Ep 49: Fractions, A Money Model Pt 2

Pam Harris  00:01

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

Kim Montague  00:09

And I'm Kim Montague.

Pam Harris  00:11

And we're here to suggest that mathematizing is not about mimicking or rote memorizing. That it's about thinking and reasoning; about creating and using mental 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 not algorithms, then what? Alright, so, today, a quick thank you to Tara Buck on Twitter who said, "I love how you all explain things, and present the Problem Strings to get me thinking." Well, thank you, Tara Buck. We love Problem Strings. And we'll be doing one in this podcast.

Kim Montague  00:55

So speaking of Problem Strings, in preparation for today, we went back and listened to the recording from last week's episode. Oh my gosh, y'all. I am laughing at how much I was on the struggle bus for that last problem. It was a real moment for sure. Sweet Pam was so patiently trying to help me with three fourths and nickels. And I was listening back, I thought, how could you not hear that? Like, how could you not just hear what she was trying to say? But the truth is, I was already trying to think of something else. and I could not even follow where you were going, right? And that actually speaks to what happens sometimes with students that we're trying so hard to guide.

Pam Harris  01:35

Yeah, so let's just replay that moment a little bit. This is honestly going to be a little bit more embarrassing for me than for Kim. Because we were talking about the problem three fourths minus three fifths. And I basically was forcing you to think in nickels. 

Kim Montague  01:54

Yeah. 

Pam Harris  01:55

Like, I was forcing it and even worse - because I wouldn't let you think. I talked too fast, too much. And as we were listening, so we just listened to the episode. I was like, "Oh, Pam, too fast let her think! Yowza." And even worse, I was forcing a strategy. So again, the problem was three fourths minus three fifths, and I was trying to get you to think in nickels. And so I said, "Well, can you think about 1/5?" 

Kim Montague  02:20

Yeah. And it's interesting, because normally, I think I would tell you, just wait. Right? I would hold up my finger to pause you. But on a podcast, I wasn't sure how that would feel. So I just went with it.

Pam Harris  02:30

Right. So we've just been talking about, you know, like, go ahead and tell me in the podcast, you know, like, we'll figure it out. But it's so important to give people time to think. And to not just funnel them down the path that you're thinking about. I was thinking about nickels. I was forcing you to think about nickels. I know when we done the problem - three fourths minus three fifths - you instantly thought, do you remember?

Kim Montague  02:55

Yeah, 75 cents minus 60 cents.

Pam Harris  02:57

I mean, bam! It was just there. And then I was like, "Ah, but let's think about nickels." And you were like, "Um okay." And then I didn't give you a chance to think. I just like funneled you down the path that I was thinking about. Instead, we're suggesting that we want to work to pull students out, sorry, we want to work to pull out what students are thinking, not take them down the path that you're doing. So hey, it's real. You watched a real moment where I didn't do what I am advocating to do with teaching. So I'm going to own that. I'm gonna admit it. We're all then going to get better, because we're going to be more clear about what we're advocating for, so that we can all get better at it.

Kim Montague  03:34

Yeah. And so it's interesting, because in retrospect, I probably should have told you that I needed time, not help. 

Pam Harris  03:41

Oh, so good. And if you don't recognize that, go check out episode 40, where we talked about homework. And we talked about that very idea. Do kids need more time? Or do they need help? And in that case, you just wanted me to breathe for a minute, and give you a chance to just think a little bit instead of cramming my thought down your throat.

Kim Montague  04:03

It's ok, you're alright.

Pam Harris  04:04

Thank you, Kim. Kim was very patient with me. And I appreciate that. Alright. So also, in the last episode, we asked you guys to think about denominators that make sense to use with money. We had been doing a lot of fraction problems where we said think money, just like I just said, I was sort of forcing Kim to think about nickels. She wanted to think about cents. And she'd already solved the problem, right? 

Kim Montague  04:26

Yeah

Pam Harris  04:26

If I could just like mention, it's also one of the things that we're kind of advocating with kids is that once they've solved the problem in their best way, then it's not always about forcing them to find just like any other way. Like then we want to do things to help them think more efficiently, more sophisticated, but recognize like that they'd already solved the problem. So we've got to kind of get out of that, "I've already got the answer" kind of thing. It's walking this fine line between recognizing that was fine, let's get a little bit more sophisticated. It's not always just about thinking about a different strategy. But anyway, blah, blah, blah. So we asked you to think about denominators that makes sense with money. So if you haven't yet, I just kind of went off on a beaten path where maybe you were thinking about it. If you haven't yet thought what denominators of fractions do make sense, to think about money, to like use coins and money to help you reason about fractions and fraction equivalence. So if you haven't thought about that yet, maybe pause and think a little bit. So which denominators do work well for a money model? Kim, you got any?

Kim Montague  05:35

10 and 20. 

Pam Harris  05:37

Okay. 

Kim Montague  05:38

100.

Pam Harris  05:39

Cents. Yeah. So in fact, let's back up so 10 was sort of dimes. 20 would be nickels. 100 was cents. Anymore? 

Kim Montague  05:47

4.

Pam Harris  05:48

 Four quarters. Okay. So those are all coins. Did we get them all?

Kim Montague  05:53

No, no. But also two.

Pam Harris  05:56

So we got the coins maybe, but now we've got other denominators that make sense. No, two, that's 50 cent pieces. You're right. Okay.

Kim Montague  06:04

Um. Five.

Pam Harris  06:07

I don't think we have a coin for 20 cents. We don't in the United States. We don't have a 20 cent coin. Maybe somebody does. Yeah, I think actually, there are 20 cent coins. I just read on Twitter about twenty cents, anyway. Okay, so five 20 cent coins? That can be a thing. Okay. 

Kim Montague  06:23

And 25.

Pam Harris  06:25

We definitely don't have a four cent coin. 

Kim Montague  06:28

Right. 

Pam Harris  06:29

Okay. But that would help if we had a denominator of 25. We could think about four cent chunks, chunks of four cents. Okay.

Kim Montague  06:37

50.

Pam Harris  06:39

We definitely don't have a two cent piece either. But we'd have fifty 2 cent pieces. And what do all - did we get them all do you think? Yeah, I think so. 

Kim Montague  06:49

Yeah. 

Pam Harris  06:50

Somebody's gonna tell us we didn't. So what do all of those have in common? Those are factors of 100. And so they're really nice to work with money because they divide evenly into 100. And so when we see those denominators and fractions, then we can think money to help us think about equivalencies.

Kim Montague  07:11

Okay, so great. Alright. Last week, you gave me problems. And we kind of started a Problem String. So today, let's wrap up that Problem String that we started with, but this time, I'm going to ask you some questions.

Pam Harris  07:23

Oh, boy. Okay, so you're gonna give me plenty of wait time?

Kim Montague  07:27

I will.

Pam Harris  07:28

Okay, all right.

Kim Montague  07:29

Do you have something to write down? You got a...?

Pam Harris  07:31

I have a pen. I do, I have a pen and pencil. Or I have a pen and paper in case I need it. 

Kim Montague  07:38

1/25 plus 3/4. 

Pam Harris  07:43

One twenty-fifth, plus 3/4. So I'm thinking 1/25 feels like four cents. Because 25 times four is $1. So I'm going to think actually, in decimals. I'm going to think about four cents. So I've just written .04 and three fourths, I can think about as like three quarters or 75 cents. And so that's 79. So four cents and 75 cents is 79 cents. So then I can go back to fractions if you want me to, and that's 79 hundredths. Yeah?

Kim Montague  08:20

I like that you were thinking about decimals.

Pam Harris  08:24

I mean, I'm really glad now that I'm done with the problem, because now that I'm looking at 79/100, I'm not gonna be able to do a whole lot in other coins. Yeah, so me too. Alright, bring it on. Give me another one.

Kim Montague  08:38

Alright, two and two fifths.

Pam Harris  08:41

Oh, mixed number. 2 and 2/5s ok.

Kim Montague  08:43

Minus 9/20.

Pam Harris  08:46

Two and two fifths, minus 9/20. So the first thing - I'm going to talk out loud. The first thing I'm considering is: Do I want to stay in mixed numbers and deal with like, the whole two and two fifths minus 9/20? Or do I want to just consider two fifths minus nine twentieths? And as I consider that, I'm thinking to myself, two fifths is almost a half, and 9/20 is almost a half. So I'm not actually really clear how those are going to pull out. I think two fifths is greater than, no, I don't know, actually. Like they're both so close to one half. 

Kim Montague  09:26

Yeah. 

Pam Harris  09:26

And now my brain is shutting off a little bit going that direction. So since they're both so close to one half, I'm not really clear if I can just do the two fifths minus 9/20. And then tack back on that whole two that we had. So I might just stay with the whole two and two fifths to begin with, and then I'll kind of reevaluate after I do that. So I don't know for some reason, maybe it's because you're on the line I'm thinking decimals again. Oh, no, no, I was gonna think decimals and then I looked at the 20ths. And so then I'm gonna backup. The 2 and 2/5 was screaming at me 2.4, but then the 9/20 I was like, ooh, but no, that's just screaming nickels. So I'm actually going to go nickels. So two and two fifths is like two... let's see, the whole 2 like $2, and then two fifths nickels, two fifths nickels, I am going to kind of go cents or decimals because I'm gonna think about point 4, 40 cents. And 40 cents is eight nickels. So it's like I've got two and eight twentieths minus 9/20. Oh, bam, now I can compare the 8/20 and the 9/20. And so the two fifths is smaller than the 9/20. That's was noteworthy to me. So let's see. 2 and 8/20 minus 9/20, I'm going to go ahead and I'm thinking about 2 and 8/20 on a number line, and I'm going to go ahead and subtract eight twentieths to get to two, but I was supposed to subtract nine twentieths. I've already subtracted 8/20 to get to two. So I've got to subtract one more 20th and two minus 1/20th is one and 19/20. So I say my answer is one and 19/20. 

Kim Montague  10:58

That is so interesting to me. 

Pam Harris  11:00

Not what you would have done?

Kim Montague  11:01

Well, I feel like I think so heavily in decimals that it's interesting because like last time, I was struggling with the nickels.

Pam Harris  11:12

And I went to nickels and you don't like that. So -

Kim Montague  11:15

When I see 9/20 I immediately think -

Pam Harris  11:17

Well, wait, wait, wait, can I think about it? 

Kim Montague  11:20

Yeah.

Pam Harris  11:21

Okay, so yeah, so two and two fifths was instantly 2.4 for me. I don't know why, I've done that enough. So 9/20 - Oh, okay. 10/20 is 50 cents. So 9/20 is 45 cents? 

Kim Montague  11:34

Yeah. 

Pam Harris  11:35

Ah, you were right there right off the bat. You were like I can compare those. Because it's .4 to .45 or 40 cents to 45 cents. Okay, so then I could take $2.40, subtract 45 cents and also get $1.95, which is like $1 and 19/20s. Okay, thanks for letting me like scream at you.

Kim Montague  11:54

That's ok. I would never have thought of 19/20, so interesting. Hey, so one of the things that you said earlier, I'm going to go back to that problem. You said 1/25. And you said that was four cents?

Pam Harris  12:09

Yeah.

Kim Montague  12:10

So it's interesting to me, because I can get on board with a fourth is 25 cents, right.

Pam Harris  12:17

That fourth of $1 is 25 cents? Hey, let me just point out really quickly, if you've listened to the episode where we talked about the five interpretations of rational numbers, we are leaning heavily on the interpretation of fractions as operators right now. Because Kim was just saying a fourth of $1. And she, like you said, "I can think of a fourth of $1 as 25 cents, like a quarter dollar is 25 cents a quarter coin is 25 cents." And you were asking about 1/25? 

Kim Montague  12:47

Yeah, yup. 

Pam Harris  12:48

So are you asking me like how am I thinking about 1/25 of $1? 

Kim Montague  12:53

So I just think it's interesting that, what I've seen sometimes kids do, is think about 1/25. And their gut reaction is to call it a quarter because it has the 25 in the denominator. Right? And so sometimes they want to call a fifth a nickel. Right? 

Pam Harris  13:11

Slow that down a little bit. Yeah, they'll look at a fifth. And they'll say to themselves, there's that five. And so they think about that as a nickel, but really a fifth of $1 is 20 cents. 

Kim Montague  13:22

Right. 

Pam Harris  13:23

And so then where's the nickel? That's the 20th. A 20th of $1 is a nickel because they're 20 nickels in $1. 

Kim Montague  13:32

Yeah. 

Pam Harris  13:33

So I think what you're bringing up is sort of what we call the 'reciprocal relationship'. 

Kim Montague  13:36

Right.

Pam Harris  13:36

If I think about fifths, and twentieths, those are related because five times 20 is 100. But I have to sort of parse out which represents the nickel and which represents a fifth of $1 or 20 cents. Yeah. And so like when we purposely did the problem, 1/25 plus three fourths, because there's those two denominators. There's that 25 in the denominator, and four in the denominator. And so parsing that out can be tricky, which is why we give students problems with both of those denominators. Now, we don't do that necessarily right off the bat, where we might give them problems for a while, not forever long, but long enough that would they get really used to dealing with thinking about fractions as operators and coins, before we put up in front of them that reciprocal relationship of fourths and 25th or that second problem we did of fifths and twentieths. This is so hard to do over like audio. I hope this is translating. You guys have to let us know on social media, if it's totally translating when we're saying these fractions because I'm like, wow, a little crazy. Hopefully it is. Cool. 

Kim Montague  14:48

Well, well done you. Well done.

Pam Harris  14:50

Cool. Anything else on that? Did I talk about that? 

Kim Montague  14:52

Yep. 

Pam Harris  14:53

Cool. Alright. So this week and last week, we talked all about using a money model to help us really think and reason about fractions. In doing that, I want to be really clear that one of our main points isn't so much getting kids great at adding and subtracting fractions, that's actually kind of a byproduct. Our goal really is to get kids thinking about equivalent fractions. We're building equivalents by having kids add and subtract fractions using a money model. Using this model that should be somewhat familiar with them, or we want to help it become more familiar with them. And so using that operator meaning where they can think about a half of $1, or a 10th of $1, or a 20th of $1. To help them think and reason about equivalencies. Our main goal here, in using this money model is to really build equivalent fractions and the sense of what it means to have equivalencies with fractions. But in these two episodes, the only denominator that we've been working with, or denominators, we've been working with, are factors of 100, because those work well with a money model. So next week, we better do some more with fractions that have denominators that aren't factors of 100. And let's do that. We will use a different model and we'll get even better at equivalencies, at getting kids really thinking and reasoning about equivalencies with yet a different model. Alright, so remember to join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday evenings where we explore problems with the world.

Kim Montague  16:28

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

Pam Harris  16:35

Yeah, don't forget that we are collecting your questions that you want answered. Send those to Kim@mathisFigureOutAble.com and we will tackle them in an upcoming episode. So if you're interested to learn more math, and you want to help yourself and your students develop as mathematicians, then don't miss the Math is Figure-Out-Able Podcast because Math is Figure-Out-Able!