# Ep 50: Fractions, A Clock Model

June 01, 2021 Pam Harris Episode 50
Math is Figure-Out-Able with Pam Harris
Ep 50: Fractions, A Clock Model

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

Jordan Using the 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:

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:

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:

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:

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!