March 07, 2023
Pam Harris
Episode 142

Math is Figure-Out-Able with Pam Harris

Ep 142: Fraction Division Part 1

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Math is Figure-Out-Able with Pam Harris

Ep 142: Fraction Division Part 1

Mar 07, 2023
Episode 142

Pam Harris

Is fraction division figure-out-able? You bet! In this episode Pam and Kim tackle another listener submitted topic and reason through a fraction division Problem String.

Talking Points:

- A sign of fake math
- An old wound for Pam and Kim
- A listener's question
- Two ways of thinking about division: partitively and quotitively
- A Problem String to develop division of fractions
- Why it helps to develop reasoning early with whole numbers
- Teacher moves to facilitate the string

See Episode 64 for another Problem String and discussion of thinking about quotitive division.

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

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Is fraction division figure-out-able? You bet! In this episode Pam and Kim tackle another listener submitted topic and reason through a fraction division Problem String.

Talking Points:

- A sign of fake math
- An old wound for Pam and Kim
- A listener's question
- Two ways of thinking about division: partitively and quotitively
- A Problem String to develop division of fractions
- Why it helps to develop reasoning early with whole numbers
- Teacher moves to facilitate the string

See Episode 64 for another Problem String and discussion of thinking about quotitive division.

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 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. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Both Pam and Kim:(laughing)

Pam:What's so funny? Because I skipped a line in the intro that I hate?

Kim:Listen, several weeks in a row, we've had a problem in the beginning. We're doing okay. Alright. Hey, everyone. In our last episode, we answer the question from a follower in the Math is Figure-Out-Able, teacher Facebook group. We absolutely love hearing from listeners. And we're super excited about what you're interested in. So, join the group, tell us what you want to hear. We can't chat about everything, but we thought we'd answer some of the questions that people in the group requested.

Pam:I mean, at least we can't chat about everything at the moment. But we'll try.

Kim:I mean, we hope to do this a while, right?

Pam:We'll keep going. Yeah. Bring it on. Bring on your questions. Alright. So, Beth Roberts, thank you for your question. Beth, you said "Division of fractions by a fraction. Without the 'Keep, Change, Flip', what would be a good model for this? How can I help teachers... How can I help teachers teach mathematize your way too..." Sorry, I'm not reading this very well. "How can I help teachers teach, mathematizing your way to division of a fraction by a fraction or mixed numbers for fifth grade?" So, let's talk about division by fractions. Division of fractions is an interesting thing. And I'm just going to mention this 'Keep, Change, Flip' thing is fascinating to me. A couple episodes ago, we talked about using words that aren't really well defined, and I will just note that 'Keep, Change, Flip' is often used for different things. So it could be used for fraction division. But I've also heard students say "Keep, Change, Change", for addition, subtraction of Integers. So, that's tricky because now we're using kind of the same words to mean drastically different things. And how about if we just don't use words at all? Because let's be clear, I'm going to call that fake math. I'm going to call that procedural math where my job is to help students get answers to questions, not the other way of thinking about mathematizing, the real way of thinking about Real Math. Which is, how can I help my students create relationships, think more sophisticatedly, actually get their brains thinking and reasoning like mathematicians with division of fractions? So, Kim.

Kim:Yep.

Pam:I'll never forget the day. I could picture where I was in my kitchen. I'm cooking dinner. My daughter storms in the house. I think she was fifth grade. It might have been sixth grade. But she storms in the house, and I'm like, "What's happening?" And she is like, smoke is coming out of her ears. And she goes, "Mom, divisions of fractions. Is division of fractions figure-out-able?" I was like, "Um, yes..." And she goes, "Oh, I knew it! I knew it! My teacher told me today, 'Ours is not to reason why, just invert and multiply'."

Kim:Yeah.

Pam:Now, I've heard that phrase a lot. But I've said it among teachers lately, and maybe it's not as ubiquitous as I thought it was. But "Ours is not to reason why, just invert and multiply" is kind of a very telling statement for me.

Kim:Yeah.

Pam:Because it says, "Don't think about math. Just Keep, Change, Flip."

Kim:Yeah.

Pam:Like, "Just invert and multiply." You can do that and get answers, but that doesn't build your reasoning about fraction division. And so, I took a deep breath, and I was like, "Yes, fraction division is figure-out-able. Let's do some work." And I did. Go ahead.

Kim:Why didn't you put her in my fifth grade class?

Pam:Oh, don't even go there. Shh! So, listeners, what you might not know, Kim is rubbing salt in a wound. What you might not know is that Kim taught my number two. She taught my number three. And when my number four. I have four kids. When my number four, who probably needed her the most, was about to have her, Kim's principal... So, Kim taught in a school that my kids went to. Kim's principal went and opened a new school. So, a new school was opening in the district. Kim's principal was leaving. We love that principal. Hey, Jodi Weimer. And Kim went with her to open the new school. And I said...

Kim:(unclear) sorry.

Pam:And I had a very serious conversation with Kim, and I said, "No, no, Kim. Kim. No, I need one more year here. Like, can you please just stay at this school for one more year, and then you can go anywhere you want. But I need my kid to have you for fifth grade. Please, pretty please, please, please, please, please, please." And she said, "Pam, Abby will be fine. There's fine... It'll be fine." And you're laughing, Kim, because then. We had a conversation recently, right? You want to tell that? You can tell that story. Do you (unclear)?

Kim:Oh, I don't remember what you're talking about, but (unclear).

Pam:About your own personal kid.

Kim:Well, I think I called you one day and said, "I get it." Like, "I'm in the same boat now, and I get it. And I'm really sorry I did that to you. And I'm really sorry I brought it up today."

Pam:Yeah, I think you started that conversation with "I'm so sorry." And I said, "What? For what? For what?" Like, "What's going on?" You go, "Oh, now I get it. I get what happened. I get why you were so desperate to have me stay and teach your daughter because..." Because, ya'll, my daughter can absolutely mathematize with the best of them, but she wants to know why, and it makes her stir crazy. It infuriates her when teachers refuse to delve into the why. And in this case... Oh, and I have to say to her every time, "Sweetheart, I don't think they know why." Like, you know, "They don't know why. They're doing the best they can. You come home, we'll talk about it. We'll do all the math together." Alright, so we did some division of fractions. And sure enough, my daughter can divide fractions thinking and reasoning like we're going to today. So, Kim.

Kim:(unclear).

Pam:That was kind of funny. So, Kim, let's do a Problem String to develop one of the major relationships in fraction division. And today, we're going to take a quotitive approach. Okay, what do I mean by that? So, division has two major interpretations, two major ways of thinking about division, and so let's just sort of be clear on what I mean by quotitive division quickly. If I said 30 divided by 5. If I'm thinking quotitively, then I'm thinking about how many 5s are in 30? That's a way of thinking about that division, and so that you can think about, "Well, how many 5s do I know?" And then, I can sort of reason. Okay, there's six 5s in 30, so I can solve 30 divided by 5 thinking quotitively. How many of the divisors are in the dividend? Okay. So, I know I didn't do. I just sort of told that. It would be much better if we'd experienced that with whole numbers.

Kim:Right.

Pam:I'm sure we did something on that earlier with partitive and quotitive somewhere on the podcast.

Kim:(unclear) Yeah, I feel like we have.

Pam:Maybe I'll search for that while you're solving one of these problems. Or not. Or we'll just let our... We have a wonderful podcast editor that will shoot me messages all the time. It's like, "Okay, you mentioned this one in the podcast, what episode is that? It's not always so obvious." Anyway. Alright, so if we're thinking quotitively. So, I'm just going to encourage you, Kim, to think quotitively about a problem like 1 divided by 1/3. 1 divided by 1/3. How would you think quotitively about that? Go.

Kim:Yeah, so how many 1/3s are in 1.

Pam:Mmhmm.

Kim:Are just three. Three 1/3s.

Pam:Yeah. And how do you know?

Kim:One-third, two-thirds, three-thirds.

Pam:Sure. So, if a kid said something like Kim just said, I would quickly draw a candy bar. I'm going to draw a rectangular candy bar, and I would say, "So, you're saying that if I cut that candy bar into 3 equal shares, that those are some thirds. You're saying there will be 3 of them?" And that's kind of the definition of thirds, right? Like, I've separated the whole, the unit into 3 equal shares, and I call each of them one-third. And if the question is, "How many one-thirds are in 1 candy bar?" we could clearly... And so on my paper right now, or on the board, or wherever I'm working with students, I've drawn this rectangle. I've cut it into 3 equal pieces. I've labeled each of them one-third. And I've sort of got that on the board. Bam! So, that's what we often do in a Problem String, right? We ask a question everybody has access to. When they say their answer, I make that thinking visible. I, the teacher, make that thing visible. Next question. Kim, what is 2 divided by 1/3?

Kim:Six.

Pam:And how do you know?

Kim:There were three thirds in 1, and so you doubled your amount of candy bars, and so now I have six 1/3s.

Pam:Cool. So, if there are three 1/3s in 1 candy bar, then you're saying there's double that. There's six 1/3s in 2 candy bars. Cool.

Kim:Yep.

Pam:So, I've just now redrawn the first candy bar. But I didn't draw it in as much specificity. I slapped up a rectangle, I quickly cut it into thirds, and then I slapped up another rectangle next to it, quickly cut it into thirds. I've labeled one of the thirds, one of those pieces, with one-third. And that's it. Like, I haven't spent time doing a whole lot of, but we can clearly see that there's 3 in the one, 3 in the other. I've doubled the number. And that's 6. Cool. What if I had a problem like 4 divided by 1/3? 4 divided by 1/3. Now, what?

Kim:That would be 12.

Pam:Because?

Kim:Because we just have six 1/3s for 2 candy bars, and we have twice as many candy bars, so we need twice as many 1/3s. So, twelve 1/3s.

Pam:Cool. And so, I've just sort of drawn quickly 4 candy bars. I've cut the first one into thirds, and I've labeled the first one 1/3. The first third, I've labeled it 1/3. And I haven't really done much with the other ones. I've got the 4 candy bars up there. I've got the first one chunked into 3, and I'm kind of thinking about that. I'm a little curious, could you have thought about that? You went from the 2 candy bars, and you've doubled. And I love that. What if you didn't have the 2 candy bars? Could you have gone from the 1 candy bar divided by 1/3?

Kim:Sure. Yeah, it would be 4 times as many pieces.

Pam:Because?

Kim:Because you went from 1 candy bar to 4 candy bars and three 1/3s to 4 times as much as that.

Pam:Which is still that 12?

Kim:Twelve thirds.

Both Pam and Kim:Yeah.

Pam:So, 4 divided by 1/3 has to, since you have 4, it would be 4 times as many thirds. Yeah, pieces. Cool. Alright, so 4 divided by 1/3 is 12. Next question. How about 4 and 1/3 divided by 1/3? 4 and 1/3 divided by 1/3?

Kim:That would be 13.

Pam:Because?

Kim:We just had 4 divided by 1/3. So, how many thirds in four, and that was 12. So, now you're just asking me one more third. (unclear)

Pam:(unclear) Yeah. Sorry, didn't mean to interrupt. Cool.

Kim:It's okay.

Pam:So, I've just slapped down 4 candy bars again, but I've tacked on kind of a ghost candy bar. And so my last, I've tacked on kind of a fifth candy bar, but I put it in dots. It's kind of like... But 1/3 of it, I've cut that into thirds. And that first third is solidly outlined. So, you can see a fifth candy bar, but I've only like sort of sketched in that third. So, you can clearly see 4 and a 1/3 candy bars. And I'm asking how many thirds are in that 4 and a 1/3. We already have the 12 from the 4 candy bars, and there's just one more. And I'm not actually going to spend a lot of time on this problem. Like, I'm just going to. Like, we have the 12. We got 1 more. Duh. We're moving on. Cool. How about 2 and 2/3 divided by 1/3? 2 and 2/3s divided by 1/3.

Kim:That would be 8 because...

Pam:Yeah?

Kim:...there were six 1/3 in two candy bars, and I have two more 1/3. So, 2 and 2/3s is eight 1/3.

Pam:Two more 1/3s?

Kim:Yeah.

Pam:So, as you were talking, I quickly slapped down 2 candy bars. And you said because we had 2 candy bars, and we had six 1/3s in those 2 candy bars. And then you said, we had two more 1/3s. So, I put another kind of ghost candy bar there, but I've solidly outlined two of those one-thirds. And the last third is kind of in dots. It's kind of just sort of sitting there. So, there's the 2 and 2/3. And then, I'm kind of circling a little bit the 2 candy bars, and I'm arrowing up to where we had 2 candy bars before that we had 6. And then I'm saying, "Oh, and then there's those extra 2/3..." So, I've written by the problem, 2 and 2/3 divided by 1/3. I've written 6 plus 2. Well, actually, I wrote 8, and then I wrote equals 6 plus 2.

Kim:Yeah.

Pam:Cool. Okay, next problem. How about 2 and 2/3 divided by 2/3? How many two 1/3s are in 2 and 2/3?

Kim:I think it's four 2/3s.

Pam:(unclear). There's four 2/3s in 2 and 2/3?

Kim:Yeah, because we just did 2 and 2/3 divided by 1/3. And that, there were eight 1/3s. And so, now instead of one-thirds, you're asking me about two-thirds, so the... Think how to say this. The size of the pieces that you're asking about have doubled. And so, I'm thinking about the size of the pieces have doubled, and so there's going to be less of them because the 2 and 2/3 didn't change. So, instead of eight 1/3s, there's only going to be four 2/3s, which would be half as many pieces because the size of the piece doubled.

Pam:Half as many pieces that are twice as big.

Kim:Yeah.

Pam:I wonder if we could do a little more candy bar talk. So...

Kim:Okay.

Pam:...in the problem before. I'm not... Yeah, well said. In the problem before, we had 2 and 2/3 candy bars.

Kim:Yep.

Pam:And I kind of said, "If I'm giving everybody a third of a candy bar, how many people can get a third of a candy bar? How many 1/3s can I get out of 2 and 2/3 candy bars?"

Kim:Yeah, that was 8.

Pam:You said there were 8. Eight portions, eight of 1/3s I could sort of hand out to people.

Kim:Yep.

Pam:In the next problem, I said 2 and 2/3 divided by 2/3? Could you tell me what that would mean in the candy bar problem?

Kim:Yes. So, now each person... I have the same amount of candy bar total, but instead of people only getting a third, they're going to get two-thirds, so they get twice as much. And so, I'm thinking about how many people can I feed now? If I'm giving twice as much, then only the first 4 people are going to get some.

Pam:Yeah, because if you give... If you start with the total amount of candy bars, and you give each person twice as much, you can only give that out to half as many people.

Kim:Right.

Pam:So, that would be 4. 2 and 2/3 divided by 2/3 is 4. And that would be some beginning reasoning that I would want to do with students about fraction division.

Kim:Yeah.

Pam:Let's maybe parse out a few things that we did. So, one, we have to be able to think about division as quotitive division, as how many of the divisors are in the dividend? If I have this many candy bars, this much of a candy bar, and I want to dole out pieces of the candy bar, sections of the candy bar, how many people can I feed? So, like the first problem, if I had a candy bar, and I'm giving out one-third pieces, how many of them are there? You said there were 3. If I double the candy bars, and I have 2 candy bars, and I'm doling out one-third of the candy bar to the people, how many people can I feed? Well, I doubled the number of candy bars, but I'm still giving out the same amount. Well, then I can give that same amount to twice as many people, so it's 6 pieces.

Kim:Yeah.

Pam:So, there's this relationship of, if I double the dividend, if I double the total, but I keep the divisor the same, I keep the share the same, then I could give that share to twice as many people. That's a big (unclear).

Kim:Wait, wait, wait. You said if you double the amount, they get?

Pam:Oh, no. Sorry. I double the total, but I keep the portion the same. I keep the portion, the amount they get, the same.

Kim:(unclear). Yep.

Pam:Then, I could give that portion to twice as many people because I doubled the total.

Kim:Yep.

Pam:That relationship, ideally, we would have developed with whole numbers.

Kim:Yeah.

Pam:Before... Notice, the way I just said that. If I double the number of candy bars, but I keep the portion the same, I can give that portion to twice as many people because I doubled the number of candy bars. That relationship, ideally, we would have developed with whole numbers before we hit this place where we're trying to reason about division with fractions. Now, if you haven't, you might want to do a Problem String with whole numbers into this Problem String with fractions. I think that would be a helpful thing. But we still can reason. Especially if I've got the candy bars up on the board, we still can reason about these chunks of candy bars with students. It's just going to be, you know, we're thinking about more complicated numbers. So, it's one of the reasons why I really encourage people to think and reason about division with whole numbers, not just do stuff, not just do an algorithm. If all we've done with division of whole numbers is stick numbers in an algorithm, and then perform a bunch of steps. "Does McDonald's sell cheeseburgers?" And, "Do dirty monkeys smell bad?" Like, whatever that algorithm is that you've done for division with whole numbers. If we've done that with division of whole numbers, thinking about division of fractions is going to be more difficult. We're going to have to do more building of "Oh, like I can think quotitively." And not only quotitively, but this idea of if I double the dividend, keep the divisor the same, then I can double the quotient. That's a big idea that I wish kids owned, so that we can sort of just like then investigate it, make it strong with fractions. We can use that same understanding. Maybe a... Well, do you want to say anything before I point out a couple other things I did?

Kim:No, it's it's just interesting that you're talking about this right now because just last night, Cooper, my younger, was asked to do something with 90 divided by 5, and he was like, "Ugh, I don't want to be about 5s." And so, he thought about 180 divided by 10. So, it's a nice example of how he thought about twice as much in a whole number setting, in order to get the answer that he needed.

Pam:Ah, very nice. I will point out that if he's going to solve 90 divided by 5 by solving the equivalent problem 180 divided by 10. Brilliant, I love your kid. That's actually the partitive meaning of division.

Kim:Yes, you're right.

Pam:Yeah. Which we would want to develop. That's not what we were messing with today, but I wouldn't want to stifle Cooper when he does that. I would want to like absolutely celebrate the thinking he's doing. In the midst of this Problem String, though, I'm going to keep the context alive about candy bars and divvying them out to kids, which will kind of keep him thinking quotitively here. But I'm not going to like, "Oh, no, we're not doing that today. Stop doing that." I'll totally support what he's doing, and I'll draw him into the context that we're using. Which leads me to a thing I was going to point out is. That we did it in context, right? So, context is hugely important. I think we need to have kids involved in contexts, and then get naked, then decontextualize, and help them think out of context. But as we decontextualize, draw back on major contexts that we've used until kids can solve contextual problems, and decontextualize problems, and go back. That we want the flexibility that they can go back and forth between them. That's the ideal goal, and we do that by starting with context and helping kids really reason in context, and then moving more abstractly. There's one other thing I wanted to point out that I did. Kim, every time we did one of those problems, I did not command you to draw a model, right?

Kim:Right.

Pam:Did I stop you from drawing a model?

Kim:No. Actually, I sketched part of the Problem String with the model, and then I just kind of stopped.

Pam:And I wonder if I had been sketching your thinking where you could see it. You know, we're just audio recording your at your house, I'm at my house. If I was sketching, so you could see it, you might not even have sketched those first couple ones, right?

Kim:Right.

Pam:Because I would have had. And you probably sketched them because you were wondering, "Hey, maybe this is going to help me later on." And as you were reasoning, you're like, "I don't need that to help me."

Kim:Right.

Pam:But I would have been. So, we're just going to suggest that with Problem Strings, often it is that we draw out students' thinking and we make it visible.

Kim:Right.

Pam:We choose the model, we make it visible, and we help put something on the board that we can point at, that we can compare, that we can have discussions about because it's visible. And that's a huge part of what we're recommending as we're developing all the things. So, Kim, this was not enough. We have not done enough work with fraction division yet. How about if we... Are you up to doing a couple more podcasts where we'll dive deeper into fraction division?

Kim:Sure, yeah.

Pam:Alright, we'll do that in our next episode. Listeners, you are not going to want to miss our next episode, or two, or three...I don't know how many we'll do...to dive into really, really developing fraction division. 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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