What if your students could think and reason about the relative sizes of fractions? In this episode Pam and Kim think about where fractions go on clotheslines to show how you can help your students compare fractions without a memorized procedure.
Take our math perspectives quiz! http://bit.ly/xyzquiz
See Episodes 24-27 for more about The XYZ Math Perspectives
Check out our social media
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC
Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam.
And I'm Kim.
And you found a place where math is not about memorizing and mimicking, where you're 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 actually mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but, ya'll, rotely repeating steps actually keeps students from being the mathematicians they can be.
So, Pam, I was looking at the podcast episode stuff on my podcast listening, and I found a review that I wanted to share with you. It's my favorite.
Oh, cool. Cool, cool.
So Prudentmama said, "I was listening to episodes 24 to 27 where Pam discusses the different kinds of math students that are commonly found in math class and beyond. I felt exactly like the person who just wanted to understand math. I knew it had to be figure-out-able, since others were doing it, but I couldn't ever see the connections and would ask why constantly." She says, "Ultimately, until I married my husband who's an X, I just assumed I didn't have the math gene. Since watching my husband play with numbers and homeschooling my kiddos, I have discovered the wonder of connections in math and can truthfully say it's enjoyable." That's super cool. "And just for giggles, Pam, I'm creative with math too through sewing. Honestly, I've shared the podcast with many and highly recommend that you listen to it." I love that. Thank you so much. If you aren't sure what Prudentmama means when she talks about her husband being an X, I would encourage you to check out episodes 24 to 27 where we talk about it. And so, thanks for the review. We love it. It's super fun for us to hear, and it helps others find the podcast.
Yeah, so thanks. If you don't mind right now, everybody, whoever is listening, pop on over to wherever you listen and just give us a quick review. Give us a quick rating. It really helps spread the podcast around, so we can transform the world, ya'll. Help everybody opened the door. Just think, Prudentmama, everybody could know that they have the math gene, the more that we can spread that. Let me just maybe note Prudentmama that you were a Y. I'm just going to suggest because now you're an R. And R for Real Math. And I know you said that your husband was an X. Maybe he's actually an R, right, for Real Math. When I did those X, Y, and Z sort of perspectives, it was an attempt to describe what you might have thought about math in school and how that could affect the way that you teach now. So, we don't want anybody to be an X, Y, or Z, now we want everybody to sort of join the Team R, the Team Real Math. So, yeah, everybody joined Team R, Team Real Math. It's totally fun.
Yeah. And if you're curious about what a team R, or what it would have been to be an X, Y, or Z, you can take Pam's quiz to find out what that perspective is that you have as a student, and you can find it at bit.ly/xyz quiz. Check it out.
Cool. Alright. So, today, Kim, we're going to have a little fun with clothesline math. Bam! So, Chris Shore thanks for making it popular. I know you didn't invent it. You give some credit to some other folks. And I had seen some other people do things. But Chris Shore has done a great job of getting a website with lots of examples of ways to do clothesline math. Clothesline math, it's literally the line that you hang up to dry your clothes on. But instead of drying your clothes on it, we're going to hang numbers on it. And the ability to be able to move numbers around as you sort of shift and organize. Kim, I remember back in the day when I was very first starting to work with elementary teachers and doing professional learning in my kids school, and you were there. Garland Linkenhoger, hey, Garland, how's it going? Shout out to Garland, was a fantastic colleague at that at that point. And she brought in some activities that we did on clothesline math, way back then. Super fun. So, we thought we would do a little clothesline math on the podcast. You're like, "Pam, how are you going to do that on the podcast?" Well, I don't know, but we're going to do our best to see if we can make this work a little bit. I'll tell you, as I'm talking, I'm grabbing my pen.
I have a pencil.
And my paper. So, I really should grab like a clothesline and numbers, but since we're audio, we're just going to kind of do it in the air here, and I'm going to probably write some things. But a thing that we could do on kind of this. If you picture kind of a clothesline hanging up, and then we're going to hang pieces of paper or cards that have a number written on them. And we're going to kind of say, hey, that's where it goes on a number line. So that's the point here is, if I were to say, "Hey, Kim? What's bigger, four-fifths or three-fifths?" And so, therefore, you would put them on the number line, so they would go in the right order. What are you thinking about to determine what's bigger, four-fifths or three-fifths. Three-fifths or four-fifths? Which one's bigger?
Four-fifths is bigger.
You didn't have think about that very long at all.
No. The way I think about those is four 1/5s compared to three, 1/5s,. So, it's a pretty easy comparison if you think about four-fifths as four of those 1/5s. So, on a number line, I kind of picture maybe where one-fifth is, but you're finding several iterations of them. So, like three, 1/5s would be at three of the 1/5s down the number line. And then, four 1/5s would be one more 1/5 further to the right. So, four-fifths is bigger.
Further to the right. Nice, nice. So, it's really predicated on this idea that you can think about a non unit fraction like four-fifths as a bunch of unit fractions four 1/5s. So, unit fractions, when the numerator is 1, when I've got 1 of the pieces that we're sharing. If 5 of us are sharing, we each get a 1/5. What's my share? I get one of those. That's a unit fraction. My share of the fair share, 1/5. And you're like, "Hey, if I can think about four-fifths is four 1/5s, then bam." I really liked how you describe that on a number line, that if you can picture where 1/5 is. What does that tell you where the 1/5? If like if you just sort of stick... In fact, I'll just tell you. On my paper, I drew a number line and I put 1/5 st the very far left. But I don't think that's quite what you meant.
Say that again. Where did I picture a fifth?
Well, so if... Yeah.
I pictured the 0 to 1.
Yeah, there was kind of the... I think you have to have the 0 in there. So, I was just looking at my paper realizing I didn't have the 0, so what I didn't have was your... When you said, "So, therefore, I can see 1/5," and then you can iterate 3 of them to be three-fifths, I didn't have the 1/5. In other words, I didn't have the length of 1/5 because I had 1/5 at the very far left of the number line, so I didn't have from 0 to 1/5 sitting there in front of me. So, as you started to describe it, I backed up my number line, I drew it longer to the left, put the 0 down, and then I was like, "Ah, here's 1/5." And then, as you said, "I need 3 of them," then I'm like, "Oh, 3 of them would be over here." There's three-fifths. "And then, 1 more of them." Yeah. So, anyway, I just thought I'd describe that because I don't just have the where 1/5 is, I have to have that length of 0 to 1/5. Yeah. Nice.
And I think it's really important is that we're talking about if we're talking about fifths, we're comparing three of those 1/5s and four of those 1/5s in order to determine which is bigger.
Nice. Nice. Yeah. Alright, Kim, next question. What is bigger? 1/4 or 1/5? (unclear).
Oh, okay, okay. So, fourth is bigger. And so, I think about it...
But isn't 5 bigger than 4?
Yeah. Yeah, that's tricky, but I think about it in context to help with that. So, if I have a candy bar shared among 4 people, then I'm going to get a bigger portion than if I have that same candy bar shared among 5 people. Okay, my part is more. But since we're talking clothesline math, and I'm going to put it somewhere on the number line, I also can think about that same 0 to 1 that we talked about.
And if I'm cutting that span of 0 to 1 into 4 chunks, then a fourth is going to be further to the right than if I cut that 0 to 1 span into 5 parts. I'm going to move further to the right if I cut it into 4 pieces than if I cut into 5 pieces. I'll hit that first mark faster with 1/5s
The first mark that is 1/4.
Yeah, it will...
Oh, I see what you're saying. Cool. So, if you were to cut that span from 0 to 1 into 5 chunks, you're saying those chunks are smaller, that bit of number line is smaller than if you only cut it into 4 chunks. For the same reason as when you were saying "If you're sharing." If you were to share that length. Say it was fruit leather. Like, Fruit Roll-Up. If I had a Fruit Roll-Up, and you're sharing it with 4 people, you'd get a bigger length of Fruit Roll-Up. Where you would cut the Fruit Roll-Up at 1/4, and you get that chunk of 1/4, would be bigger than where you would cut it. If you were sharing with 5 people at 1/5, that would be a smaller strip of the Fruit Roll-Up Is that right?
For 1/5. So, as I've drawn the number line here on my clothesline math, I've got 0 to 1. I've got 4 marks where I cut it in half for a half. And then, cut that in half for a fourth. And that's where I mark the fourth. How are you marking the fifth?
That's a good question.
Like, where does it? If that's the clothesline, and I put the fourth there, where's the fifth?
I just eyeballed it to be honest with you.
Right, but just tell me where.
Oh, the fifth would be to the left of the fourth.
Closer to the 0, yeah. To the left of the fourth. Cool. So, as I look at the number line, 0 first, then there's the one-fifth, and shortly after that is the one-fourth.
Cool. Alright. Let's see. What is bigger, three-fourths or three-fifths.
And maybe I shouldn't have said it quite that way. Three-fourths, or three-fifths? Or maybe I should have said it... How do of I say it without any emphasis? Three-fourths or three-fifths? (unclear).
Well, either way you say it, my first inclination was to think 75% and 60%. (unclear)
(unclear). I love it. You're such a percent girl.
I know three-fourths is bigger.
Well, so stay there for a second. So, if you stay at 75% and what was the other one? 60%. Then, you could put that on a clothesline math, right? What would be... What would you... How do you envision the clothesline math there?
I thought it was going to be snarky for just a second. Sorry.
Three-fourths down the number line. So, between 0 and 1, I would put three-fourths. If I need to put it on a number line, I would find the halfway mark, which would be 50%. And then, I would go halfway between the 50% and the 1 for the 100%, and I would put 75%, which is three-fourths. Smack dab in the middle of the of the halfway mark and the 1.
Okay, so I'm just going to slow you down for just a second. So, do you have on your number line 0 to one?
So, then, in clothesline math, as soon as you said 100%, because you did, then I would give you an additional card, and we could clothes pin that 100% right underneath the 1. And underneath the 0, we could clothes pin 0%. So, now I have kind of some markers, 0 to 1, and those are also equivalent to 0% to 100%. And when you cut it in half, I think you actually said that's where the 50% went, I think. But we could also then clothes pin the one-half to that. So, right now we've got 50% and one-half. And then, you said you cut that in half. And I think you said, "So, you put 75% there, so therefore it's the three-fourths." So, again, we would hang one. And I don't care which one we'd hang first. We could hang the 75%, and then clothes pin to it. And the reason I'm saying clothes pin because then it just goes right underneath it. So, it's in the same position. So, now I've got, in the same position, 0/0%, 50%/one half, 75%/three fourths, and 1/100%. And that's part of the beauty of clothesline math is that we can have all these different kind of representations. And since you like percent so much, why not use percent? I like it. Absolutely. Do you want to talk about it without percents? Or do you want me to?
Well, we already determined that 1/4 Is bigger than 1/5. That's the previous problem you gave me. So, if I have three of those 1/4s or three of those 1/5s, then three-fourths is still going to be bigger. I just now have 3 of each of them.
Oh, nice. So you actually use that unit fraction, kind of like you talked about before, three-fourths is three 1/4s. And three-fifths is three 1/5s. Since the fourths are bigger than the fifths, you got three of something bigger is going to be further along than three of something smaller. Yeah, cool. Nice. Nice. I think you have one more way to think about that one. Just because I know you, Over girl.
Well, yeah. I mean three-fourths is one-fourth away from a whole. And three-fifths is two-fifths away from a whole. So, if I'm a fourth away from the whole, that's... I went back to percents for a second. So, 25% away from the whole. And two-fifths is 40% away from the whole. Yeah. So, then the three-fifths is smaller.
Cool. Hey, so maybe I missed it. Did you actually tell me where to put three-fifths.
Oh, I didn't. No, I didn't. (unclear).
You just said 40% away from the whole, so I could do that.
(unclear) and we had, so I could... Here's the thing, I didn't put four-fifths and three-fifths as the first one you gave me. I didn't draw it on number line. But if I had, then I would have already had the three-fifths on there. The first problem you gave me was four-fifths and three-fifths. But it would be three of those 1/5s down the number line.
Cool. Or 2 back from the 100%. Nice. And so now we could also have three-fifths hanging about where we would put 60%. And we could also clothes pin to that 60%. Either one could have come up first. Cool. Nice. Nice. I like it. Okay, so interesting. It feels like we started with a problem today, "Four-fifths and three fifths. Which is bigger?" where you talked about... How do I say this? Could you kind of generalize when you said like four-fifths and three-fifths? What was the same in that problem?
The size of the piece was the same. And in one of them, I just had more of them. So, like I had fifths in both of those cases. But I had 4 of them versus 3 of them, so I could compare.
Cool. And I love how you just said that because what you didn't say first was, "Well, they had the same denominator, so I could just compare the numerators." I hear that a lot when teachers are talking about comparing fractions, and it's too... I would encourage everyone to talk about that same denominator, so four-fifths and three-fifths, as what's actually happening. "Oh, I've got the same kind of pieces. I've got fifths," like Kim just said. And so then, I can say, "Well, if I have the same kind of pieces, which one's more? Four-fifths therefore is greater than three-fifths." So if I said to you, Kim, what's bigger? Now, this is going to be tricky over just audio, but let's try it. So, I've got written A/C. So, like A "fraction bar" C. So, I've got A, 1/Cs. Does that makes sense? Or B/C. So, if you guys can picture. Let me just say for listeners. I've got A divided by C or B divided by C. A/C or B/C. A/C or B/C. How are you reasoning about that? Which one is bigger?
Such a weird thing to say. Okay, so in both of those cases, I have Cths.
I know it's terrible.
So, the one that's going to be bigger is if A is more or if B is more. So, in other words, if, A is a bigger number than B, than A/C is going to be bigger than B/C.
Cool. So, once you have the same kind of pieces, you can compare how many of them you have.
Nice. Cool. Okay, then the second bit of numbers that we compared three-fourths, or three-fifths, could you kind of generalize the way you did that?
Well, that one was based off of the understanding of the one-fourth and the one-fifth, so let me mention that the first. So, one-fourth and one-fifth. In that case, I had the same amount of something, the fruit strip, the candy bar, the whatever, and I'm sharing it with less people or sharing it with more people. So, the smaller amount of people I'm sharing it with, the more I get.
So, when you originally said you had the same amount, I think you meant you had the same whole, the same unit, the same like. And you did say fruit strip, right,. You had the same whole. Fruit ship or candy bar. And then, say the rest of it. You had the whole? I kind of got stuck there. So, you have the whole. The whole is the fruit strip.
(unclear) and I'm either sharing it with a smaller group or a bigger group. So, the more people I share it with, the less I get.
The smaller your share is. Yeah. So, that's why 1/5 was smaller than 1/4.
Right. Sharing with more people, so I get a smaller portion.
Cool. So the ones were the same, right? It's my one share out of sharing it with the 4 (unclear).
The whole's were the same. Yeah.
(unclear) my one share sharing with 5 people. And so, then, I asked you three-fourths or three-fifths.
And in that case, I considered the fact that we had three of the pieces, but I considered the size of those pieces. Was I sharing the hole with 4 people and I get 3 of them, or was a sharing the hole with 5 people, and I get 3 of them.
That's where the three-fourths, three 1/4s or three 1/5s
So, it goes back to understanding that unit fraction.
Yeah, nice. So, if I just got very general again, and said alright, this time Kim, you've got A/B. So, like A fraction B, A/B. Or A/C. A/C? Which one's bigger? How would you generalize that? I've got A/B or A/C.
Right, so we have the same amount, the same number of pieces, so I could compare is Bths a bigger portion or is Cths a bigger portion?
And when you do that, how do you know if Bths or Cths are a bigger slice of the Fruit Roll-Up?
Well, I don't think I would know if Bths or Cths are bigger.
Does it have anything to do with B or C?
Well, yeah. So, if B is a larger number like 4, and C is smaller, so it's like a 3, then I'm sharing with more people. So, I have to think about am I sharing with more in B or less in B? It's kind of weird to say. Sorry.
So, let's say B is is really big. It's like 50.
So, you've got A/50 compared to A/4. Then, what does the bigger number tell you?
The bigger number means I'm sharing it with more people, so I get less. So, the larger the number in the denominator, the more that I'm sharing it with, the smaller I get.
The smaller the pieces. So, if you're comparing A/B or A/C, then you can compare B and C, you compare the denominator. And like you just said, the bigger that denominator is, the more people you're sharing it with, therefore the smaller the fraction would be. Whoa! Alright, ya'll, we want students to be thinking about the meaning of numerators and denominators, and sharing, and using this equal partitioning idea of how we can sort of split things into these equal places. And that is super important. One thing you need to know is your perspective. You might have thought that fractions are all about memorizing to compare, that there's some sort of sequence of steps that you have to follow in order to compare. Maybe a butterfly? But you can actually think and reason through them. Also, you might have had a different perspective, where you might have been reasoning about fractions, but you're not quite sure why others just didn't naturally do it, and so I guess they need the rules to reason. In which we would say, "No, no, no, no. No, they just need to know it's a thing." Like as a student, if you would have said, "Oh, you can actually reason through these. You don't have to use this rule." I just bought into it. When you said, "Use this rule," I was like, "Oh, okay, that's what I'm supposed to do," so I did it. If you say "No, what you're supposed to do is think," then I'll think. And obviously help me build the relationships to be able to do that. So, ya'll, it's about perspective. And once we acknowledge what perspective we had as students, then we have the option to shift that perspective and teach more and more Real Math.
Yeah, I'll just pop in that we can give them the quiz one more time. They can find out about the perspectives by taking the quiz at bit.ly/xyz quiz.
Absolutely. Take our quiz, and we can keep gathering data on those perspectives. And you also, then, will get a print out, a report of how your perspective might be impacting the way you teach today, and that can help you then decide how you want to keep teaching. Ya'll, thank you 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. And keep listening to the podcast. Let's keep spreading the word that Math is Figure-Out-Able!