March 14, 2023
Pam Harris
Episode 143

Math is Figure-Out-Able with Pam Harris

Ep 143: Fraction Division Part 2

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Math is Figure-Out-Able with Pam Harris

Ep 143: Fraction Division Part 2

Mar 14, 2023
Episode 143

Pam Harris

Fraction division does not have to be scary, and it certainly does not have to be memorized. In this episode Pam and Kim show how to reason through non unit fraction division problems.

Talking Points:

- What is a unit fraction?
- What does it mean to think quotitively about division?
- A Problem String building on unit fraction understanding for division of non unit fractions
- How to model this Problem String
- Using whole number relations among fractions
- Fraction division is accessible!

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

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Fraction division does not have to be scary, and it certainly does not have to be memorized. In this episode Pam and Kim show how to reason through non unit fraction division problems.

Talking Points:

- What is a unit fraction?
- What does it mean to think quotitively about division?
- A Problem String building on unit fraction understanding for division of non unit fractions
- How to model this Problem String
- Using whole number relations among fractions
- Fraction division is accessible!

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Pam:

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

Kim:And I'm Kim.

Pam:And you have found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim:In our last episode, we started answering a question from a follower of our Math is Figure-Out-Able, teacher Facebook group, and so today, we've decided we're going to continue to answer Beth's question. And, Pam, I wanted to tell you that this is also timely because not too long ago, I got a text from a good friend of mine that you know well, and she said, "Hey, Kim! Here's a problem that my kid has to solve. How can I think about this?" And I loved her question because it wasn't "What's the answer?" Like, "Just tell me what to tell my kid." She's interested in thinking. So, I think there are probably plenty of people out there who want to learn how to think about division of fractions.

Pam:Absolutely. And so, Beth Roberts, thanks for asking the question about how to teach division of fractions. So, we started last time. We did some whole number divided by fraction. We also did some mixed number divided by fraction. Let's dive in more to some reasoning that we would want to develop about fraction division. Last week, we did some problems like 3 and a 1/3 divided by 1/3, and we thought about how many unit fractions are in a bigger number. So, all of the problems last week had dividends, or the sort of first number in the division problem, that were bigger than the divisor. Like, 3 and 1/3 divided by 1/3. So, 3 and 1/3 is bigger. The divisor's 1/3. How many 1/3s can we get out of 3 and 1/3. It's very quotitative approach to thinking about division. In a way what we did last time was really focus on the definition of a unit fraction. What's a unit fraction? It's when the numerator's 1. So, it's like my fair share when I'm thinking about sharing. So, if I'm sharing a candy bar, and I'm going to share it with 4 other people. So, 5 of us are sharing, and I share it fairly, then what do I call my piece? I call my piece one-fifth because 5 people are sharing. Well, what if 7 people were sharing? Then I call it one-seventh. The definition of a unit fraction is sort of one person's fair share. If you share the thing fairly with that many people. That's important. We need to know what one-twelfth looks like, one-eighth looks like, one-half looks like. We need kids to reason about the relationships between that unit fraction and a whole, the unit that were sort of cutting up and dividing fairly. Yeah?

Kim:Yep.

Pam:So, today, we're going to continue to take a quotitive approach, how many of the divisors are in the dividend? So, for example, if I was dividing a whole number problem like 210 divided by 10. If I'm looking at quotitively, that's asking the question, "How many 10s are in 210? So, if I was going to reason about how many 10s are in 210, I might reason about how many 10s are in 100 because I know that, and then I could double that to think about how many 10s are in 200, and I can just tack on that extra one more 10 in 210. So, that would be a way of me sort of reasoning about the number of 10s is 210. I wonder if we could use similar reasoning that we just did with whole numbers. Could we use similar reasoning to think about division fractions? So here we go. Alright, Kim. If I had a candy bar, and I'm going to split it up, and share it. And I'm thinking about sharing it with 4 other people, so five people are sharing. What is 1 candy bar divided by 1/5? How many 1/5s can I get out of 1 candy bar?

Kim:Well, your going to have 5 because you just told me how many people you have. So, one-fifth. There are five 1/5s in 1.

Pam:Did I do too much there? Too much of your own thinking? So, I want to start in a place where kind of everybody can access. Again, we're kind of reminding ourselves what we did last time. And so, you're saying that the answer to 1 divided by 1/5 is 5 because there's five 1/5s in one. So, I've quickly sketched a candy bar. I've sort of eyeballed dividing it into 5 equal pieces. I've kind of dotted those, put some dotted lines, so we can see 5 chunks of that rectangle. And I've labeled the first chunk. I've labeled it one-fifth. So, a little bit less labeling then I did in the Problem String that we talked about in the last episode. Okay, next problem. I've only got half a candy bar, Kim. So, what is 1/2 divided by 1/5? And what does that mean with our candy bar scenario?

Kim:So, when you had a whole candy bar, 1 divided by a 1/5, then you had 5 people or 5 pieces. But now you have only half as much candy bar, so you're going to have half as much people or per person. Let me think about what you... I can't remember exactly what your scenario was. 1/2 divided by 1/5 is 2 and a 1/2. Because in the previous problem, you said 1 divided by 1/5, and now we have 1/2 divided by 1/5, so the answer is going to be half as much.

Pam:Since the answer to the first problem is 5, the answer to the next problem is going to be half as much, which is 2 and a 1/2. 1/2 of 5 is 2 and a 1/2. Yeah.

Kim:Yeah.

Pam:So, you were like, "What was the scenario again?" So, that's a good question. Like, I said I have a whole candy bar, and I want to share it with 5 people, so they each get... Well, I guess what I should say is, I've got a whole candy bar, and I've cut it into 5 equal sections.

Kim:There you go. Yeah. (unclear).

Pam:I've cut it into 5 sections. Cut it into fifths. How many people can I give a fifth of a candy bar to?

Kim:I think that's why I was like, "Wait a second..." because in this scenario, it'd be 2 and a 1/2 people.

Pam:Yeah. So, that's kind of weird, right? How many fifths can I get out of 1/2 a candy bar? (unclear).

Kim:Two and half 1/5s. Yeah.

Pam:I can only get two and a half of those 1/5s. So, that's not so great, right, for 2 and a 1/2 people, so... Yeah, that's kind of weird to think about. But I can think about. I can only give that same amount to half as many people.

Kim:Yes.

Pam:Cool. Yeah. Okay. So, what if I had a problem like... I'm back to a whole candy bar, but this time I'm going to split it into tenths.

Kim:Okay.

Pam:How many tenths can I get out of that candy bar?

Kim:Ten.(unclear)

Pam:Because?

Kim:One-tenth because there's ten-tenths in a whole.

Pam:Ten-tenths in a whole. So, 1 divided by 1/10 is 10. How does that relate to the problem 1 divided by 1/5?

Kim:Oh, good question. I hadn't thought about that.

Pam:I like it when I get Kim to think. Bam! That's a good day.

Kim:Let me think if I can say this right. I still have 1 candy bar, but instead of fifths, now the size of the piece is half is big. Right, so instead of fifth, now I have tenths. The size of the piece is half as big, so I'm going to have twice as many of those pieces.

Pam:Right? Yeah. Because before I had five 1/5s, but I've cut the, now I'd have tenths. And tenths are half as big as fifths. That's like the definition of a tenth, right?

Kim:Yep.

Pam:In fact, often if we say to kids, "Hey, draw a rectangle, and cut it into tenths." Some of them will cut it into fifths, and then they'll cut each of those fifths in half.

Kim:Yep.

Pam:And then, they've got tenths. Cool. So, you're saying that if there's five 1/5s in 1, 1 divided by 1/5 is 5, then there's ten 1/10s in 1 because you've cut the pieces in half, so there's twice as many. Nice. What if... Oh, so on my candy bar, I went ahead and cut the candy bar into fifths again. So, I've kind of underneath the one I had before. I've cut it into fifths, and then each of those... Actually, I've only cut the first one in half to be honest with you, and I've labeled that piece as 1/10, and I've labeled the second one-tenth as 1/10. So, I have 2 little pieces labeled as 1/10. And I kind of have an overarching one of those bracketty things above the two 1/10s and I've labeled that as one-fifth.

Kim:Okay.

Pam:Does that make sense? So, I kinda have a one-fifth labeled, and I've cut it in half, and I've labeled each of those halves as one-tenth. Alright, cool. Next problem. How about if I ask you for 3/5 divided by 1/5?

Kim:Three. Three 1/5s in 3/5.

Pam:There's three 1/5s in 3 is your answer? That's your final answer?

Kim:Yep.

Pam:Cool. So, I might draw that whole candy bar, and I might divide it into fifths, and then I might circle three-fifths, three of those 1/5s. And then, I might say, "So, that's what you're focused on is those three 1/5s?" Because you just were like, "Bam! 3." And in those three 1/5s, you can see three 1/5s. So, it's interesting because I actually want to call... So, the problem is 3/5 divided by 1/5. I actually want to say, three 1/5s divided by 1/5. How many 1/5 are in three 1/5s. And I think you said that. How many 1/5 are in three 1/5s. There's 3 of them. So, listeners, you might recognize that I'm calling the non-unit fraction three-fifths. I'm actually naming it three-fifths, but I'm also naming it three 1/5s. Not 3 and 1/5, but three 1/5s. And I'm simply writing the fraction 3 "fraction bar" 5, but it can have two names, and one of the names is three-fifths and one of the names is three 1/5s. And we find that helpful. We find it helpful to talk about these non-unit fractions in terms of their unit fractions. That can be helpful in some cases. Alright, cool. How about the next problem? The next problem is 3/5 divided by 1/10.

Both Pam and Kim:(unclear)

Pam:Yes. I want to know what you really thought about, and then I'm going to ask you a question. Go ahead. What are you really thinking?

Kim:So, what I really thought. I know that we're living in land of fractions, and I see a relationship between...

Pam:Oh, no. Oh, no. I know where you went.

Kim:In a previous problem, but I really thought $0.60 divided by $0.10. Like.

Pam:Because you're just like this percent girl. You were...

Kim:Well, I mean, percents, or dimes, or whatever. Like, all of it. Like, it's all connected. So.

Pam:Yeah.

Kim:Yeah, I thought about how many dimes were in $0.60.

Pam:Because three-fifths is $0.60. And one-tenth... I have to say it out loud. And one-tenth is $0.10. So, it's like you said $0.60 divided by $0.10. How many $0.10 are in $0.60? And so, you said the answer's 6. Cool. Okay. I have no problem with that. I just realized, I totally leaned away from my microphone. Sorry about that. If I was working with kids, and I saw someone with that strategy. I'm circulating around. I'm listening and stuff. Or even if a kid... Well, let's say I'm circling around, and I saw that strategy, I might say to that kid. I might give him a fist bump and go, "Nice thinking. Hey, can you think about it in terms of fractions and the candy bar I have on the board?"

Kim:Yeah.

Pam:So, I'm going to say like, "Nice thinking. Can you also think about it in terms of like the candy bar I have on the board?"

Kim:Yeah.

Pam:So, Kim, I'm curious. Can you? Can you think about it in terms? Especially like the problem before? So, I know you guys... A lot of listeners are probably listening. So, the problem before was "How many one-fifths are in three-fifths?

Kim:Yeah.

Pam:How would you say that quotitively for the problem we just said? How would you say the problem we just did quotitively?

Kim:So, when it was 3/5 divided by 1/5, then there were three 1/5s. Now, I have smaller pieces. The pieces are half as big, and so I have...

Pam:Because you have one-tenth.

Kim:Yeah. One-tenth. They're half as big as a fifth, and so I'm going to have twice as many pieces.

Pam:And so, you're still getting 6 dimes and $0.60, but you're also getting 6... I'm going to try to let you finish that sentence. You're also...

Kim:Oh, sorry. I zoned out.

Pam:That's okay. Alright, so you're getting 6 dimes... It's okay. It's a real thing. You're getting 6 dimes in $0.60. Now, I'm looking at the fractions, and if I was pointing to the board, you see I was pointing at?

Kim:Six 1/10s in three-fifths.

Pam:Six 1/10s in three-fifths.

Both Pam and Kim:Yeah.

Pam:Cool. And you kind of compared that to the problem before. So, if we have three 1/5s in 3/5. Now, you have smaller pieces. You've cut the pieces in half. You'd have twice as many pieces, right? What if you just were thinking about this problem? If I literally just said for cold one-tenths? How many 1/10 in 3/5? I'm just curious if there's any other thinking that comes to mind. And I'm pointing. You can't see. I'm pointing at the picture that I had on the board when we were talking about how many. I'm interrupting your thinking. Sorry, Kim.

Kim:No, go ahead.

Pam:You want me to tell you where I'm pointing? Okay, I'm going to say where I'm pointing. So, on the board, when we did 3/5 divided by 1/5, I drew a whole candy bar. I cut it into fifths, and I circled three of the 1/5s. And I said, "How many 1/5 are in 3/5." Now, I'm still looking at a candy bar that's cut into fifths, and I've circled three of the 1/5s. And now I'm asking how many 1/10 are in 3/5?

Kim:So, within each one of those fifths, there's two-tenths. Is that what you're asking?

Pam:Mmhmm. So, if there's...

Kim:I don't think I understand your question.

Pam:Yeah. So, where you just said. So, in each of those fifths. I've circled three-fifths, right? And the

question is:How many tenths are in three-fifths? And you just said, "Well, in each of those tenths, there's two 1/5s." So, you've kind of got two 1/5s, two 1/5s, two 1/5s.

Kim:The other way around. In each of those fifths, there's two-tenths.

Pam:Ha, I said it backwards. You're right. In each of those fifths is two-tenths. So, you've got two 1/10s, two 1/10s, two 1/10s, so you've got six of those 1/10s in 3/5.

Kim:Yes.

Pam:Just kind of sort of looking at the picture.

Kim:Oh, the model. Yeah.

Pam:The model where I kind of had 2/10 in each of those 1/5s.

Kim:Yes.

Pam:And so, I could sort of like. Well, if there's two-tenths in one-fifth, and I've got three of those 1/5s, then I have three times those two would be?

Kim:Six 1/10s.

Pam:Yeah, I wish I was pointing. It's so much better pointing. Maybe, Kim, we should do a video podcast one day, and we can actually be pointing at stuff. Ya'll, we thought about doing a video podcast, and every time we do, it's me that's the holdout because I'm like, "I'd have to like brush my hair, and..."

Kim:Stop it. That's not true.

Pam:Like, I mean, right now what am I wearing? I'm wearing jeans and a hoodie? And my glasses.

Kim:(unclear).

Pam:I did brush my hair this morning, but I didn't do anything else to it. Anyway, okay, so.

Kim:We would just laugh at each other the whole time. (unclear)

Pam:Well, you know the other reason we've pushed back on doing any video podcast. We would laugh at each other the whole time. But is we worry that... Well, like right now, I'm really cognizant that I'm talking to people who are just listening.

Kim:Yes.

Pam:And if we could point, we would probably say less because we would be able to point, and then anybody listening. So, hey, podcast listeners? We'd be super interested to know would it be worth it? Would you be like, "Oh, really do a video one because we'd understand so much better!" Or you're like, "No. Really, I'm listening in my car, and I would not be working, so keep..." Yeah, I'd be super curious. Alright. Our last problem was how many tenths are in three-fifths. Or 3/5 divided by 1/10?

Kim:Yeah.

Pam:The next problem is 1 and 3/5 divided by 1/10? 1 and 3/5 divided by 1/10. What are you thinking about?

Kim:$1.60.

Pam:Alright, how many? How many dimes? Well, go ahead. How many dimes are in $1.60?

Kim:How many dimes are in $1.60?

Pam:Yeah, how many?

Kim:So, 16. But also, I've been writing down the problems. I have not been drawing them all model, but I've been writing down the problems. And you asked me 1 divided by 1/10, and you asked me 3/5 divided by 1/10.

Pam:Mmhmm.

Kim:So, when it was 1 divided by 1/10, it was 10. And 3/5 divided by 1/10 was 6. So, if you put those together, 1 and 3/5 divided by a 1/10 would be the same 16.

Pam:Would be that 10 plus the 6 is 16.

Kim:Yeah.

Pam:Yeah, nice. Nice. I love that reasoning. Super. And if I'm doing this Problem String with students those things are on the board, so students have access to those. I wonder then if I might grab that problem 1 divided by 1/10 and that model that we used to see that one 1/10 in that whole candy bar. And then, I might also grab the model where I had the 3/5 divided by 1/10 and grab that piece of it, and show those six 1/10s. And I've got the ten 1/10s, and the six 1/10s would be the sixteen 1/10. Nice. Next question. How about if I asked you 1 divided by 1/3?

Kim:That would be 3. Three 1/3 thirds, in 1 whole.

Pam:Cool. Because we just got a whole, and you're asking how many one-thirds are in 1. That's just three. I don't know why I repeat what you say every time, but it helps me think about what... It really helps me think about what you just said. So, I know sometimes listeners, you might be like, "Why does she repeat what Kim says?" Well, it's because I have to think about what Kim said, and it helps me think about it. Okay, so you said that was 3.

Kim:Yep.

Pam:Now, I'm going to ask you two-thirds divided by one-third. Two-thirds divided by one-third.

Kim:Two.

Pam:And how do you know?

Kim:I feel like we're going somewhere fun. I'm excited.

Pam:Okay.

Kim:Two 1/3s is 2/3.

Pam:So, you're thinking quotitively. You're saying how many one-thirds are in two-thirds? And like there's two of those. There's two. And we mentioned earlier that we could say "two-thirds" as two 1/3s. Yeah. So, when I say that problem like that two 1/3s divided by 1/3. You could be like, "Well, how many 1/3 are in two 1/3? Two." Alright, so 2/3 divided by 1/3 is 2. Cool. Next question. How about two-thirds divided by one-sixth? Two-thirds divided by one-sixth?

Kim:That's going to be 4.

Pam:Because?

Kim:Because just like we were talking about with the fifth and the tenths, when you asked me about how many 1/3 are in 2/3, then that was 2. But now, the size of the piece is half as big, so there's going to be twice as many pieces.

Pam:Cool. I love that relationship.

Kim:Yeah.

Pam:That's a little bit more sophisticated than maybe I might ask a kid to describe first. I might have a kid say "Well, I know there's two-sixth in every one-third."

Kim:Yeah.

Pam:So, if there's two 1/3s, I can sort of picture those. That 1/3 and that 1/3. I know there's two of them in two-thirds. In each of those one-third, there's two-sixths. And so, yeah. Kind of like your twice as many. But I might be looking a little more additively at first. There was a reason I mentioned that is I want to make sure people are understanding. Yes, I want kids thinking multiplicatively about these. And that's what Kim just did. "Well, if the pieces are half as big, then there's going to be twice as many." But I also can allow kids to enter as they're sort of thinking about it as additively. I want to help kids enter thinking about additively, and then also help them transition to thinking about it more multiplicatively. Alright, what is two-thirds divided by one-twelfth?

Kim:Eight.

Pam:And how do you know?

Kim:When it was 2/3 divided by 1/6, that was 4. So, now again the pieces are half as big, so there's going to be twice as many, which is 8. But also I could.

Pam:Oh, go ahead. Well, let me just repeat that a slightly different way.

Kim:Yeah.

Pam:So, if I'm asking how many 1/6 are in 2/3, and you're saying there's 4. And then, I ask how many there is of something half as big in two-thirds. How many one-twelfths are in two-thirds, then you're like, "There's got to be twice as many. There's got to be 8." Okay. Okay, what else we're you going to do?

Kim:Well, you had asked me earlier, two-thirds divided by one-third.

Pam:Mmhmm.

Kim:So, I'm thinking that I know the relationship between one-twelfth and one-third, and it's going to be... It's a fourth as big. So, we just did the half, half, and double relationship.

Pam:Mmhmm.

Kim:But there's also a quadrupling, quartering relationship to the previous problem, two-thirds divided by one-third.

Pam:So, if a twelfth. Oh, keep going,

Kim:If a twelfth is one-fourth as big as a third, then I'm going to need four times as many pieces.

Pam:So, instead of getting two 1/3s in two-thirds, you're going to have eight 1/12s in two-thirds. Nice. And I didn't talk about the modeling here, but I would also be making sure that I have a whole candy bar. And when we talked about thirds, I would have, you know like, sketched out the two-thirds of the candy bar. So, I've got the whole candy bar. I've sort of circled the two-thirds because that's the whole that we're dealing with, that's the unit we're dealing with every time. And then, the first question I asked: How many one-thirds were in there? And you were like 2? And then, I asked how many one-sixth were in there. We would have cut each of the thirds in half to show those one-sixth. And you're like, "There's the four." And then, I would have said, "Like where are twelfths? Like, how are you thinking about twelfths? How would you even cut a candy bar into twelfths?" And we would have talked about how twelfths were half as big as the sixths like you said, but they're also a fourth of the thirds. And there's that quarter, quadruple relationship. Nice.

Kim:I have a question.

Pam:(unclear)

Kim:I don't want to derail this because I'm afraid we'll go a little too long. But I wonder what you think about this.

Pam:Okay.

Kim:Sometimes, I think that when we have numbers like two-thirds and one-twelfth, sometimes that can be maybe a little bit overwhelming or a little bit like, "Oh my gosh. I don't really know" kind of situation for kids until they have, you know, a lot more experience or at least some more experience. But I wonder what you think about going back to the unit fraction of instead of two-thirds saying "Okay, do you know how many 1/12 are in 1/3?" And then, once you know that, getting back to the unit fraction stuff, and then going "Okay, but now we don't need one-third divided by one-twelfth. We need 2/3 divided by 12. Do you have any Problem Strings that...

Pam:Let's do one next week! (unclear).

Kim:Yeah! Okay, okay, okay.

Pam:(unclear) get that relationship. Maybe. Okay. So, we'll see what happens next week. But absolutely. We would absolutely want to develop the relationship that if we can think about how many one-twelfths are in one-third.

Kim:Yeah.

Pam:How can that help us think about how many one-twelfths are in two-thirds? And let's like actually answer that question. So, how many one-twelfths are in one-third?

Kim:Four.

Pam:How do you know?

Kim:I'm trying to think of how I know. Because if you have a third, you would have to cut it. Half would be a sixth. Then a fourth would be a twelfth.

Pam:Yeah. And as you said that, I drew a candy bar, and I cut it into 3 pieces. Because you said, "Well, if you have a third..." So, I drew a candy bar and cut it into 3 pieces. You said, "Well, if I cut that third in half, I'd have sixths." Right?

Kim:Yeah, yep.

Pam:And you're like, "Okay, so there's the sixth." And then you said, "But then, I'd have to cut those in half to get twelfths." And I've only really cut that one-third. I've just that one 1/3, I cut it into two-sixths. And I cut each of those sixths into two-twelfths. So, in that third, I now see four-twelfths. So, that one-third has four-twelfths in it. Right?

Kim:Mmhmm.

Pam:Yeah.

Kim:But we don't need one-third. We need two-thirds. (unclear).

Pam:Because the problem is two-thirds divided by one-twelfth. Right, so we don't just have. So, 1/3 divided by one-twelfth is 4, so 2/3, twice as much, divided by 1/3 is twice as much, 8. Nice. Well, I almost wonder if your problem was a better... Well, is that your problem or is that just a different way of thinking about? Yeah. Anyway.

Kim:A different way of. Yeah.

Pam:A different way of thinking about that last problem. I'm going to give you one last problem for this string.

Kim:Okay.

Pam:Okay. So, I'm just going to note the recent problem we did was two-thirds divided by one-twelfth, and you got 8. Now, I'm going to ask you 1 and 2/3 divided by 1/12. In other words, how many 1/12 are in 1 and 2/3?

Kim:So, we just did two-thirds divided by a twelfth a couple of different ways, and talked about how that was 8. And so then, I'm going to also think about the extra 1, the leftover 1 divided by 1/12 is 12. So, if two-thirds divided by 1/12 is 8. And 1 divided by 1/12 is 12. Then, 8 and 12 is 20. Twenty 1/12s.

Pam:There are twenty 1/12s in 1 and 2/3.

Kim:Yeah.

Pam:Super thinking. And there's a few different ways that we could think about that, but I think at this point, I think we've done some nice fraction reasoning, fraction division reasoning. Teachers, I wonder if any of you, or listeners I wonder if any of you, are thinking, "Whoa, I mean... That's a lot of thinking. Like, I got kids that don't even know their multiplication facts, and you want us to be..." I hear you. I hear you. I acknowledge that the thinking that we just did was not trivial. Like, we were thinking. But in a big way, we were almost thinking less about fraction division, and we were thinking about the definition of unit fractions. Like, as we're asking ourselves, "How many 1/6 are in 2/3?" Two-thirds divided by one-sixth? If we're thinking about it quotitively, how many 1/6 are in 2/3, we're asking ourselves, "What's the relationship between one-sixth and one-third?" And so then, therefore, what's the relationship between one-sixth and two-thirds? Like, we're really helping students think about fractions, the definition of fractions. What does it mean? What is the relationship between one-twelfth and one-sixth? And one-twelfth and one-third? And one-twelfth and two-thirds? The quotitive meaning of division can be super helpful to help the kids actually reason about what fractions mean. And we need that. In other words, you might be like "Pam, in my section for fraction division, I got a day. I got a day to say, 'Ours is not to reason why, just innvert and multiply.' I don't have this time to help kids reason about this." Well, then we get what we've gotten. Which is kids not reasoning about fractions, kids just memorizing a bunch of rules, and then mixing and matching the rules, and getting it all screwed up. And we get the mess we've had. What we need is kids to actually understand the meaning of fractions. So, I'm actually suggesting we could ask some of these questions that we asked today, and in last week's, and what we'll ask next week. We could ask some of these questions in grade 3. Like, as kids are thinking about the meaning of one-third, I could ask the question, 1 divided by 1/3. Now, I probably wouldn't write it that way. I would say things like, "Hey, how many 1/3 are in 1?" And third graders should say, "Duh, 3." Right? And then, maybe I'll write it as 1 divided by 1/3, but I want to like have them think, so that I might say, "Well, how many one-thirds are in two-thirds?" Now, you might be like "Pam, that's a fifth grade standard! That's fraction division!" Or is it? Like, of course it is, but there are the meaning of fractions we can get at. We can help develop these relationships by asking questions like we did today in a quotitive kind of way...How many of the divisors are in the dividend?...to really help students reason about the relationships of fractions. And then, maybe we can develop everyone to think more and more like Kim, like mathematicians, like really reasoning about fractions. Anything else you want to say about that?

Kim:I don't think so.

Pam:Like, I'm going to push a little bit on this, Kim. I'm wondering if you could tell me as we were doing this Problem String, how much division were you thinking about, and how much like relationships between fractions are you thinking about? Do you know what I mean?

Kim:You know what, last week I think you mentioned that working with some of the strategies with whole numbers can be really useful. And I found myself thinking about that a little bit. That I know some relationships that now I'm just being asked to use those relationships with unit fractions.

Pam:Nice.

Kim:So, I'm drawing back on things that I know and have had experience with whole numbers. So, if we have an opportunity to do that as well with students, then it doesn't feel quite so big a hurdle.

Pam:So overwhelming here.

Kim:Yes, yes.

Pam:Yeah, absolutely. That's an example I would say of kind of the finger thing that we've done before. Where we're students can be kind of all over the map, and if we do a Problem String like this, they should be able to access it kind of where they are. For example, I just talked about how a third grader could think about several of these problems just by using the definition of fractions, while Kim is simultaneously using these really nice multiplicative relationships in division that she's already developed with from whole number division.

Kim:Yeah.

Pam:That's a great example of if we mathematize with students, students can enter where they are, they all learn and grow, and we all think more and more like mathematicians. Nice. Alright. So, ya'll, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!

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