June 08, 2021
Pam Harris
Episode 51

Math is Figure-Out-Able with Pam Harris

Ep 51: Fractions, a Clock Model Pt 2

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Math is Figure-Out-Able with Pam Harris

Ep 51: Fractions, a Clock Model Pt 2

Jun 08, 2021
Episode 51

Pam Harris

We've had so much fun talking about fractions! In this episode Pam and Kim do another Problem String to demonstrate the effectiveness of a clock model.

Talking Points:

- What denominators work well for a clock model?
- Problems that work with both money and a clock model
- Today's students need the experience from learning with a clock model

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We've had so much fun talking about fractions! In this episode Pam and Kim do another Problem String to demonstrate the effectiveness of a clock model.

Talking Points:

- What denominators work well for a clock model?
- Problems that work with both money and a clock model
- Today's students need the experience from learning with a clock model

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 or rote memorizing. It's about

reasoning:about creating and using mental relationships. That mathematics class can be less like it has been for so many of us, and more like mathematicians working together, learning together. We answer the question, if not algorithms, then what? Alright, y'all on Facebook, we got a really cool message from Amanda Mctavish Watson, who said, "I got a teacher-appreciation card from a young lady this week that said, 'Thank you for helping me THINK math' made me cry. Honestly, I always praise my girls for thinking math. It's not a favorite for many girls and men dominate the STEM world. So here's one for Pam Harris and teaching kids to THINK math, rather than do math". Amanda I love that. Thank you so much for sharing that. That is amazing. Super cool. Super cool that your student recognized you during appreciation week when we got this, it was a little while ago, but we're so appreciative that you sent that on. And way to change these young women's lives. That's excellent, excellent, excellent.

Kim Montague:I love, love, love when people share what they're doing and how their kids are thinking based on what they're learning from you. It's so fantastic to hear.

Pam Harris:I think math not do math, yah.

Kim Montague:Yeah. Okay. So in the last episode, we introduced the idea of a clock model for adding and subtracting fractions. It is such a great tool. And we asked for which denominators would this model be helpful? We asked y'all to think about what those would be.

Pam Harris:So pause, if you haven't thought about it. Pause first. Okay, go ahead.

Kim Montague:All right. So that would be anything that is a factor of 60. Right? Thinking about minutes.

Pam Harris:Clock model, minutes, that sort of makes sense. So with money, we get 100 - factors of 100. And with clock, we get 60 - factors of 60. Totally cool. When we started talking about fractions, I quoted a dear friend and colleague Garland Linkenhoger, who said, if you really understand fraction equivalence, you do not need any rules for fraction operations, you can just reason your way through the operations. So true. So cool. Garland, I appreciate you throwing that out for me years ago. Let's get right at continuing to do that.

Kim Montague:So today, let's do another problem string to help build equivalence using addition and subtraction. Are you ready?

Pam Harris:I'm on the hotseat, alright.

Kim Montague:Alright. So you're thinking about a clock model today. And I'm gonna ask you the first problem, which is three fourths minus 1/3.

Pam Harris:3/4s minus 1/3. So instantly, I'm thinking not money, because 1/3 money yuck. Clock, nice, I can think about a third of an hour. So that's going to be really brilliant. So let's see the first thing that comes to mind... I'm going to do minutes first, even though actually five minute chunks came to mind first, but I'm going to do minutes in an attempt to kind of keep everybody with us today. Okay, so I could think about well, and actually, in our last episode, we promised that we would do fractions that were more than unit fractions. So I'm just going to remind you, that in the last episode, if you haven't listened to that one, go check it out, because we really talked about unit fractions, and today we're going to talk about how we can think about non unit fractions. So three fourths, that's a non unit fraction, the numerator is not one - three fourths. So I can think about 1/4 on a clock. And then I can think about three of those one fourths. And so that sort of puts me at 1/4 is 15 minutes. So three of those one fourths would be 45 minutes, so I just wrote 45/60, 45 minutes out of the 60 minutes. And 1/3 I can think of on a clock is a third of an hour. 60 divided by three is 20. So that's like 20 minutes out of the 60 minutes. So 20/60s, so now I've got 45/60 minus 20/60, 45 minus 20 is 25. So 25/60. So that would be one way of solving the problem three fourths minus 1/3. I'd like to do another way. For fun, and this is actually the way that came to mind first. I'm going to use five minute chunks. So where are the five minute chunks. just to review a little bit, five minute chunks are sort of the numbers on the clock. Every time that minute hand goes to one that's five minutes to two, that's 10 minutes, right, another five minute chunk. Three, that's 15 minutes, 3 5 minute chunks. So I'm gonna think about three fourths of an hour in terms of five minute chunks, three fourths of an hour's is over there on the nine. And so that's nine out of 12 so that's like nine twelfths. And then that 1/3 we decided was on the 20 minutes, or the four, so that's four twelfths and so 9/12 minus 4/12 is five twelfths. I can kind of ask myself 5/12, where am I on a clock? 5/12 that's like 25 minutes, right? Hey, that's what we got when we did minutes, we got 25 sixtieths. 25 minutes out of 60 minutes. So 5/12 and 25/60 is sure enough the same place on the clock. Equivalent. Bam!

Kim Montague:Nice. All right, here we go. Three fourths minus a sixth.

Pam Harris:Cool I've already thought about three fourths a little bit. So one sixth. So y'all, if you're listening to the podcast you might pause it right now, and see what you're thinking about. In fact, you could have paused it before. I'm a little late in the game to tell you that, but maybe think, maybe pause a bit and think about three fourths minus one sixth. Think about how you would do it. And really push yourself if you've never used a clock model to think about it using a clock, and maybe even a couple of different chunks of time, and then come back and hear how I'm thinking about it. So let's see three fourths minus a sixth, I'm gonna do minutes first. So three fourths of an hour is 45 minutes out of 60 minutes, and 1/6 of an hour, if I think about an hour, 60 minutes divided by six, that's like 10 minutes out of 60 minutes. A sixth of an hour, 10 minutes out of 60 minutes. So now I've got 45/60 s minus 10/60 is 35/60. Cool. So I'm going to hope that when I do this with five minute chunks in a minute, 35/60 that I land on the 7. 35 minutes is on the seven. Okay, cool. So I'm going to think about this in five minute chunks. Three fourths of an hour again over there on that 45 that's like the nine. So nine out of 12, nine five minute chunks out of 12 five minute chunks. And then a sixth. A sixth we decided was at the 10 minute mark, because 60 divided by six is 10. So 10 minutes, what is that in terms of five minute chunks? 10 minutes is like two five minute chunks. So that would be two five minute chunks out of the 12 five minute chunks. Two twelfths. So I've got nine twelfths minus two twelfths . That is seven twelfths. And bam, I'm on the seven o'clock, which is what I was hoping for. Because I'd also gotten 35/60, that's at seven o'clock and now 7/12. There we go. All right, cool.

Kim Montague:That's good. All right.

Pam Harris:Bring it on!

Kim Montague:One more problem, you're on a roll. What is 11/12 minus 5/6.

Pam Harris:11/12 minus five sixths. Y'all on a clock model it's just screaming at me. But I'll go ahead and talk through

Kim Montague:Yep. One more. All right, we go 4 and 1/4 minus a my thinking process here. Remember, remember, pause a little bit, think about 11/12 minus 5/6 and see see what you're thinking about. See if you can push yourself to use a clock model. Okay. 11/12 is screaming at me that I'm at the 11 out of the 12 five minute chunks. So it's kind of like if I'm thinking about - you know what actually comes to mind? I'm thinking about a TV show. And I'm saying to myself, I got five minutes left. Now there's dating myself, because I know nobody these days actually watches TV on the hour anymore, right? Like, it's all like, whenever you're streaming, whatever you're streaming. Which is true for me. But when I was growing up, everything was you know, on the hour, so I would have a clock, and anyway you don't care. Alright, so 11/12 means I'm at the 11 o'clock. And 5/6. I can think about 1/6 of an hour of 60 minutes is 10 minutes. And so five of those 10 minute chunks puts me at the 50 Minute. So I'm kind of at the 55 out of 60, subtract 50 out of 60 is just five minutes out of 60 minutes or 5/60. But I was actually thinking about that as the 11 o'clock minus the 10 o'clock, and 11 o'clock - or 11/ 2 minus 10/12 is just 1/12. Coo . So I hope you can see everybo y that as I'm sort of solving th s using different chunks of tim , that one of them usually fal s out to be the most simplifie , not all the time, but often t does. But that's not the mo t important. I think we put w y too much emphasis in the Unit d States on simplifying a d finding the most simplifi d answer. And I'll be frank wi h you. I think it's a lazy teach r thing. I think it makes t easier to grade. And so we ha e just overtime demanded that ki s always simplify their answer. t makes it easier to grade a d teachers have just been sort f lazy. Now, I'm not suggesti g that maybe that's why you do i . Maybe you're not lazy. May e that was just the way you gr w up. Your teachers asked you to give the most simplified ans er. Textbooks do it. And so we ust kind of have gotten this con ention that we always do it. I' pushing back on that conven ion. I don't think it's necess ry. I think it's far more import nt that we find equiva ent fractions that we really develop equivalencies, and th n we use a denominator that m kes the most sense. Makes the m st sense for us to solve the p oblem and makes the most sense in the context for the answe . The answer might actua ly want one that is not the m st simplified, so I'm gonna push back on this idea that as to be the most simpl fied while pointing out that f I am flexible, I can often use a different chunk of time nd get the most simplified. Or t ke the answer that I've gott n and then use a chunk of time to simplify it. Let me give you n example. If I had just done the problem that we just did 1/12, subtract five sixths. T e answer we got the first time I did it with minutes was 5/60. nce I got that five sixtieths, I can actually stay there on that 5/60 and just - what's he word I want? Just hang there a minute and think about 5/60 n terms of chunks of time. I cou d say, well, 5/60, that's, let's see, five minutes out of 60 mi utes. That's like I'm on t e one o'clock, right where am I when five minutes have gone by, I'm on the one. And so I could say, well, that's just like on out of the 12. Bam! And I've s mplified. I've simplified using my understanding of e uivalencies. But I'm gonn suggest you don't go there oo fast with students. In other words, we're trying to give yo an example of how we are s ggesting you use problem strings use addition and subtract on of fractions to help stu ents build equivalencies. You're going to work against yourself a little bit if when you do th t, you always push for them o find the most simplified versi n of their answer every time. If you push every time, okay, I aw that you just use pennies, bu now simplify it. I'm bac in money. Or I saw that you used minutes, now simplify it. O I saw that you used 10 minute c unks, now simplify it. If that's always your gut, like go o, like thou shalt emphasis. hen kids are going to lean aw y from the idea of just like, oh, how can I use this cloc model to think about equivalenc es? And they're gonna be like i 's all about the answer. It's a l about the answer. You're gonna do one and only one way they're going to do the way that com s quickest and easiest and us a calculator if they can. We do 't want to emphasize answe at the expense of thinking abo t equivalencies and relationsh ps. Okay, sorry, I totally just ike broke into that string. I th nk I have one more problem, righ ? third.

Pam Harris:4 and a fourth, subtract a third. So we promised you guys a mixed number, we wanted to make sure we got at least one of those in there. And really nothing changes, I'm still going to use a clock to help me think about what's going on. Let's see. So four and a fourth. It's almost like I've got four hours and a quarter of an hour. And I'm going to subtract a third of an hour. Hmm. I'm thinking about that. I yeah, it's screaming at me, I'm going to do minutes first, even though the twelfths are screaming at me, I've gotten much better at the 12ths at the five minute chunks over time. I'm going to do minutes first. So I've got sort of four hours and a quarter is four hours and 15 minutes. So I'm going to write four and 15/60. Subtract a third of an hour, a third of an hour is like 20 minutes out of 60 minutes. So I've got four and 15/60 minus 20/60. I'm thinking about that on a number line. I couldn't do it on a clock, I could think about - nah I'm gonna do it on a number line. 4 and 15/60 and if I subtract 20/60, I'm gonna subtracts 15/60 first and land on the four and now I've got to subtract another 5/60 or five minutes. And so four hours minus five minutes. That's like three hours and 55/60, that would be one way to solve that problem. 3 and 55/60, right?

Kim Montague:Yeah.

Pam Harris:Yeah. Okay, cool. Now I'm going to go ahead and do five minute chunks. So four and and a fourth is like four and - a fourth of an hour is like on the 15. So that's like the three o'clock or three five minute chunks out of 12 five minute chunks. So four and three twelfths. I'm supposed to subatract a third. A third is like at the four o'clock, so that's four five minute chunks out of 12 five minute chunks. Four twelfths. So I've got four and 3/12 minus four twelfths. So again, I'm sort of at four and three twelfths. I'm going to subtract three twelfths to get at four. I've got to subtract one more twelfth. What is four minus 1/12? That would be three and 11/12.

Kim Montague:Nice.

Pam Harris:Yeah. And I'm sort of I'm looking back. You guys can't look back, unless you're writing along with me. But I had three and 55/60 or three and 11 twelfths. And sure enough, those put me at the same place on the clock right? 55 minutes and the 11 minute - yeah, yeah, we're good.

Kim Montague:All right, so great. So we today used, actually last week or the week before, used money to work with denominators that were factors of 100. And then we've been working with a clock to create equivalence with factors of 60. But what about when there's denominators that are both factors of 100 and 60?

Pam Harris:Bam.

Kim Montague:Right? Well, that's when we give kids choice, right?

Pam Harris:Yep, yep. Yep. In fact, I'll tell us a little bit. We were thinking about the prompts trying to use for today and we had a couple problems that were too choosy. And we thought, for this one, let's just make it like you have to use a clock model. You can't use either one. Maybe in the future someday we'll do some work in an episode where we have some choose your model problems. And maybe I'll just throw out an example, where you could sort of choose your model. An example would be something like a half minus a 10th. Like I can use money and money sort of screams. But I could also use a clock, no problem. Maybe make it a half minus a 20th. And maybe you feel even more like being on a clock. But anything that is a denominator of both 100 and 60, then you have a choice. And you could use either model, whichever one feels sort of right to you at the point. And that's the real power. The real power is when people have the power to choose, not just one way to solve a problem that traditional way where we just only thought about one way. So you might be asking yourself, okay, what about problems that have denominators that aren't factors of either 60, or 100? Well, we've built enough experience around equivalence by working with these models regularly that now we're ready for some double numberline work and ratiotable work to help students make sense of any denominators. So stay tuned to the podcast, we'll do more on that in later episodes. But the work with money and clock models lay such a great groundwork. And now students fraction sense is so much better. And to be clear, once you have students really good at fractions and money using those models, so many of the problems they're ever going to hit in their life could be solved with those. Maybe not the problems that they hit in higher math, though, honestly, often they are. There are other denominators, and we do need to work with them. But so many of the problems that they're ever going to hit, they could just think clock or money and bam, they're done. They don't have to reach into rote memory to do anything, nor do they even have to reach into anything more sophisticated than just using clock or money. So here's a way to summarize what we've been doing in these last few episodes. Our friend Kathy Hale said, "it's like you're using common situations, contexts, models to help students develop common denominators or the need for common denominators.". Yep, that's exactly it. Thanks for that, Kathy, we were in the middle of a workshop and she raised her hand and she's like I get it, I get it. We're using clock and money because they are these common situations, contexts that kids deal with every day, to help motivate the need for them to find common denominators. Now, teachers, you might be thinking, Pam, you don't understand our kids these days. Money and clocks, those are not common contexts. Those are not uncommon situations. They don't deal with change anymore. They don't read analog clocks. Okay. Okay. Kim, and I will grant you that. We will grant that maybe our students - okay, not even maybe - that our kids have less experience with money and reading, especially reading analog clocks, telling time and having a sense of the relationships among 60. All the more important that we do this work with them, so that we teach them both money, a sense of money and time and equivalencies with fractions at the same time. Like this is going to be important, we cannot have students lose the sense of these important denominators of 100, or factors of 100 and factors of 60. So we agree with you, yes, your students have less experience with those, then let's give them experience. If you're a teacher of younger grades and you've made it this far in the podcast, absolutely wonderful. Do more work with your students with money and time. If you're a teacher who's

teaching fractions:do work with the money model and the clock model to help your students create both relationships, the relationships with money and time and the relationships with equivalencies with common denominators. So we get it, we get it, but it is all the more important to do this kind of work. Alright, fabulous. Remember to join us on MathStratChat on Facebook, Twitter, Instagram on Wednesday evenings, as we explore interesting problems with the world.

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

Pam Harris:So if you're interested to learn more math, and you want to help yourself and students develop as mathematicians that don't miss the Math is Figure-Out-Able Podcast because math is figure-out-able.

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