. Alright. I think maybe we're ready for you to keep going.

Kim Montague  20:34

Yeah, so maybe what I -

Pam Harris  20:36

No, sorry. 

Kim Montague  20:37

Oh, you've got more? Alright. I'm listening.

Pam Harris  20:39

So we had the ratio of seven candy bars to eight kids. That's like taking seven divided by eight and getting seven eighths of a candy bar per kid. That's the quotient meaning bam!

Kim Montague  20:49

Which is actually probably where I'd go straight to. And that's what I was about to say. If I'm considering that it's seven bars divided by eight kids, they each get seven eighths of a bar. And one of the things I didn't necessarily mention is that if I'm in real life, right, I might think about it as 7/8. And I'm probably, realistically, if I'm going to cut those bars, I'm going seven eighths for the first kid, seven eighths for the second kid, and then continuing on.

Pam Harris  21:18

Let's picture that a little bit, because I know what you mean when you say that. Instead of taking seven 1/8s and sticking them together. You guys picture that, an eighth from each candy bar. And then you're like handing seven pieces of little tiny one eighths to a kid. Instead, you're thinking to yourself, if I already know that a kid's gonna get seven 1/8s or 7/8. So you're taking the whole candy bar, and you're like only hacking off 1/8 and giving the rest the seven eighths, the seven out of those eight pieces - ooh, there's the part-whole interpretation - to a kid right? You give them the whole contiguous piece of candy bar. So instead of giving them all these little tiny pieces, you're like, "Since I know it's seven eighths, I'm going to give them that whole chunk together." Except Kim, if you do it that way. You get that little one eight left over seven times. So one kid's getting the shaft, right? One kid gets all those little tiny pieces. 

Kim Montague  22:09

Yup. 

Pam Harris  22:09

I mean, if you're gonna do it that way, so at least everybody else got the whole, you know, kind of pieces together instead of kind of apart. Yeah. Okay. So the only meaning that we didn't maybe talk about was operator, but you were kind of using operator as you said, it was seven eighths of a candy bar. Like when you decided that they were going to get seven eights. And you had to like look at the whole candy bar and say, "Well, where is seven eighths of this candy bar?" And you cut off that little one eighth to give to the poor kid that gets all the little one eighths. But in order to do that you had to have been thinking about the operator meaning of fractions. The big point here is that it's the shifting back and forth. And it's using all of these interpretations. It's kind of like the optical illusion with the you know, if you look at it, and you're like, do I see the old woman do I see the young woman? And it kind of depends on like, what you're focusing on. It's not that they aren't both there, all of these interpretations could be there in the problem. It's just you kind of focus on one or the other, which is so fascinating. 

Kim Montague  23:09

It is, right? It's so fascinating. We know that there's these five interpretations. And that's why it's so important that we don't just focus on part-whole relationship, even in the very early grades.

Pam Harris  23:20

Yeah. And speaking of early grades, you are not going to want to miss next week's episode, where we address ways to start building fractions with our youngest learners. So tell all of your colleagues who teach prek-two or really like any younger grade, but preK-two students for sure to check out next week's episode. Hey, and also remember, check out the March 31, 2021 episode of the Two Teachers in Texas podcast where high school history teacher, Todd, and fourth grade teacher, Melissa, and I do some great fraction works, super cool. Alright, remember to join us on hashtag MathStratChat on Facebook, Twitter or Instagram on Wednesday evenings where we explore problems with the world.

Kim Montague  24:08

If you find this podcast helpful, we would love it if you would rate us and give us a review. That way more people can find it wherever they get podcasts. And if you really wouldn't mind we'd love it if you leave your Twitter handle and we can tag you. Don't forget that we're collecting your questions that you want answered. You can send those to Kim@mathisFigureOutAble.com and we will tackle them in an upcoming episode.

Pam Harris  24:31

Which will be so fun! So get those questions in because we are going to tackle your most asked questions in that upcoming episode. So if you're interested to learn more math and you want to help yourself and students develop as mathematicians then don't miss the Math is Figure-Out-Able Podcast because Math is Figure-Out-Able!