Math is Figure-Out-Able with Pam Harris

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

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 familiar circumstances. In this episode Pam and Kim use coin denominations to demonstrate how students can learn to reason about fractional equivalence. 
Talking Points

  • If you understand fractional equivalence, you don't have to memorize any rules for operators of fractions
  • Interchanging use of decimal and fraction notation in a part-whole scenario
  • Kim's uncomfortable relationship with nickels
Show Notes Transcript

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 familiar circumstances. In this episode Pam and Kim use coin denominations to demonstrate how students can learn to reason about fractional equivalence. 
Talking Points

  • If you understand fractional equivalence, you don't have to memorize any rules for operators of fractions
  • Interchanging use of decimal and fraction notation in a part-whole scenario
  • 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 memorizing, but it's 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, we answer the question, 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 in part three we talked all about fair sharing. Today we're going to talk about something a little bit diferent.

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, thanks 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 afternoon. Yeah. So the gamut she ranged, that K-12, she could kind of see the whole perspective. One day Garland said to me, okay, this is 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 you know, 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 the equivelance of fractions, 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 quotes 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 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 and own 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, see, he really taught me a lot. I was like, What do equivalent fractions mean to you? If I say equivelant fractions, what does that even mean? Because I was reaching you guys. I was like, What was 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 he goes, you know, like if you have a pizza, and you've got it cut into these 7 pieces. But if you double the number of pieces, like you cut them all in half, but you only get a slice, you only get half as much. Again, brain blown. I was like you do understand equivalent fractions! Like it was so fascinating to me that his response had nothing to do with the procedure, and everything for him to do with really understanding equivalence. Now, that's not to say that in that sentence, all equivalence was explained. But he really had a sense and a feel for kind of 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 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 and 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. I have 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 stream with Kim -

Kim Montague:

Who hasn't seen the problems yet.

Pam Harris:

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? All right, thinking about money, 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. 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, nope, nope, this time, it's a fourth of $1.

Kim Montague:

Okay,

Pam Harris:

Hey, that notice there? There's that operator, meaning coming in. One 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.

Pam Harris:

Okay, one quarter out of the four quarters of a dollar is 1/4. Okay?

Kim Montague:

Yep.

Pam Harris:

What's another one?

Kim Montague:

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 onehundreths. But I didn't say 25 onehundreths. As I repeat back what you say I'm just saying, Ah, 25 pennies out of the 100 pennies. Okay, any other fractions? For a fourth dollar?

Kim Montague:

In 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 haven't thought about nickels for a while. I don't know if it's because 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. Okay, how many? In fact, maybe let's do dimes and then back back up to nickels. Can you make a quarter of $1? Oh, no, that's not so good with dimes. So let's say nickles, sorry. But how many dimes are in a dollar, I asked the seventh graders. And they said, well, yeah, that's 10. Pretty much they could do that. But I said, Okay, so 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, 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?

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? 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 got 10 dimes in $1.

Kim Montague:

Yeah.

Pam Harris:

Then 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 said. 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, where 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:

So it's 25 cents out of 100 cents or $1.

Pam Harris:

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

Kim Montague:

So you mentioned dimes. Can we can we go to dimes? In $1? If I'm thinking about a quarter, I think you can think about dimes.

Pam Harris:

A quarter like a quarter dollar? But are you referencing the coin, a quarter, that's 25 cents?

Kim Montague:

No. A 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. So you're thinking that I could represent the fraction one quarter, 1/4, one over four. And I'm 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. Yep. Cool. So on my paper I literally have written down now: one divided by four 1/4 equals 25 out of 100, 25 over 100 equals five out of 20, equals 2.5 out of 10. Yeah, 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, 2.5 out of 10. Nice place value shift there, right? Where i could 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 one half plus 1/4? But I want you to think about money. When 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 the dollar is 50 cents.

Pam Harris:

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

Kim Montague:

And a fourth of $1 is 25 cents.

Pam Harris:

So 25 out of 100.

Kim Montague:

And I'm actually writing in decimals do you want me to-? I'm writing it out as I'm thinking because I wrote 50 cents plus 25 cents equals 75 cents. And then I thought about -

Pam Harris:

When you said cents .5 + .25 = .75

Kim Montague:

Uh huh. Yah. Are you ok with me keeping it in in decimals or do you want me to talk out loud in fractions?

Pam Harris:

I mean, I wrote it in fractions.

Kim Montague:

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

Pam Harris:

Yah, actually. And we want kids to be able to do both. And say we're 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, five dimes out of 10 dimes, ok.

Kim Montague:

Plus two and a half dimes.

Pam Harris:

2.5 out of 10.

Kim Montague:

That's seven and a half dimes,

Pam Harris:

Ah, seven and a half times. 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. 7.5 out of 10. Again, that nice place value shift. Or if we'd written it in decimals, I'd have point five 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 wrote the decimals wrong because I was trying to translate straight from the tens instead of the hundreds. And I can't do that when -

Kim Montague:

Talking and writing at the same time.

Pam Harris:

And so I tried to I tried to translate directly from tense 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 a podcast. This is live here we are alive. It's all good. Okay, so you just did pennies.

Kim Montague:

Yeah.

Pam Harris:

And you did dimes? 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. You're making me think.

Pam Harris:

Excellent. I like it when you have to think. Alright, here's the next problem. 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, we probably should say and we didn't say

this earlier:

once I give the problem, you all pause the podcast. Think about these numbers before you hear just kind of how Kim's thinking about them. Okay Kim a 10th plus a 20th go ahead.

Kim Montague:

Okay a 10th is a dime. So one of 10. 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 were really think in terms of decimals?

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 five pennies. 15 pennies. So 15 out of 100.

Pam Harris:

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

Kim Montague:

I kind of want to take dimes next.

Pam Harris:

Okay, let's go dimes.

Kim Montague:

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

Pam Harris:

So the 1/10 doesn't really change. That's the number of dimes, ok.

Kim Montague:

And then for the 20th that 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. 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. I like that. So let's see. Can you do nickels?

Kim Montague:

Oh yes. So 1/10 is two 20ths, 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 notice that we won't ever use the word reduced to mean simplified because that means like it 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 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 the simplified either, but it's better, 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 how you made sense of it. But let's use simplified. My point was, sorry, I just digressed. My point was that we are not emphasizing the simplified version. Because when students understand equivalence, then if on a standardized test or 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, they'll especially be able to recognize it. So we don't make a big deal of that. Maybe ever, but especially not at the beginning. All right, 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 I can just give you one way to think about it and listeners can think about a couple others.

Pam Harris:

Okay. But it's killing me because I wanted to do...

Kim Montague:

Okay, go. Just pick one.

Pam Harris:

Ok, three fourths minus three fifths. Wow, subtraction! Yep. Because once you understand equivalents, nothing changes. You just keep thinking of the same pieces.

Kim Montague:

All right. So three fourths is 75 pennies. 75 cents, okay. 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. So 75 hundreths and 60 hundreths, what is that?.

Pam Harris:

And when we're subtracting,

Kim Montague:

Oh, thank you. So 75 hundreths minus 60 hundreths is 15 hundreths.

Pam Harris:

And so your final answer is like 15 hundreths. Or point one five if you're in decimals. Point 15. I really liked, 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 will let you guys think about that in other ways? You totally did pennies. But you could have done dimes you could have done nickels. In fact, should we do? No? Yeah. Can we do nickels? Can we 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. Three fourths?

Pam Harris:

We're letting Kim think.

Kim Montague:

Right. I hate thinking 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 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 are a half and 20 nickels are a hole isn't 15 right in between for three quarters?

Kim Montague:

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

Pam Harris:

Yeah. And working numbers is real, right?

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. So if you know half of $1 is 10. Yep. And a full dollar is 20.

Kim Montague:

Okay, 15 20ths, I'm with you now.

Pam Harris:

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

Kim Montague:

3/5s of a dollar in nickels?

Pam Harris:

Can you talk out loud? What do you think?Are you thinking about 1/5 of $1?

Kim Montague:

Yes, so that's 20 cents. So 1/5 is 20 cents. That's four nickels. So thats 4 twentieths.

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 three 3/20. Nope, nope, minus 12/20s for 3/5s. 12/20. Right. Chiminey cricket, I'm not helping our listeners at all. 15/20 minus 12/20 is 3/20. Which 15 cents.

Pam Harris:

Yeah. And that was the 15. And Kim's like, Why did you make me work in nickels?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 equivalent, its all about equivelance.

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 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. All right, totally cool. I have a couple more problems. So Kim and I will go offline and we'll talk about whether we're going to do those or what we have 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? Because it's not for all denominators.

Kim Montague:

Right.

Pam Harris:

But there are some and can you kind of like maybe choose some, like maybe name some off, but also kinda try to get general? You know, like, if you can sort of generalize, what denominators is money a really helpful model for? Yeah? All right, y'all remember to join us on hashtag MathStratChat on Facebook, Twitter, or Instagram on Wednesday eves 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@mathisFigureOutAble.com because she's organized and I'm 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 then don't miss the Math is Figure-Out-Able Podcast because math is figure-out-able!