Pam Harris  00:02

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

And we're here to suggest that mathematizing is not about mimicking or rote memorizing, but it's about thinking and reasoning, about creating and using mental 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  00:33

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  00:53

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  02:21

Yeah. 

Pam Harris  02:21

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  04:54

Who hasn't seen the problems.

Pam Harris  04:56

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  05:15

Okay.

Pam Harris  05:15

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  05:39

Okay. 

Pam Harris  05:39

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  05:49

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

Pam Harris  05:55

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  06:06

Yeah. Okay. 

Pam Harris  06:07

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

Kim Montague  06:09

Yep. 

Pam Harris  06:09

What's another one?

Kim Montague  06:10

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

Pam Harris  06:17

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  06:37

Yeah, nickels would be five nickels of 20 nickels.

Pam Harris  06:42

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  07:36

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

Pam Harris  07:41

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

Kim Montague  07:45

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  07:54

Uh-hum.

Kim Montague  07:55

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

Pam Harris  08:02

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

Kim Montague  08:06

Yeah. 

Pam Harris  08:07

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

Kim Montague  08:11

Yeah. 

Pam Harris  08:11

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  08:19

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  08:31

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

Kim Montague  08:34

Sorry.

Pam Harris  08:34

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  08:41

Yep. 

Pam Harris  08:42

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

Kim Montague  08:45

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

Pam Harris  08:48

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

Kim Montague  08:50

So you mentioned dimes. Can we go to dimes?

Pam Harris  08:53

Let's go to dimes. Absolutely.

Kim Montague  08:53

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

Pam Harris  09:00

And a quarter like... 

Kim Montague  09:02

A quarter of a dollar. 

Pam Harris  09:03

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

Kim Montague  09:06

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

Pam Harris  09:10

Okay. 

Kim Montague  09:11

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

Pam Harris  09:16

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  09:34

Sure. 

Pam Harris  09:34

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  09:42

Yep. 

Pam Harris  09:43

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  09:56

Yeah. 

Pam Harris  09:56

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  10:33

Yeah, so half of a dollar is 50 cents.

Pam Harris  10:37

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

Kim Montague  10:39

And a fourth of a dollar is 25 cents. 

Pam Harris  10:43

So 25 out of 100.

Kim Montague  10:45

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  10:55

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

Kim Montague  10:59

Yeah,

Pam Harris  11:00

Uh huh. 

Kim Montague  11:00

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

Pam Harris  11:05

I mean, I'm I wrote it,

Kim Montague  11:07

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

Pam Harris  11:08

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  11:36

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

Pam Harris  11:40

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

Kim Montague  11:42

Plus two and a half dimes. 

Pam Harris  11:45

2.5 out of 10.

Kim Montague  11:46

Is seven and a half dimes.

Pam Harris  11:48

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  12:19

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

Pam Harris  12:21

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  12:39

Talking and writing at the same time.

Pam Harris  12:41

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  12:58

Unscripted.

Pam Harris  12:58

Okay, so you just did pennies. 

Kim Montague  13:00

Yeah. 

Pam Harris  13:00

And you did dimes. 

Kim Montague  13:01

Yeah.

Pam Harris  13:01

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

Kim Montague  13:03

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

Pam Harris  13:08

So 10 out of 20. 

Kim Montague  13:09

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

Pam Harris  13:17

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

Kim Montague  13:22

Yeah.

Pam Harris  13:22

Nice. 

Kim Montague  13:23

You're making me think.

Pam Harris  13:25

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

Kim Montague  13:28

Okay.

Pam Harris  13:29

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

Kim Montague  13:37

Okay, so a 10th is 10 cents.

Pam Harris  13:42

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  13:52

Okay 10th is dime. So one of 10. 

Pam Harris  13:56

Okay. 

Kim Montague  13:57

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

Pam Harris  14:04

So you are really think in terms of decimal. 

Kim Montague  14:06

I was. Yeah. 

Pam Harris  14:07

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

Kim Montague  14:11

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  14:18

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  14:30

I kinda want to take dimes next. 

Pam Harris  14:32

Okay, let's go dimes. 

Kim Montague  14:33

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

Pam Harris  14:38

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

Kim Montague  14:41

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

Pam Harris  14:46

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  14:53

Yeah. 

Pam Harris  14:54

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  15:04

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

Pam Harris  15:08

Two nickels out of the 20 nickels.

Kim Montague  15:11

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

Pam Harris  15:16

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  17:06

Sure. 

Pam Harris  17:07

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  17:13

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  17:28

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  17:39

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

Pam Harris  17:43

Okay. 

Kim Montague  17:45

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  17:58

Okay. 

Kim Montague  17:59

So 75/100s and 60/100s.

Pam Harris  18:04

We're subtracting,

Kim Montague  18:05

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

Pam Harris  18:11

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

Kim Montague  18:15

Yeah. 

Pam Harris  18:15

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  18:48

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

Pam Harris  18:59

We're letting Kim think. 

Kim Montague  19:00

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

Pam Harris  19:06

How many nickels are in $1.

Kim Montague  19:08

20. 

Pam Harris  19:09

How many nickels are in half $1.

Kim Montague  19:11

10. 

Pam Harris  19:12

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

Kim Montague  19:16

Say that again? 

Pam Harris  19:18

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  19:28

I'm blank, I can't. 

Pam Harris  19:30

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  19:37

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

Pam Harris  19:39

Yeah. And working memory is real.

Kim Montague  19:42

I started to and I walked away. Go ahead. 

Pam Harris  19:44

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

Kim Montague  19:48

Yep.

Pam Harris  19:48

So if you know half of $1 is 10. 

Kim Montague  19:51

Yep. 

Pam Harris  19:51

And a full dollar is 20. 

Kim Montague  19:54

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

Pam Harris  19:56

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  20:05

Three fifths of a dollar in nickels? 

Pam Harris  20:10

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

Kim Montague  20:16

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

Pam Harris  20:21

Four nickels. 

Kim Montague  20:23

So it's 4/20th.

Pam Harris  20:25

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

Kim Montague  20:27

So it's 12 nickels.

Pam Harris  20:29

So 12 nickels out of 20 nickels is three fifths.

Kim Montague  20:32

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  20:44

Nope, nope, minus 12/20. 

Kim Montague  20:45

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  20:57

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

Kim Montague  21:01

My goodness. 

Pam Harris  21:02

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  21:13

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

Pam Harris  21:19

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  22:15

Yeah. 

Pam Harris  22:15

Because it's not for all denominators. 

Kim Montague  22:16

Right. 

Pam Harris  22:16

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  22:32

Yeah. 

Pam Harris  22:32

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  22:40

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  22:47

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.