Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

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  01:23

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  01:31

I think math, not do math, very cool.

Kim Montague  01:33

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  01:50

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

Kim Montague  01:55

Alright. So that would be anything that is a factor of 60. Right? Thinking about minutes.

Pam Harris  02:02

Yeah. 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  02:36

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

Pam Harris  02:46

I'm on the hotseat, alright.

Kim Montague  02:48

Alright. So you're thinking about a clock model today. And I'm gonna ask you the first problem, which is 3/4 minus 1/3.

Pam Harris  02:57

3/4 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 go ahead and 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. 

Kim Montague  03:17

Okay.

Pam Harris  03:18

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 3/4, 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 20/60. So now I've got 45/60 minus 20/60. 45 minus 20 is 25, so 25/60. That would be one way of solving the problem 3/4 minus 1/3.

Kim Montague  04:19

Nice.

Pam Harris  04:20

I'd like to do another way, for fun. 

Kim Montague  04:22

Ok.

Pam Harris  04:22

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, three 5 minute chunks. So I'm gonna think about 3/4 of an hour in terms of five minute chunks. 3/4 of an hour is over there on the nine. And so that's nine out of 12. So that's like 9/12. And then that 1/3 we decided was on the 20 minutes, or the four. So that's 4/12. And so 9/12 minus 4/12 is 5/12. 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 25/60. 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  05:19

 Nice. Alright, here we go. 3/4 minus a sixth.

Pam Harris  05:26

Cool I've already thought about 3/4 a little bit, so 1/6. 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 for me to tell you that. But maybe think, maybe pause a bit and think about 3/4 minus 1/6. 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 3/4 minus 1/6. I'm gonna do minutes first. So 3/4 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 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. 3/4 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, 2/12. So I've got 9/12 minus 2/12 . That is 7/12. 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. Alright, cool.

Kim Montague  07:10

That's good. Alright.  

Pam Harris  07:12

Bring it on!

Kim Montague  07:13

 One more problem. You're on a roll. What is 11/12 minus 5/6.

Pam Harris  07:20

11/12 minus 5/6. Y'all on a clock model, it's just screaming at me. But I'll go ahead and talk through my thinking process here. Remember, pause a little bit, think about 11/12 minus 5/6 and 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, 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 5, 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 11/12  minus 10/12 is just 1/12. Cool. So I hope you can see everybody that as I'm sort of solving this using different chunks of time, that one of them usually falls out to be the most simplified, not all the time, but often it does. But that's not the most important. I think we put way too much emphasis in the United States on simplifying and finding the most simplified answer. And I'll be frank with you. I think it's a lazy teacher thing. I think it makes it easier to grade. And so we have just overtime demanded that kids always simplify their answer. It makes it easier to grade and teachers have just been sort of lazy. Now, I'm not suggesting that maybe that's why you do it. Maybe you're not lazy. Maybe that was just the way you grew up. Your teachers asked you to give the most simplified answer. Textbooks do it. And so we just kind of have gotten this convention that we always do it. I'm pushing back on that convention. I don't think it's necessary. I think it's far more important that we find equivalent fractions that we really develop equivalencies. And then we use a denominator that makes the most sense. Makes the most sense for us to solve the problem and makes the most sense in the context for the answer. The answer might actually want one that is not the most simplified. So I'm gonna push back on this idea that it has to be the most simplified, while pointing out that if I am flexible, I can often use a different chunk of time and get the most simplified. Or take the answer that I've gotten and then use a chunk of time to simplify it. Let me give you an example. If I had just done the problem that we just did 11/12, subtract 5/6. The answer we got the first time I did it with minutes was 5/60. Once I got that 5/60, I can actually stay there on that 5/60 and just - what's the word I want? Just hang there a minute and think about 5/60 in terms of chunks of time. I could say, "Well, 5/60, that's, let's see, five minutes out of 60 minutes." That's like I'm on the 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 one out of the 12." Bam! And I've simplified. I've simplified using my understanding of equivalencies. But I'm gonna suggest don't go there too fast with students. In other words, we're trying to give you an example of how we are suggesting, use Problem Strings, use addition and subtraction of fractions to help students build equivalencies. You're going to work against yourself a little bit if when you do that, you always push for them to find the most simplified version of their answer every time. If you push every time, "Okay, I saw that you just use pennies, but now simplify it." Or, "I saw that you just used..." pennies, look I'm back in money. Or, "I saw that you used minutes, now simplify it." Or, "I saw that you used 10 minute chunks, now simplify it." If that's always your gut, like go to, like thou shalt emphasis, then kids are going to lean away from the idea of just like, "Oh, how can I use this clock model to think about equivalencies?" And they're gonna be like, "It's all about the answer. It's all about the answer." And they're gonna do one and only one way. They're going to do the way that comes quickest and easiest and use a calculator if they can. We don't want to emphasize answer at the expense of thinking about equivalencies and relationships. Okay, sorry, I totally just like broke into that string. I think I have one more problem, right? 

Kim Montague  11:45

Yep. One more. All right, we go 4 and 1/4 minus a third.

Pam Harris  11:53

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, actually.  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 subtract 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  13:18

Yeah.

Pam Harris  13:18

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 4 and 3/12. I'm supposed to subtract a third. A third is like at the four o'clock, so that's four five minute chunks out of 12 five minute chunks, 4/12. So I've got 4 and 3/12 minus 4/12. So again, I'm sort of at 4 and 3/12. I'm going to subtract 3/12 to get at 4. I've got to subtract one more twelfth. What is 4 minus 1/12? That would be 3 and 11/12. 

Kim Montague  14:02

Nice. 

Pam Harris  14:03

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 3 and 55/60 or 3and 11/12. 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  14:18

Alright, 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  14:38

Bam. 

Kim Montague  14:39

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

Pam Harris  14:43

Yep, yep. Yep. In fact, I'll tell us a little bit. We were thinking about the Problem String 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 tenth. 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 number line work and ratio table 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. 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, 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  19:06

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  19:15

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.