June 01, 2021
Pam Harris
Episode 50

Math is Figure-Out-Able with Pam Harris

Ep 50: Fractions, A Clock Model

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Math is Figure-Out-Able with Pam Harris

Ep 50: Fractions, A Clock Model

Jun 01, 2021
Episode 50

Pam Harris

How important is fractional equivalence? Important enough for another model! These models are fantastic for helping students develop a natural understanding of equivalent fractions. Listen in as Kim and Pam discuss how to use a clock model for denominators uncommon to the money model.

Talking Points:

- Use adding and subtracting fractions to help think and reason about fraction equivalence
- Using the word "over" for fractions is NOT helpful
- Equivalent fraction examples on a clock model
- A real student's math journey

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How important is fractional equivalence? Important enough for another model! These models are fantastic for helping students develop a natural understanding of equivalent fractions. Listen in as Kim and Pam discuss how to use a clock model for denominators uncommon to the money model.

Talking Points:

- Use adding and subtracting fractions to help think and reason about fraction equivalence
- Using the word "over" for fractions is NOT helpful
- Equivalent fraction examples on a clock model
- A real student's math journey

Pam Harris:

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

Kim Montague:And I'm Kim.

Pam Harris:And we're here to suggest that mathematizing is not about mimicking and rote memorizing, but it's about thinking and reasoning, about creating and using mental relationships, that math class can be less like it was for so many of us and more like mathematicians working together. We answer the question, if not algorithms, then what? Alright, y'all very excited today for our episode. I got an email recently from a colleague who said, "I had the most amazing time this morning working through partitive and quotative division with two 6th graders who just wanted to work with division today. One asked me where she could buy Building Powerful Numeracy. She had her teacher's copy and was reading about division.". Can you believe that? Alright, so cool. I appreciate receiving that email. That was kind of fun to hear about that. So yeah, I have a book called Building Powerful Numeracy. Um, you might want to check it out. It's a good read. I don't know, that I ever had intended for students to read it. But bam, by all means anybody's welcome. I think it's a pretty good read.

Kim Montague:So good. Okay, so in the last few weeks, we've been talking about fractions.

Pam Harris:Fractions. Wahoo!

Kim Montague:And in the last two episodes, we dove into using a money model to help students find equivalent fractions. And with equivalent fractions, addition and subtraction becomes doable, more Figure-Out-Able.

Pam Harris:Yeah. And so you might have noticed that we were talking about equivalence, and helping students develop equivalence with fractions. But then we quickly started doing addition and subtraction with fractions. It might have looked like or sounded like, 'teach equivalence, and then teach addition and subtraction of fractions'. However, what we're actually suggesting is that you use adding and subtracting fractions to help teach equivalence. So we did a little bit to think about fractions with money. But then we quickly dove into adding and subtracting fractions for which you had to think about equivalence in order to solve. That gives all sorts of experience with why we even need equivalent fractions, and in finding equivalent fractions, and then using them to add and subtract. That might be a little bit of a new idea for you. In my traditional experience, we sort of taught kids the rule about how to find equivalent fractions because they needed a common denominator. And we did that rule and they memorized that and then we sort of said, "Okay, now when you add and subtract, thou shalt find a common denominator." Instead, we're just giving them a problem and we're asking them to think about money. And then we're generating a bunch of equivalencies. And we're using those equivalencies to then do the addition and subtraction. So yeah, kind of cool.

Kim Montague:Yeah. So today, we're going to do a similar thing, but with a different model. Which is a clock. Using time, right?

Pam Harris:Alright. So let's get at it. Let's do a little bit of finding equivalencies. And then we'll dive right into addition, subtraction again.

Kim Montague:Okay, so I'm gonna start you off with just thinking about one half using a clock model.

Pam Harris:Alright, so back to the operator meeting, or the operator interpretation of rational numbers. I'm going to think about one half of a clock or one half of an hour, or one half of 60 minutes, or one half of sort of that clock face, as I think about it. So let's see. I just wrote down the fraction one half equals, and I could think about that one half as 30 minutes out of the total 60 minutes. And what I wrote was 30/60. So, 30 fraction bar 60. Now, as I describe what I'm doing today, I know that I'm going to struggle a little bit to describe the fractions, because I'm going to use some words, but actually write something different, perhaps. And so I want you to know that when I'm saying some words, what I'm writing, in other words, when I say 30/60, I didn't write the word thirty and then the word sixtieths, I wrote 30 fraction bar 60. I'm going to be tempted because of old habits, to say 30 over 60. So before we move too much further on, I'm going to explain why I'm not using that positional word "over" to describe the fractions. Because we're all learning mathematics as a second language, right? So as we're learning mathematics as a second language, we want to use vocabulary words that mean what we say, not have connotations or meanings that could trip us up or get us thinking about other things, or that aren't particularly helpful. "Over" is one of those words that isn't particularly helpful in describing fractional relationships. It might be helpful in describing what the fraction looks like. Like the numbers, the numerals I'm writing on the page, the way the fraction itself, the representation looks like, but not the... It's so complicated to use these words! Not the actual fraction, not the actual relationship. So the relationship of one half, or the relationship of 30/60, is all about one half of the object or one half of the set, or 30 minutes out of 60 minutes. It's not about 30 "over" 60. "Over" doesn't have anything to do with the fractional relationship I'm trying to talk about. It has everything to do with position, you know, like, I'm looking around my office, my laptop is over the desk, that's even a dumb thing to say, like, give me a better one Kim. Like, what's a good word to describe with over?

Kim Montague:Gosh, I got nothing.

Pam Harris:My... Wow, my pen is over the paper? It describes position, right? It doesn't describe "out of." It doesn't describe the part-whole relationship of fractions. Alright, we'll move on. As soon as we get off this podcast, I'm gonna think of a much better "over" relationship, but whatever. So one half could be 30 minutes out of 60 minutes. And I write that as 30/60. It could also be thought of in terms of other chunks, not just minutes, but I could think about, say, a quarter of an hour, how many quarters of an hour are in a half of an hour? Well, there's like two of them, right? Like sort of picture the minute hand going down to the 15 minutes, and then going down to the 30 minutes. And there's sort of those two 15 minute chunks in there, or two quarter hours. So I could also say one half is equal to two quarter hours out of the four quarter hours in a whole hour. So I just wrote two out of four or 2/4. But that 2/4 isn't just me going, let's say 2/4 is equal to 1/2. It's me thinking about two quarter hours out of four quarter hours. What other chunks can I think of? I can think of in terms of 10 minute chunks. So if I'm thinking about 10 minute chunks, and I want to think about half of an hour, how many 10 minute chunks are in a half of an hour. Well, I can think of three 10 minute chunks out of the six 10 minute chunks in the whole hour. I just wrote 3/6, three fraction bar six, because I've got three 10 minute chunks out of the six 10 minute chunks. Let me see if I can think of any more. One more. I can think of five minute chunks. So five minute chunks, if you think about a clock, five minute chunks are kind of where that minute hand hits every five minutes, or the numbers on the clock, right? So if the minute hand goes to the one, five minutes has gone by. If the minute hand goes to the 3, 15 minutes have gone by or three five minute chunks, right? If I'm on the 15. But I want to be on the six, right? Because we decided we want a half an hour. So the half an hour, if that minute hand has gone all the way down to the six, how many five minute chunks? 1, 2, 3, 4, 5, 6, right? I'm on the six. So I can think about one half of an hour as six five minute chunks. Out of the total five minute chunks, 12 five minute chunks. So I just wrote 6/12. But I meaning six five minute chunks out of 12 five minute chunks. So I can think about different chunks of numbers. You might be a little bored with one half because one half is like so obvious. So we probably better do a more interesting one. Are you ready, Kim?

Kim Montague:Sure.

Pam Harris:I'm going to give you 1/3. How can you think about 1/3 on a clock? What are some equivalent fractions? But you can think about with time or with a clock.

Kim Montague:Okay. I'm going to use some of the chunks that you just mentioned if that's okay. So I'm going to go with okay, 1/3. And I'm going to think that is equivalent to 20/60, Because one 20 minute chunk out of three 20 minute chunks makes that hour.

Pam Harris:And 60 minutes is total minutes, right, so 20 minutes out of 60 minutes is equivalent to 1/3.

Kim Montague:Yep. And that's the equivalent fraction. I wrote down 20/60. I'm going to go with it's also equivalent to 2/6, because two 10 minute chunks out of the six 10 minute chunks in the hour.

Pam Harris:And two 10 minute chunks because you're on that 20 minutes. That's two 10 minutes. Uh huh. Okay.

Kim Montague:And let's go five minute chunks. So five minute chunks are nice for me because it's marked clearly on the clock. And so I'm going to say, four five minute chunks out of the 12 five minute chunks.

Pam Harris:And you can kind of think of that...

Kim Montague:It's a third of the clock.

Pam Harris:Yeah, sorry, sorry.

Kim Montague:That's ok.

Pam Harris:And a third of a clock you can almost think of as a peace sign, right? And if you sort of draw that peace sign on a clock, it's kind of up at the 12 and then down at that four. Down the four and you just said it was four out of the total 12 five minute chunks. So 4/12. Cool. I like it.

Kim Montague:Alright, let's, let's do one more. If I gave you a fourth.

Pam Harris:All right, 1/4. So again, using that operator meaning I'm gonna think about a fourth of an hour. So fourth of an hour is 15 minutes out of the total 60 minutes. So I just wrote down 15 out of 60. 15/60. Let's do five minute chunks. A fourth of an hour is on a 15. Like we just said, so that's the three that's like the three o'clock, three five minute chunks. So that's three - you should see my hand. I'm like ticking over 5, 10, 15. I just realized I was doing that with my hand. 5, 10, 15. So I'm at the 15. That's the three out of the total 12. So I just wrote down 3/12. Okay, let's see, we also did 10 minute chunks. You might if I go there?

Kim Montague:Sure.

Pam Harris:Okay, so a fourth of an hour, 15 minutes, but I want to do 10 minute chunks. How many 10 minute chunks is 15 minutes? I'll let everybody think about that for a second. 10 minute chunks. 15 minutes? Is that one and a half 10 minute chunks? And how many 10 minute chunks in the whole thing? There's six of them. Right? So is that 1.5/6? Yeah. And that can actually be kind of helpful. And we might see that pop out as we do some more clock equivalencies to help us think about adding and subtracting fractions. Sometimes it can be fun to have something like one and a half sixths. Now, let me just talk about that for a second. Sometimes in traditional education, we have told students, "No, no, no, you can't have a decimal on a fraction in the same representation". I'm going to submit that that is completely by convention. Like we have decided that as mathematics, maybe just even teachers, because it's easier to grade if we don't do that? I don't know. I don't think there's anything wrong with writing one and a half sixths, or thinking about one and a half 10 minute chunks out of the six 10 minute chunks. I think it's brilliant. And it's going to help us build Proportional Reasoning. And so I want to do it. I'm going to give credit to very first seeing that to Cathy Fosnot. When I was learning about using a clock to think about fractions, and she sort of threw that out as a possibility. And I was like, "Yeah, absolutely." Like we can absolutely think that way. So I would agree with her, especially from my higher math perspective, I think it's a fine thing to do. Okay, so now that we have some equivalencies, with a few fractions, we better get right at doing some adding and subtracting.

Kim Montague:Yeah. Okay. So let's use what you just came up with. Okay. So let me ask you 1/3 plus 1/4.

Pam Harris:Okay, so I just totally wrote those down. So I can kind of hang on to what I'm doing. So if I'm going to think about a third plus a fourth, maybe the first thing I'm going to note is notice how money would not be a great model right now.

Kim Montague:Oh, yeah.

Pam Harris:Because a third of 100. A third of $1? Not so delightful. So bam, I want to go to a clock because it's gonna be much nicer. Okay. So a third on a clock, we decided that was sort of the 20 minute mark. So I'm gonna think about that in terms of minutes, so that's like 20 minutes out of the 60 minutes. 20/60. And a fourth of a clock, or fourth of an hour is like 15 minutes, so 15/60. So I've got 20/60 plus 15/60 is 35/60. That'd be one way to think about it. But let's be flexible and see if we can think about it not just in terms of minutes. However, notice that I'm not - and in fact, notice in all of this, that what we're not doing is pushing for the least common denominator. We're actually letting students use whatever denominator makes sense. And then what we're pushing for is, can you be flexible? Like can you think about it in terms of other denominators, so that you get better at the other denominators, really which means we're pushing for equivalencies. Like we're trying to get kids better and better at different equivalencies and thinking about different denominators. So let's see a third plus a fourth. I did minutes. Let's think about five minute chunks. Yeah, let's go five minute chunks. So a third of an hour is like at the 15. So that's like, I'm doing it again, I'm ticking my hand down. That's like at the three o'clock. So that's three out of the 12 five minute chunks. And a fourth of an hour that's like - Oh, wait, that's a fourth of an hour, sorry. A third of an hour's at the 20. So that's four twelfths. 4/12. The fourth of an hour is 3/12. So that's funny because there's a little interesting reciprocal relationship kind of going on there.

Kim Montague:Right.

Pam Harris:So I've got a third of an hour is 4/12. And a fourth of an hour is 3/12. And so that would be seven, five minute chunks out of the 12 five minute chunks or 7/12. That was fine. But what I really want to do is have a little bit of fun with the 10 minute chunks quickly. We're doing a third plus a fourth. So a third of an hour, that's at the 20. So that's like two 10 minute chunks out of the six 10 minute chunks. So I've written 2/6. And then a fourth of an hour, that was that fun one and a half 10 minute chunks out of the six 10 minute chunks. So now I've got 2/6 plus 1.5/6 is 3.5/6. Nice, nice. And if I can think about three and a half 10 minute chunks, I can think about three 10 minute chunks which puts me on the six, right, because we're on the 30 minutes, and then a half of a 10 minute chunk is another five. So that's like 35 minutes. And that's what I got. I got 35 out of 60. And also that seven out of 12 also lands me on that three and a half sixths. Pretty cool.

Kim Montague:Well done.

Pam Harris:So Kim - thank you. Thank you. Thank you. You know that we have a mutual acquaintance named Jordan, and we love Jordan. And Jordan, if you ever listen to this, thank you so much for letting us study your thinking over time. Jordan was Kim's student in third and fourth grade. He had Kim in third grade. And then Kim looped with her students, and had him in fourth grade. And we were able to sort of magically - or by the grace of God, which is just kind of the way I fall - get video of him in second grade because his teacher videoed her class in second grade. And then we got video of Jordan in third and fourth grade because Kim had him and we were videoing her kids at that point. And then later when I realized I had him in second grade, and in third and fourth grade that I went back and grabbed him in fifth grade and sixth grade and we even have video of him in 10th grade. And so I want to tell you about that video that we took of him in 10th grade today a little bit because I realized right around that time that I had video of Jordan in all these places, but I didn't ever have him doing any fraction work. So I realized 10th grade was a little bit old, you know, to like do just fraction work. But, who cares? I decided to go grab him. So I knew his mom from church. And so I called her up and I said, "Hey, do you mind if we video Jordan?" She's like, "Yah no problem." And so I did. I literally just popped in on this poor 10th grade kid and said, "Hey, do you mind if we video again?" He's a little shy. He's got this great kinda shy grin? And he's like, "Mo, it's fine." And so I said, "Hey, do you ever remember doing a clock model with fractions?" And he looked at me, he goes, "Miss Harris, it's been a long time since I've done fractions." And I was like, "It's okay. Totally. Don't worry about it." We turn the camera on. And I said, "Hey, Jordan, what can you tell me about a third-" how did I even ask it? I think I just said the fraction 1/3, but on a clock. And he kind of thought about a third on the clock a little bit, kind of like we just did. And then I said, how about a fourth on a clock? And he thought about it a little bit like we just did. And I said, "So can you use that to help you think about a third plus a fourth?" And I'll never forget what he did. He kind of looked up, you know, how you sort of look up when you're kind of thinking, and he goes, "Well, let's see a third, four, and a fourth, three, and three plus - seven, seven twelfths." I mean, he just like thought through it. Just 7/12. And then I was like, "Whoa, dude, what did you just do?" And then it took him a sec to describe, he's like, "Well, a third of a clock is like 20 minutes. And so that's like, on the four. And a fourth of a clock is like 15 minutes. So that's like on the three, and three plus four is seven, so 7/12." And it took him far longer to describe what he was thinking about than to just actually do it. It was brilliant to watch him just think and reason using relationships. And one of the reasons he can do that is because he learned from the Master. Kim, and our colleague in second grade to think and reason about math. And so he learned he could just think and use what he knew about math. And it's an awesome video, maybe we'll put in the show notes. Why not? We'll put in the show notes. The video that we have posted on YouTube where you can go watch Jordan, actually do that. Notice when you watch the video, we are in his house. You'll see the vacuum behind him. Like we are totally right there in his house. Thank you, Jordan, for being willing to let us study you and how you think as you were growing up, we really appreciate that. So very cool. Alright, we got a lot more to talk about with fractions on a clock. So we're gonna do more of that next week and have a lot of fun with not just unit fractions, you heard us talk about unit fractions today where the numerator is one. Let's do some work next week on a clock where we think about non unit fractions, and even mixed numbers and all the things. So stay tuned for next week where we're gonna have more fun with fractions because fractions are our friend, especially when we think about money and a clock model.

Kim Montague:But before we end, I want to ask what denominators do you think would work well, for a clock model? We talked about denominators for money models. So be thinking what would work well for a clock model, and we'll talk more about that next week.

Pam Harris:Excellent. Alright. Remember to join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday evenings, where we throw out a problem and check out strategies from around the world.

Kim Montague:If you find the podcast helpful, would you please rate it and give us a review? That way more people can find it wherever they get podcasts.

Pam Harris:And don't forget that we are collecting your questions that you want answered. Send those to Kim@mathisfigureoutable.com. We're going to tackle them in an upcoming episode coming soon. So send them in quickly. So if you're interested to learn more math, and you want to help yourself and students develop as mathematicians, don't miss the Math is Figure-Out-Able Podcast. Because Math is Figure-Out-Able!

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