What are some important relationships that students can use to do Real Math with fractions? In this episode Pam and Kim discuss some important relationships that can help develop deeper understanding about fractions for fractional equivalence and comparison.
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Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, and I'm laughing. Sorry.
And I'm Kim, and I'm laughing too.
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 laughter and 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 really not very helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. Kim, I got to tell you. The other day I was talking. I think it was Chris Hogbin. Hey, Chris! The fantastic creator of Number Hive. And he said... Yeah, we love Number Hive, right?
In fact, my kid and I played it a ton the other day. And he was so funny. This is the one who's going into AI, computer science research. And he's like, his question was, I wonder how he created his hexagons?
I was like, "What?" He goes, "Yeah, yeah. Programming hexagons is super tricky." Anyway, it just went on. Which is kind of funny, because I had just created a hexagonal grid in illustrator. Anyway, so I was kind of... I was feeling his pain. But, Chris, back to the point here. I'm all over the place. Chris said to me, "You don't pre-record your intros and outros.
[Kim laughs] Nope!
And I was like, "No, we re-say them every time." He's like, "I love that because I'm always listening. Like, is it going to..." Well, there you go. Chris. Today, that one was for you because I was laughing as the recording was starting, and then I just couldn't (unclear).
Most days we're just trying to pull it together. Hey, so we got another review. Which I super love. It's my favorite thing just to see.
Yay, reviews! People, more reviews, more reviews. Spread the word. Spread the word.
(unclear) But leave nice ones (unclear).
Okay, so this one is short. And it says, "This is an AMAZING..." capital letter "...podcast. I love it so much, and it really helps my fifth grade brain." Which, Pam? Oh, and it's from 3.... Sorry, 78359737 is the user who shared that with us. So, but listen, when I saw that I thought, "my fifth grade brain", now it could be... You know, it's summer, maybe, for some people, and so maybe it's my fifth grade teacher brain. But what if it's a fifth grader who listens? That happens, right? We have several people who say, "I listen in the car with my kid."
Oh, that would be cool. Yes.
So, 78359737, I kind of hope you're a fifth grader. And we love that you listen. Thank you so much.
That will be super cool. Yeah. So, and it's fantastic that your name is 78359737.
You say that fast! Good job!
Well, it helped that you put spaces between them. Like, in my notes right here I've got them written, so I can actually see. But I wonder if there's like a pattern in those numbers. So, contact us, 78359737, and tell us why you chose those numbers.
Super curious. Especially if you're a fifth grader. We still like you if you're a fifth grade teacher, but it will be you know. And so, we would love to hear from everybody. It helps others find the podcast, and then we can keep spreading the word. Hey, Kim? I think we hit... In fact, I know we did. We hit half a million downloads the other day.
That's really cool.
That was really fun. That was a good benchmark. Bam!
It was actually quite a... It wasn't just the other day. It was a few days. Yeah, a while ago. But yeah. Yay for that. Cool.
Super cool. Okay, so let's get started. In the last episode, we started talking about two major ways to compare fractions - by looking at the numerator, and by looking at the denominator. And I think maybe those aren't super uncommon for teachers to share with their students. And those examples are doable, their figure-out-able. But we're going to suggest that some others are as well.
Yeah, absolutely. So, let's talk about some other ways that we can compare fractions. And then, if you want, you can hang them on a clothesline. We talked about clothesline math a little bit last time. I think we might emphasize that a little less today. More, just the reasoning. So, Kim, I'm going to give you a few problems.
Okay, first problem. What is bigger nine-eighths or six-sevenths? Nine-eighths or six-sevenths. Which one's bigger and how do you know?
Nine-eighths is bigger because nine-eighths is more than 1 whole. Eight-eighths is a whole, so nine-eighths is more than that. And six-sevenths is a little bit less than a whole because seven-sevenths would be a whole. So, six-sixths is less than a whole.
So, six-sevenths is less than a whole.
Mmhmm. So nine-eighths is bigger.
And nine-eighths is greater than a whole. Yeah, so nine-eighths is bigger than six-sevenths.
Cool. Next problem. So, Kim, what's bigger? Two-fifths or four-sevenths? Two-fifths or four-sevenths? Which one's?
Oh, those are kind of ugly. Yeah. So, two-fifths...
Thanks for the ugly problem. Oh, two-fifths is a little bit less than a half. So, 2.5/5 would be a half, equivalent to a half. So, two... I like litterally wrote an L. Two-fifths is less than a half, and four-sevenths is more than a half. Because 3.5/7 would be equivalent to a 1/2. So, two-fifths is a little less than a half. Four-sevenths a little more than a half. And I think you asked me which ones more. So, four-sevenths would be more.
Nice. So, kind of similar to the problem before where nine-eighths and six-sevenths you compare to 1, the number 1. This time, you compare two-fifths and four-sevenths to one-half. And since one was smaller than one-half and the other one was bigger than one-half, then you could compare it that way based on that comparison of one-half. I have to tell you. When you were talking, I was kind of writing like a clothesline math like we did in the last episode. And I wrote... As soon as you said, one-half, I wrote it down. And you said something about two-fifths being less, so I wrote it to the left. And then, you said, "Because 2.5/5 is equal to one-half." So, I wrote 2.5/5 under the one-half. So, I can kind of see two-fifths, and then one-half and 2.5/5 are in the same place. And then, further to the right, I wrote four-sevenths. And then, you said, "Because 3.5/7 is one-half," so I wrote that underneath the one-half as well. So, now my one-half looks like one-half, 2.5/5, and 3.5/7. Yeah? You know, Kim, that really brings out a point that came out in the workshop last week where people said, "This is not only... We're not only learning fractions, but we're also getting better at multiplication, and division, and even additive thinking." And big time equivalence! But being able to like think about 2.5/5 and 3.5/7 it's not just fractions. Like, we're really thinking about fractions multiplicatively. Which is so important because they are multiplicative. Not about counting the numerator and counting the denominator and saying, "We're done with fractions." We got to do more than that. Okay, cool. Next question. How about... Okay, I can't even read my own writing. five-eighths, or thirteen-sixteenths? How would you compare those? Five-eighths or thirteen-sixteenths?
You know what? The first thing I thought when you said five-eighths was like immediately I went, "Okay, we're a little more than a half." And then, when (unclear) thirteen-sixteenths. I went, "Okay, we're still more than a half." So, the half is not super...
Trying to compare to a half. Yeah. Okay.
But if I think about comparing to three-fourths. Six-eighths is equivalent to three-fourths. Five-eighths is a little bit less than three-fourths.
And twelve-sixteenths is three-fourths, so thirteen-sixteenths would be a little more than three-fourths. So, I have one that's a little less than three fourths, and one that's a little more than three-fourths.
So, which one's greater?
Thirteen-sixteenths is greater.
Cool. So, if you were on a number line, which one would you see first coming from the left?
Because that's smaller. So, five-eighths. So, on my number line, the clothesline that I've kind of written on my paper, I've got five-eighths. And then next, I have three-fourths, and I have six-eighths, and I have twelve-sixteenths all lined up in the same spot. And then next to that, I have thirteen-sixteenths. So, I can kind of compare the five-eighths to the six-eighths, and the twelve-sixteenths to the thirteen-sixteenths. Nice. So, it seems really helpful in all three of these problems that you could compare to a common, nice fraction, a benchmark fraction. Does that sound like kind of what you did?
Yeah, absolutely. Yep, little more, little less.
Yeah, nice. Super cool. Okay, Kim, cool. Next problem. How would you compare five-sixths and four-fifths. Five-sixths and four-fifths? Which one's greater? Or how would you compare them? How would they relate?
Okay, so the first thing I noticed is that they are... Ooh, how to say. Both of them are a unit fraction away from the whole. So, four-fifths is just one-fifth away from a whole. And five-sixths is just one-sixth away from a whole.
And how is that helpful?
So, if I put them on a number line, and I think about starting at a whole.
And I want to go back the fifth. Right, I'm drawing. So, if I'm going to go back a fifth to record where four-fifths would be. Or I go back a sixth, to record where five-sixth would be. Then, five-sixths would be closer to the whole. It's only a sixth away from a whole.
Mmhmm, and It sounds like to be able to do that...it's kind of like what we did in the last episode...you needed to be able to compare one-fifth and one-sixth.
To see how far back from the whole you were going to go. And then, once you kind of land, if you went back a smaller bit. One-sixth is smaller than one-fifth. If I'm sharing with 6 people, I get... If 6 of us are sharing, I get less than if 5 of us are sharing, right? So, one-sixth is smaller. Then, that smaller one-sixth went not as far back from the 1, therefore the five-sixth was closer to the 1.
Yeah, and there's so many good things that are happening here. Because, you know, at the same time, I'm thinking about four-fifths as four 1/5s and five-sixths as five 1/6. So, I'm thinking about, basically a scale up from the unit fraction. That's one thing that's happening. Then, I'm also comparing a fifth and sixth. So, I'm getting a really good understanding of the relationship between amounts of unit fractions. And then, I stuck it on a number line in order to think about like proximity to 1, and what it looks like to be a little bit less than 1 in varying. Like, in the versus the sixth of it and a fifth of it away. So, within that one question of comparing fractions, I'm kind of wrestling with a lot of fraction understanding.
And building in that wrestle, that grapple, that fracture, understanding even better, and now we can have kids that can actually reason with fractions and not just repeating those crazy rules. Nicely done. Alright, Kim, I think I have one more for you. How would you compare 3/4 and 29/30?
So, I actually like this problem better than the one you gave me. So, 3/4 is 1/4 away from a whole. And 29/30 is is 1/30 away from a whole. So, they're both almost a whole. But 29/30 has to be bigger because it's just like a little sliver, a thirtieth, to get to that whole. And it's like it's just a tiny amunt. Yeah.
It's 1/30 away.
And the three-fourths, it's a whole fourth away from the whole. That three-fourths plus that 1 more big, whole fourth to get to the unit. Nice. It's a bit of a mind twist.
Yes, it is. It is. So, you're thinking about which one has a smaller amount to get to the whole?
Yeah, or another way of thinking of it is, which unit fraction is smaller, therefore it's back less from the whole. Therefore, the 29/30. Since, 1/30 is smaller than 1/4, then 29/30 is going to be bigger than three-fourths because it's so much closer to the. It's only a smaller unit fraction away from the whole, from the 1. Woah!
And this is like a perfect example of how I'm super clear in my head about... Like, I can picture it. I know it without question. But it's easy to get tangled up in your words when you're trying to justify it to somebody else.
Mmm Mmhmm. And so, models can be really helpful, and verbalizing it, and get the words out tangled and have the person go, "Wait, what?" and back and forth and forth and back until we all get more, gain more clarity. And let's be clear, that is one reason why we use symbols. We use symbols to represent these complex ideas. What doesn't work to mentor mathematicians is to just give kid the symbols and expect that all this deep meaning will come with the symbol. We have to build that deep meaning and represent it with the symbol. Okay, cool. Hey, Kim, I got to tell you. When we were preparing to film Building Powerful Fractions, my husband and I, we have a weekly date night. We go to our little, handy dandy Tex-Mex place around the corner. It's called Garcia's. Super good chips and salsa. We actually had some queso the other day. I know you'd appreciate that, Kim. Right? (unclear).
I want it all. Chips, salsa, queso. Not guacamole.
There you go.
No? Yeah, that's right. You don't like guacamole. I'll take the guac. Really, I'll take it all. You won't have to. Anyway, as we were eating our chips, and salsa, and guac, and queso the other day at Garcia's, he and I were talking about some of the things that we were doing in the workshop. And he reminded me that when I met him, he said something like, "Fractions are stupid. I turn everything to decimals, and I do it all in decimals." But the more that he and I have been married, and the more that we have talked, and as he did some of the fraction tasks. Because he's like curious what we were doing, and so we were doing over chips and salsa at dinner. The more that we were doing, it was so interesting, that he said, "So, I can actually think in fractions. Now, I don't have to think in decimals. I can. I can think in decimals. But I can also think in fractions." So, ya'll, if that's you, if you've kind of been stuck in one way, that fractions have not been so figure-out-able for you. Bam! Like, let's continue to think and reason like last episode, where we really focused on, if you have the same numerator, the same number of pieces, then you can talk about the size of the piece. And if you have the same denominator, the same size of piece, then you could talk about the number of pieces. Or today, where we really compared to a benchmark fraction, a benchmark number, and used that comparison. Or, where we can think about if you're just one unit fraction away from the whole, then you can use that to compare. You can use these big ideas to really reason about fractions because fractions are figure-out-able. One thing you need to know that it's not Real Math when you're just doing a bunch of rote memorized procedures. But fractions are actually reasonable and figure-out-able. 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 helping us spread the word that Math is Figure-Out-Able!