Ep 48: Fractions, A Money Model

May 18, 2021 Pam Harris Episode 48
Math is Figure-Out-Able with Pam Harris
Ep 48: Fractions, A Money Model

It's time to talk about everyone's two favorite things: fractions and money! Fraction equivalence is a super important concept in mathematics, and it is best developed naturally in a familiar context. In this episode Pam and Kim model using coin denominations to  reason about fractional equivalence.

Talking Points:

• If you understand fractional equivalence, you don't have to memorize any rules for operators of fractions
• Examples of using the money model to think about fraction equivalence, and bonus, fraction operations as well:
• Using decimals and fraction notation interchangeably
• Kim's uncomfortable relationship with nickels
Pam Harris:

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

Kim Montague:

And I'm Kim.

Pam Harris:

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 relationships. That math class can be less like it has been for so many of us and more like mathematicians working together,

if not algorithms, then what?

Kim Montague:

So welcome back to our series on fractions. In week one, we talked about the five interpretations of rational numbers. Week two, we talked about fractions for young learners. Last week is part three and we talked all about fair sharing. Today we're going to talk about something a little bit different.

Pam Harris:

A little bit different. So let me just introduce this with telling you about a story from a while ago. I'm going to give a shout out to a mentor of mine. I learned so much from Garland Linkenhoger. Garland shout out to you. Thank you so much. We had a blast working with Garland. I literally learned a lot. She was a unique individual because she taught kindergarten in the morning. And calculus across the street in the other end. Yeah. So the gamut she ranged like that K-12 really well and could kind of see the whole perspective. One day Garland said to me, okay, this is a little embarrassing, because it tells you a little bit about where I was in the moment. She said, "Pam, um, do you realize that if you understand fraction equivalence, you don't actually have to know any rules for operations of fractions." And y'all, all I heard in that was 'operations of fractions'. I said, "What?" She goes, "You don't have to memorize any rules for fraction operations if you understand fraction equivalence. If you really can find equivalent fractions and understand equals the fraction, you don't have to memorize any rules." Y'all I'm doing that brain blowing up motion with my hands. Like I looked at her, I said, "I want to believe you. What does that even mean?" Because in that moment, fraction equivalence for me was way too proceduralized. Like I knew how to find an equivalent fraction, I just did air quote with my fingers, find an equivalent fraction, which was all about this procedure. How many this goes into that times this?

Kim Montague:

Yeah.

Pam Harris:

But I didn't really feel, I didn't have a sense for equivalent fractions. And, oh I do now. And now that I do, Garland you were correct. You are right, that when we understand equivalence, where we really can feel its own equivalence, equivalent fractions, we don't have to have any rules. Let me give you an example of that. I went home that day. To my, and I've talked about Cameron before. He's my oldest. He's the one who really started me on this journey. I owe so much to him. Love, love my son. And I said to him, and I want to say he was around fifth grade at this point, I think, and I said something like, "Cameron, what do you think, Like..." Like, see, he really taught me a lot. I was like, "What do equivalent fractions mean to you? If I say, equivalent fractions, what does that even mean?" Because I was reaching you guys. I was like, what does she even talking about? And I'll never forget what he said, he looked at me, he goes, "It's kind of like this mom, if you double the number of pieces, you only get half as much." And I was like, "What?" And it goes, "You know, like if you have a pizza, and you've got it cut into these however many pieces. But if you double the number of pieces, like you cut them all in half, and then, but you only get a slice, you only get half as much." Again, brain blow. I was like (exploding sound). You do understand equivalent fractions. Like it was so fascinating to me that what his response had nothing to do with the procedure, and everything for him to do with really understanding sort of equivalence. Now, that's not to say that he understood that in that sentence, all equivalence was explained. But he really had a sense and a feel for kind of what equivalent fractions what it means to double the number of pieces and how that related to the size of the fractions and etc, etc. So another thing that we think is really important to understanding equivalence of with fractions is that models are important. And so today, we're going to have this podcast be all about one particular model that we can use to help students really understand and feel and develop fraction equivalence. And as a nice side benefit, we're going to get some fraction operation out of it as well. So isn't that interesting that we're going to sort of get fraction operation, and in this case, addition subtraction, out of work that we're doing really to understand fraction equivalence. So we've done some episodes before that have been less planned. And this is one of them. Y'all, I have like a few things written down like literally four sentences. And that's it. And here goes the podcast where I'm going to do a problem string with Kim.

Kim Montague:

Who hasn't seen the problems.

Pam Harris:

Yep, hasn't seen the problems. So this is a lot fresh and we're going to hear how Kim thinks about these problems. And I'll kind of interject as we go so that you get a feel for work that we could do with students to help them build a sense of equivalence and towards fraction operations. So, Kim, I'm going to ask you today to think about money. Okay?

Kim Montague:

Okay.

Pam Harris:

Alright, thinking about money. And we're going to use money as a model to help us understand fractions. So we're going to start today with the fraction 1/4, one quarter, but I want you to think about 1/4 or one quarter, not as a fourth of a pizza, or a fourth of your brownie pan, or a fourth of a deck of cards, because there's four of us are playing cards and we're going to split, but nope, nope. This time, it's a fourth of \$1.

Kim Montague:

Okay.

Pam Harris:

Hey, notice that there's that operator, meaning

coming in:

fourth of \$1. So if I want you to think about a fourth of \$1, tell me what first comes to mind, go.

Kim Montague:

First thing I think of is an actual quarter, like a fourth of \$1 is one quarter piece.

Pam Harris:

Okay? And I'm going to write down 1/4. But I'm going to say as I write it down one quarter out of, as I draw the fraction bar, four quarters.

Kim Montague:

Yeah. Okay.

Pam Harris:

One quarter out of the four quarters in a dollar is 1/4. Okay.

Kim Montague:

Yep.

Pam Harris:

What's another one?

Kim Montague:

Like another fourth? Oh, 25 pennies, 25 of 100 pennies.

Pam Harris:

And I just wrote down '25 out of 100'. So notice how I'm using the part/whole representation of fractions as I'm writing them down, 25 pennies out of the 100 pennies. And so I have the fraction written down 25/100. But I didn't say 25/100. As I repeat back what you say I'm just saying, Ah, 25 pennies out of the 100 pennies. Okay, any other fractions for fourth of a dollar?

Kim Montague:

Yeah, nickels would be five nickels of 20 nickels.

Pam Harris:

Okay, let's slow that down just a little bit. So I've noticed that not a lot of people have thought about nickels for a while. I don't know if it's we don't deal with change so often, or whatever. But I actually did this work with seventh graders not too long ago. And when I said nickels, then nothing. I was like, how many nickels are \$1, nothing. I said okay, how many? In fact, maybe let's do dimes and then back up to nickels. Do you make a quarter of \$1? Oh, no, that's a good dimes. So let's stay nickels. Sorry. But how many dimes are the dollar? I asked the seventh graders, and they said, "Well, yeah, that's 10." Pretty much they could do that. Then I said, "Okay, so how many nickels are in a dime?" Maybe I was too funneling when I was asking this. Maybe that's why they didn't hang on to it. Then I said, "How many nickels are in a dime?" They're like, "Two." And I said, "Well, then how many nickels in \$1?" Nothing. I was like, "Come on guys, hang with me, like, could you, if you can think of 10 dimes in \$1. And I've got twice as many nickels is that?" We finally got to the point where there were 20 nickels. So you just said-

Kim Montague:

That's kind of what Cameron was saying. Right? And it might be something they hadn't really thought about before.

Pam Harris:

Mmm, nice. Can you say more about that like go back to Cameron.

Kim Montague:

So Cameron talked about, I'm trying to remember exactly what he said. But he talked about if you have twice as many pieces, you only get half as much. Is that what he said?

Pam Harris:

Uh-hum.

Kim Montague:

So if that was true with dimes, if you have twice as many nickels, you only get half as many.

Pam Harris:

Well, let me say it this way. So if I've 10 dimes in \$1.

Kim Montague:

Yeah.

Pam Harris:

Then if I have 20, if I cut those in half,

Kim Montague:

Yeah.

Pam Harris:

Then now I have 20 nickels, but they're only worth five, half as much. They're only worth five cents. not 10 cents. Yeah, I think that would follow what he says. Cool.

Kim Montague:

So a thing that I think about is each nickel is five cents. So five of them makes the 25 cents. So 25 sorry, five of 20 that make a whole dollar.

Pam Harris:

Oh, nice. You just did lots of reunitizing in those sentences.

Kim Montague:

Sorry.

Pam Harris:

It's okay. So five nickels, nickels is the unit, out of the 20 nickels, so I've written down five out of 20.

Kim Montague:

Yep.

Pam Harris:

And you were saying? I can't repeat what you said.

Kim Montague:

It's the 25 cents out of 100 cents in a \$1.

Pam Harris:

Oh, yeah. Nice. Nice. Okay, cool.

Kim Montague:

So you mentioned dimes. Can we go to dimes?

Pam Harris:

Let's go to dimes. Absolutely.

Kim Montague:

What's interesting in \$1, if I'm thinking about a quarter, I think you can think about dimes.

Pam Harris:

And a quarter like...

Kim Montague:

A quarter of a dollar.

Pam Harris:

But are you referencing the coin a quarter that that's 25 cents?

Kim Montague:

No, fourth. You said think about a fourth and think of money.

Pam Harris:

Okay.

Kim Montague:

I think that two and a half dimes is a fourth.

Pam Harris:

Two and a half dimes out of the 10 dimes. So I just wrote down 2.5 divide over with the fraction bar 10. 2.5/10s. So you're thinking that I could represent the fraction one quarter, 1/4, one over four. And I'm the only using the word 'over' so you guys can sort of picture it.

Kim Montague:

Sure.

Pam Harris:

One over four, that 1/4 is equivalent to 2.5 out of 10. Because it's like 25 cents out of 100 cents.

Kim Montague:

Yep.

Pam Harris:

Cool. So on my paper I literally have written down

now:

one divided by four, 1/4 equals 25 out of 100, 125 over 100, equals five out of 20, equals 2.5 out of 10.

Kim Montague:

Yeah.

Pam Harris:

Yeah. Nice and I would want to do that work. Hey, because of that you guys can't see my paper, maybe you've written that down. Notice that I have 25 out of 100, and 2.5 out of 10. Nice place value shift there, right? Where I sort of scale up by 10, or scale down by 10, divide by 10. Okay, cool. Once we've kind of established maybe one or two fractions like that with money, then I'm going to give students a problem like this. And I might say, "Hey, guys, how do you think about 1/2 plus 1/4? But I want you to think about money. Want you think about money." So Kim, a half plus a fourth? half plus a quarter? How do you think about that with money?

Kim Montague:

Yeah, so half of a dollar is 50 cents.

Pam Harris:

Okay, so I'm going to write down 50 out of 100.

Kim Montague:

And a fourth of a dollar is 25 cents.

Pam Harris:

So 25 out of 100.

Kim Montague:

And I'm actually writing in decimals. I'm writing a lot as I'm thinking because I wrote 50 cents plus 25 cents equals 75 cents. And then I thought about-

Pam Harris:

When you just said cents, like, .5 plus .25 equals .75.

Kim Montague:

Yeah,

Pam Harris:

Uh huh.

Kim Montague:

Are you okay with me keeping it in decimals? You want me to talk out loud in fractions?

Pam Harris:

I mean, I'm I wrote it,

Kim Montague:

Or do you love me because I'm doing both?

Pam Harris:

Yeah, I kinda like both. Actually. And we want kids to be able to do both. And teachers, if you were doing this string with students, you would want to get a kid who sort of did it in fractions, which is what I wrote down, 50 out of 100 plus 25 out of 100 equals 75 out of 100. And the point five plus point two five equals point seven five. I want both of those representations. Maybe not the same time, but yeah, cool. You got any more coins? That was pennies, what else you got?

Kim Montague:

Oh, goodness, okay, I've got five dimes.

Pam Harris:

Because that's the half, so 5 dimes out of 10 dimes, okay.

Kim Montague:

Plus two and a half dimes.

Pam Harris:

2.5 out of 10.

Kim Montague:

Is seven and a half dimes.

Pam Harris:

Seven and a half dimes. So I've got 7.5 divided by 10. 7.5 out of the 10, seven and a half dimes out of the 10 dimes, which is now on my paper, very close to the 75 out of 100 and 7.5 out of 10. Again, that nice place value shift. Or if we'd written it in decimals, I'd have .5 plus 2.5, is that right? Yeah. Is 7.5. Cool. No.

Kim Montague:

I didn't hear the last thing you said, sorry.

Pam Harris:

Well I totally wrote that wrong. It's still point five plus point two five. Because I just wrote the decimals wrong. Oh, that's funny. Okay, so I yeah, I wrote the decimals wrong because I was trying to translate straight from the tenths instead of the hundreds. And I can't do that when-

Kim Montague:

Talking and writing at the same time.

Pam Harris:

Well, and so I tried to translate directly from tenths to decimals, not realizing that in our decimal representation it has to be out of hundreds. So just ignore the fact that I was trying to write that incorrectly in decimals. Okay. Pam on the podcast. This is live. Here we are live. It's all good.

Kim Montague:

Unscripted.

Pam Harris:

Okay, so you just did pennies.

Kim Montague:

Yeah.

Pam Harris:

And you did dimes.

Kim Montague:

Yeah.

Pam Harris:

So let's do nickels. Can you do nickels?

Kim Montague:

Sure. So half \$1 in nickels is 10 of 20 nickels.

Pam Harris:

So 10 out of 20.

Kim Montague:

And a quarter in nickels is five of 20 nickels. So that's 15 out of 20 nickels.

Pam Harris:

Cool. So a half plus a quarter you can think about is 15 nickels out of 20 nickels.

Kim Montague:

Yeah.

Pam Harris:

Nice.

Kim Montague:

You're making me think.

Pam Harris:

Excellent. I like it when you have to think. Alright, here's the next problem.

Kim Montague:

Okay.

Pam Harris:

What if I just wrote down the fraction 1/10 plus 1/20. How would you think about those in money?

Kim Montague:

Okay, so a 10th is 10 cents.

Pam Harris:

Oh, you know what, we probably should say and we didn't say this earlier. Once I give the problem, y'all pause the podcast. Think about these numbers before you hear just kind of how Kim's thinking about. Okay, Kim a 10th plus a 20th, go ahead.

Kim Montague:

Okay 10th is dime. So one of 10.

Pam Harris:

Okay.

Kim Montague:

And a 20th is like a nickel. So it's like five cents. So 10 cents and five cents is 15 cents.

Pam Harris:

So you are really think in terms of decimal.

Kim Montague:

I was. Yeah.

Pam Harris:

That's okay. And so that's like 15 out of 100 could be a fraction representation.

Kim Montague:

Yeah. So if you want to think pennies, then it's 10 pennies and 5 pennies is 15 pennies. So 15 out of 100.

Pam Harris:

And when you said pennies, I wrote 10 out of 100 plus 5 out of 100 is 15 out of 100 which is 15 pennies. Yeah, it's really cool. What if you were to think, so let's see, we just did pennies.

Kim Montague:

I kinda want to take dimes next.

Pam Harris:

Okay, let's go dimes.

Kim Montague:

I do like dimes. So one dime out of 10 dimes.

Pam Harris:

So the 1/10 doesn't really change. That's representing dimes. Okay.

Kim Montague:

Right. And then for the 1/20 it would be point five dimes, like half a dime.

Pam Harris:

So I wrote point five out of 10. So now we've got one out of 10 plus point five out of 10 is one and a half out of 10.

Kim Montague:

Yeah.

Pam Harris:

So that's like one and a half dimes, which totally works. Right? Is 15 cents one and a half dimes? Absolutely. I like that. So let's see. Can you do nickels?

Kim Montague:

Oh yes. So 1/10 is two 20th, two nickels.

Pam Harris:

Two nickels out of the 20 nickels.

Kim Montague:

And then we have one more nickel 1/20. So that's three twentieths, which is 15 cents.

Pam Harris:

So three nickels of the 20 nickels is 3/20. And in that case, using nickels actually got you to the most simplified version of the answer. You might notice that up till now we have not talked about the most simplified version of the answer. And I'm not necessarily going to do a lot of that, especially at the beginning with kids. It's all about equivalence. Now, eventually, we can talk about the most simplified version, even though most of us don't like that term, 'simplified'. What I really don't want is 'reduced'. So I noticed that well, we won't ever use the word 'reduced' to mean simplified, because that means like you got smaller. And you guys were talking about equivalence. It's so important that we strike that word. I cannot believe the number of textbooks we see that have the word 'reduced'. Oh, it's so terrible. Can you imagine learner, yeah, learners of English as a second language when we use the word reduce? Like that has everything to do with the connotation of get smaller, and we're not getting smaller. No, no, no, we are keeping them equivalent. So we don't really like 'simplified' either, but it's better, it's tons better than 'reduce'. So we would strike the word 'reduce'. I did have somebody say to me once, "Yeah, but you can use reduce, because you're reducing the number of common factors in the numerator and the denominator." That to me is making sense of a bad term. So I think we just strike 'reduce'. I like a made sense of it. But let's use simplified. Yeah, my point was sorry, I just digress. My point was that we are not emphasizing the simplified version. Because when students understand equivalence, then if on a standardized tests are on a high stakes test, if they have to then put their answer in simplified form, they'll be able to do it because they understand equivalence so well, especially to be able to recognize it. So we don't make a big deal of that. Maybe ever, but especially not at the beginning. Alright, so we just did a couple of problems really nicely. I'm looking at the time. Kim,I think we have time for one more. Can we do one more?

Kim Montague:

Sure.

Pam Harris:

Okay. What about a problem like, oh, I have to choose. Maybe we're gonna have to do more of these in the next episode.

Kim Montague:

Okay. Or maybe you can, I can just give you one way to think about it and listeners can think about a couple others. Oh, other ways, okay. Okay. But it's killing me because I wanted to do- Okay, go.

Pam Harris:

Three fourths minus three fifths. Wow, subtraction. Yep. Yep. Okay and once you understand equivalence, nothing changes. You just keep thinking of the same. Okay.

Kim Montague:

Alright. So three fourths is 75 pennies, 75 cents.

Pam Harris:

Okay.

Kim Montague:

And three fifths I know is 60 cents, 60 out of 100. And I know that because I know 1/5 is is 20 pennies. So three fifths would be 60 pennies.

Pam Harris:

Okay.

Kim Montague:

So 75/100s and 60/100s.

Pam Harris:

We're subtracting,

Kim Montague:

Oh, thank you. So 75/100s minus 60/100s is 15/100s.

Pam Harris:

And so your final answer is like 15/100s. Or point one five if you are in decimals.

Kim Montague:

Yeah.

Pam Harris:

Uh huh. Point 15. I really like, I just want to emphasize a little bit how you thought about three fifths as 1/5 times three. And you knew three fourths was point seven five. But kids if they didn't could think about 1/4 and scale that up to get three one fourths. Totally cool. Okay, maybe we do have time for one more. So we'll let you guys think about that in other ways. You totally did pennies. But you could have a dimes, you could have done nickels. In fact, should we do? No? Yeah. Can we do nickels? Will you do three fourths minus three fifths in nickels. I know I'm pushing you there because you don't usually do that.

Kim Montague:

Nickels, nickels, nickels. Okay, so let me think for a second, nickels. Three fourths, I'm writing down now, three fourths-

Pam Harris:

We're letting Kim think.

Kim Montague:

Right. I never think in nickels. This is why I should think in nickels more, right?

Pam Harris:

How many nickels are in \$1.

Kim Montague:

20.

Pam Harris:

How many nickels are in half \$1.

Kim Montague:

10.

Pam Harris:

So how many nickels are right in the middle of that for three quarters of \$1.

Kim Montague:

Say that again?

Pam Harris:

So you had 10 yet half at 10 nickels and a full dollar at 20 nickels, but I want three fourths of the dollar which is right in between a half and a full dollar.

Kim Montague:

I'm blank, I can't.

Pam Harris:

Oh, I love it. So if 10 nickels or half and 20 nickels or a whole, isn't 15 right in between for three quarters?

Kim Montague:

Say it again, because I'm not writing anything down is my problem.

Pam Harris:

Yeah. And working memory is real.

Kim Montague:

I started to and I walked away. Go ahead.

Pam Harris:

So we're trying to get three quarters of \$1 in nickels.

Kim Montague:

Yep.

Pam Harris:

So if you know half of \$1 is 10.

Kim Montague:

Yep.

Pam Harris:

And a full dollar is 20.

Kim Montague:

Okay, 15/20. I'm with you.

Pam Harris:

Okay, cool. So three, this was real, right? Live, here we go. So three quarters of dollars is 15 nickels out of 20 nickels. What's three fifths of \$1 in nickels?

Kim Montague:

Three fifths of a dollar in nickels?

Pam Harris:

Can you talk out loud? What are you thinking? Are you thinking about 1/5 of \$1 in nickels?

Kim Montague:

Yeah, so that's 20 cents. So 1/5 is 20 cents. So that's four nickels.

Pam Harris:

Four nickels.

Kim Montague:

So it's 4/20th.

Pam Harris:

That's 1/5. But we want three fifths.

Kim Montague:

So it's 12 nickels.

Pam Harris:

So 12 nickels out of 20 nickels is three fifths.

Kim Montague:

So I've got 17 nickels is 65 cents. Yep. Wait, oh, I'm adding again. I'm adding again. So 15/20 minus 3/20.

Pam Harris:

Nope, nope, minus 12/20.

Kim Montague:

12/20. Right? Jiminy cricket, not helping our listeners at all. 15/20 minus 12/20 is 3/20, which is 15 cents.

Pam Harris:

Yeah. And that was the 15 cents you got earlier. And Kim's like, "Why did you make me work in nickels?"

Kim Montague:

My goodness.

Pam Harris:

Yeah. And so Kim doesn't often work in nickels, because she doesn't have to, because she's got other great strategies. But if we can get kids to sort of think in terms of nickels, it can be really helpful, again, towards the equivalence. It's all about equivalence

Kim Montague:

And we should. Right? Like, I lean towards decimals and percents. But I need to be more well rounded. Right?

Pam Harris:

I love it. And you were willing to go there. And it was really cool. And you guys, working memory is at a premium. It's so much harder to do stuff when you don't write things down. And you heard Kim just go, "I gotta write this down." And when you write it down, it frees up your working memory to think and to reason. And we want to encourage kids to think and reason. Just because kids have a pencil in their hand doesn't mean do an algorithm. No, it can free you up to just think and be able to use that working memory to use the relationships that you know to solve problems. Alright, totally cool. I have a couple more problems. So we're gonna, Kim and I will go offline and we'll talk about whether we're going to do those or what we had planned to do next in our next podcast. Thank you so much for joining us today, as we have thought about using money to help us think about equivalence with fractions. We're going to end with a really interesting question that we want you guys to think about. If we use money to help us think about equivalence, for what denominators is money useful?

Kim Montague:

Yeah.

Pam Harris:

Because it's not for all denominators.

Kim Montague:

Right.

Pam Harris:

But there are some and could you kind of like maybe choose some, like maybe name some off, but also kinda try to get general? You know, like, if you were sort of generalize, what denominators is money, a really helpful model for? Yeah?

Kim Montague:

Yeah.

Pam Harris:

Alright, y'all remember to join us on hashtag MathStratChat on Facebook, Twitter, or Instagram on Wednesday nights where we explore problems with the world.

Kim Montague:

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:

And also don't forget that we're collecting your questions that you want answered. Send them to Kim at mathisFigureOutAble.com because she's organized or not, and we'll tackle them in an upcoming episode. We would really appreciate it. Y'all 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.