# Ep 189: The Operator Meaning of Fractions

January 30, 2024 Pam Harris Episode 189
Ep 189: The Operator Meaning of Fractions
Math is Figure-Out-Able!
Math is Figure-Out-Able!
Ep 189: The Operator Meaning of Fractions
Jan 30, 2024 Episode 189
Pam Harris

Finding a fraction of something can be a tricky concept. In this episode Pam and Kim run through two strings that help reason about using fractions as operators.
Talking Points:

• How are unit fractions related like halves, fourth and eighths?
• Scaling and Over Strategy with unit fractions
• Fractions as operators
• Why is scaling a unit fractions critical? Knowing that three fourths is the same as three 1/4s?
• Knowing non-unit fractions from fractions is a precursor.
• Thinking of fractions only as part-whole is not sufficient.
• A different problem string using the operator meaning of fractions.

Check out our social media
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education

Finding a fraction of something can be a tricky concept. In this episode Pam and Kim run through two strings that help reason about using fractions as operators.
Talking Points:

• How are unit fractions related like halves, fourth and eighths?
• Scaling and Over Strategy with unit fractions
• Fractions as operators
• Why is scaling a unit fractions critical? Knowing that three fourths is the same as three 1/4s?
• Knowing non-unit fractions from fractions is a precursor.
• Thinking of fractions only as part-whole is not sufficient.
• A different problem string using the operator meaning of fractions.

Check out our social media
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education

Pam  00:01

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

Kim  00:10

Totally spaced out.

Pam  00:12

And you're Kim.

Kim  00:13

I'm still Kim. Oh, my gosh.

Pam  00:16

And, ya'll, you found a place where we're a little punch happy here. Where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. And laughing when appropriate.

Kim  00:30

Oh, my gosh.

Pam  00:30

We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps isn't fun and actually keeps students from being the mathematicians they can be. Hey, Kim.

Kim  00:45

Hi, guess what?

Pam  00:47

Oh, gosh. Yes?

Kim  00:48

We're going to talk fractions again today.

Pam  00:50

Whoo!

Kim  00:51

I love fractions. Not as much as percents, but I really do like fractions.

Pam  00:55

You know, it's funny because when I first met you, I think you said something like, "Oh, I really prefer percents." But the more I worked with you, I was like, you really love fractions. Like, I know you love. Like, you love percents more, but that doesn't mean you like fractions any less.

Kim  01:06

No.

Pam  01:06

Yeah, you like fractions. It's good fractions are good.

Kim  01:09

Yeah.

Pam  01:09

The fractions are our friends.

Kim  01:11

I hear you have some strings for me today.

Pam  01:12

Yes. Let's string. So, everybody, pick up your pencil, pen.

Kim  01:16

Yeah!

Pam  01:17

Whatever (unclear).

Kim  01:17

Oh, you said pencil first.

Pam  01:19

Well, I said whatever your favorite writing utensil is. Who was just telling me they were going to send me a box of mechanical pencils.

Kim  01:25

Bleh.

Pam  01:26

I haven't gotten it.

Kim  01:26

You will hate those. Don't do it. (unclear)

Pam  01:31

I'll tell you what, if I'm going to do it, it will be mechanical. It will not be.

Kim  01:35

Do you actually own any Ticonderogas?

Pam  01:37

I do.

Kim  01:38

Okay.

Pam  01:38

I do, but none of them are sharp.

Kim  01:41

(unclear).

Pam  01:41

I'll tell you what. When my kids were in school, we had this really, really nice pencil sharpener, and so we always had sharp Ticonderogas around. And that made it doable. And I don't know what happened to the pencil sharpener. I honestly don't.

Kim  01:56

Sounds like you need a Christmas present.

Pam  02:01

Sounds like somebody should have gotten me for... No, it's fine. I'll put it on next year's list. Okay, here we go. Ready? Kim?

Kim  02:06

Yeah.

Pam  02:06

What is one-half of 36?

Kim  02:10

That would be 18.

Pam  02:11

And how do you know? I know. Do you just know have 36?

Kim  02:17

Yeah.

Pam  02:18

Okay. Could you? If a kid was like.

Kim  02:21

Sure. Half of 30 and half of 6. So, half of 30 is 15 and half a 6 is 3. Cool.

Pam  02:26

Good enough.

Kim  02:27

Or you could do Over. Half a 40.

Pam  02:29

Oh.

Kim  02:30

And then, backup? Half of 4.

Pam  02:33

Nice. Wait, half of 40? Yeah. I had to think for a second.

Kim  02:38

That would be 20.

Pam  02:38

Okay.

Kim  02:39

Okay.

Pam  02:39

Okay, what is one-fourth of 36?

Kim  02:44

That would be 9 because one-fourth is half of a half, and so half of the 18, which was half a 36, gives me 9.

Pam  02:58

So, we could think about a fourth of something as half of the half.

Kim  03:01

Yeah.

Pam  03:02

Of that thing. Cool. Is there another way that you might think about one-fourth of 36?

Kim  03:07

I mean, I could divide 36 by 4.

Pam  03:12

Mmhm.

Kim  03:12

And so, 36 divided by 4 is 9.

Pam  03:14

Yeah. And if you know 4 times 9, and you're thinking about 36 divided by 4, that can be. Cool. So, kind of two different ways to find a fourth. You can divide by 4. But you can also think of half a half. Yep. Cool. What is three-fourths of 36? Three 1/4s of 36?

Kim  03:32

Yeah. If one-fourth of 36 was 9. Then, three 1/4s is 3 times 9. So, that's 27.

Pam  03:42

Cool. I have confidence you have an Over strategy for that one in you somewhere.

Kim  03:48

Yeah, I could. Four-fourths of 36 is 36. So, I want a fourth of 36 less than that. And you just asked me one-fourth of 36, which was 9. So, 36 minus 9 is 27.

Pam  04:05

Bam!. So, you can think of three-fourths as three 1/4s. Or you could also backup a fourth from four-fourths.

Kim  04:12

Yep.

Pam  04:13

Nice. What is one-third of 36? Yes?

Kim  04:22

A dozen eggs.

Pam  04:23

Oh. Okay.

Kim  04:25

It's 12.

Pam  04:27

Do you buy eggs in 36s?

Kim  04:30

No, but sometimes we buy three 12s.

Pam  04:34

Oh.

Kim  04:35

The big ones don't fit in the fridge well, but we can stack three dozen.

Pam  04:39

Okay.

Kim  04:40

And so, if my whole amount is all the 36 eggs, then a third of them would be 12 eggs.

Pam  04:49

Cool. And so, you're thinking about 36 divided by 3 a little bit?

Kim  04:53

Yep.

Pam  04:53

Kind of?

Kim  04:54

Yeah.

Pam  04:54

And 3 times 12 is 36. Cool. You got anything else? I'm just curious. I don't know that I do.

Kim  05:03

A third of 30 and a third of 6.

Pam  05:05

Oh. Yeah. Okay. A little partial quotients there. Alright, how about two-thirds of 36?

Kim  05:12

That's just 24. So you could have two 1/3s.

Pam  05:18

Okay.

Kim  05:19

So, two 12s. Or if you want me to go over, then I can say all of it would be 36. That's three-thirds. And back the one-third that you just asked me. So, 36 minus 12.

Pam  05:32

Is also 24. Cool. How about one-sixth of 36?

Kim  05:37

Oh, that's 6.

Pam  05:38

How do you know?

Kim  05:39

Because 6 times 6 is 36.

Pam  05:42

Bam! Is there a relationship between any of the other ones we've done? I have them all written down. I don't know if you were writing them down?

Kim  05:50

I did. I did write them down.

Pam  05:51

Okay.

Kim  05:51

So, I could get from one-third to one-six by dividing in half. So, a sixth is a half of a third. And so, 6 is half of 12.

Pam  06:05

Nice, nice.

Kim  06:06

I'm looking to see if there's any other.

Pam  06:09

Yeah.

Kim  06:10

I could go back to a half of 36. And it's kind of like backwards of what I just said. One-third of a half is one-sixth. So, 18 divided by 3 is 6.

Pam  06:29

Cool. And how do you know a third of a half is a sixth?

Kim  06:35

Say it out loud, Kim. So, if I have a half of something, and I want a sixth of them, then I have to cut each one of those half's into thirds.

Pam  06:49

Nice. And I'm actually picturing fraction bars. If I had a bar, and I wanted to cut it into halves, thirds, sixths. To cut it into sixths, I could do it two ways. I could fold it in half, and then fold that into 3. Once it's in half, fold that into thirds. And when I unfold it, like you just said, each of those halves would be in 3 pieces.

Kim  07:14

Mmhm.

Pam  07:14

And so, that would give me sixth. So, that's a half split, divided by 3.

Kim  07:19

Mmhm.

Pam  07:19

But I could also have folded the first original one into thirds. Kind of that, you know, as you sort of jury rig it until it's like equal and they're thirds. And then, cut that in half. So, half of a third is a sixth or a third of a half is a sixth. Nice. Cool. What if I wanted five-sixths of 36. I can't say sixths without spitting. Five-sixths.

Kim  07:40

Yeah, I don't think anybody can. Six-sixths is 36. So, I want to go back a sixth. Which is those 6 eggs in my mind still. So, 36 minus 6 is 30.

Pam  07:58

So, five-sixths of 36 is 30. Cool. Can you also think about it as Five 1/6s of 30?

Kim  08:07

Oh, yeah, yeah. Scale up from 1. So, that would 6 times 5.

Pam  08:12

Yeah, so (unclear).

Kim  08:12

One-sixth. Mmhm.

Pam  08:13

Kim  08:15

One-sixth is 6. One-sixth of 36 is 6. I need five 1/6s, so I'm scaling up by 5. (unclear).

Pam  08:23

(unclear). Nice, nice. Because five-sixth is five 1/6s. Okay. Almost done. What is one-twelfth of 36? One-twelfth of 36?

Kim  08:36

I went back to one-sixth of 36, and I know it's going to be half as much as that, so that's going to make it 3. Half of a sixth is a twelfth.

Pam  08:49

Nice, nice. Got anything else for a twelfth of 36? I like the half of a sixth. I like it a lot.

Kim  08:55

Yeah. If I go from a third to a twelfth, then I'm dividing by 4. So, on the other side of what I'm writing down, I could be 12 divided by 4.

Pam  09:10

Which is 3. Okay, cool. Cool. Anything else?

Kim  09:14

Half to 12, divided by 6.

Both Pam and Kim  09:18

18 divided by 6.

Pam  09:19

Cool. Yeah. How about from a fourth?

Kim  09:22

You can go from a fourth to a twelfth divided by 3.

Pam  09:26

Nice.

Kim  09:27

So, 9 divided by 3 is 3.

Pam  09:28

And what if you didn't have any of those, and I just said one-twelfth of 36 (unclear).

Kim  09:32

Then, I could do 36 divided by 12.

Pam  09:35

And that's also 3. Cool last question. eleven-twelfths of 36. Eleven-twelfths of 36.

Kim  09:41

Twelve-twelfths of 36, so eleven-twelfths is 3 less than that, so 33.

Pam  09:51

Nice. Nice Over to think about eleven-twelfths. You could also have thought about eleven 1/12s, right?

Kim  09:59

And that's not bad. because it's 11, so 3 times 11. But sometimes scaling up that much can be funky.

Pam  10:06

Yeah. And we would probably do the scaling times the unit fraction first.

Kim  10:11

Yeah.

Pam  10:12

But today, we thought we would kind of do a little bit of both. Scaling the unit fraction, but also backing up just that one unit fraction from the whole to get things like three-fourths, two-thirds, five-sixths, and eleven-twelfths.

Kim  10:26

Yeah.

Pam  10:26

All of those were one unit fraction away from the whole. So, purposeful ordering of problems in a string. Yeah. Nice. Alright, cool.

Kim  10:36

So, I knew we were going to talk about operator today. (unclear).

Pam  10:40

And you're saying that. Let me actually just say that for a second. Sorry for interrupting. So, that idea of thinking about one-fourth of something, three-fourths of something, one-twelfth of something is kind of treating fractions as operators. And that is one way. It's one of the five interpretations of rational numbers that we need to help students deal with and have experience with. And so, we could do a Problem String like we just did to help students really think about this operator meaning of fractions. And then, Kim, go ahead. You wanted to talk about how you did some just the other day with it.

Kim  11:18

(unclear). Yeah, not too long ago, I did a string similar to this in a fourth grade class. Just to see, you know like, what could happen. And so, the string was half of 16. No problem. And then, we did one-fourth of 16. Also fine. And then, I asked them three-fourths of 16. And what I realized, as they were kind of fumbling around about it is that it wasn't the operator meaning of fractions that they were having some difficulty with. And it wasn't necessarily thinking about multiplication. The struggles were absolutely knowing that three-fourths was three 1/4s. And I know they had done some work with fractions. And, you know, there was probably some assumption on my part because I hadn't been in the classroom before. But we definitely needed unit fractions like we had talked about in some of the other podcasts we've done. That in order to do some of this work, kids have to really make sense of three-fourths not as 3 out of 4, but as a scaled up from one-fourth. So, I think there's... It just occurred to me in that moment that there was a lot of... I don't know if it was a missed opportunity or just not knowing what the kids knew or didn't know at that time. But, you know, we are saying here on the podcast three-fourths and three 1/4s. But it was incredibly challenging to attempt to do this particular string because they did not see three-fourths as three 1/4s. And so, then the next thing I did was one-eighth. Not so much of a problem. But they didn't see that that was 16 divided by 8. Sorry, 16... Yeah, 16 divided by 8. You know, kind of like cutting things apart and seeing what they could find. But then again, when we get to three-eighths. It felt like a whole brand new problem.

Pam  13:25

They weren't able to. Once they had fussed around with one-eight of 16, and they found that was 2, they couldn't then use that to think about three of those 1/8s

Kim  13:34

Yeah. And so, some of the precursor of knowing non-unit fractions from unit fractions was something that I made a suggestion with the teacher that they could continue to work on. Because some of this operator meaning stuff that we're doing has everything to do with scaling up from the unit fraction. Anyway, so I just... You know, I knew that we were going to do this operator meeting of fractions. And, you know, I wanted to just share some of the challenges that I had had, and some of the work that, you know, I feel like can be done in order to make this string that you and I did more successful.

Pam  14:13

Yeah. So if students have only ever had a part-whole. If they only worked with fractions as a part-whole representation, and so they're looking at a problem like three-eighths of 16, they're like, "3... 1, 2, 3 out of 8... 1, 2, 3, 4, 5, 6, 7, 8 of 16..." Ah! Like, we need the multiplicative relationship, thinking about three 1/8s. Like, you kept saying, like scaling that unit fraction three 1/8s. The unit fraction of one-eighth, if I know that's 2. One-eighth of 16 is 2 Then, I can scale that multiplicatively. And that's another example of how fractions really are multiplicative in their nature.

Kim  14:52

Yeah.

Pam  14:52

And then, they're not only multiplicative three 1/8s. That's multiplicative. But then, three-eighths of 16 is really proportional. And we're really dealing with three-eighths is a proportional relationship of 16. And we can get at it as sort of breaking into that multiplicative thing. Yeah. Nice. That doesn't mean that the work that you did that day was all for naught, right? Like (unclear).

Kim  15:14

No! Oh, my gosh. We had the best conversations. Absolutely. And I think that's what happens, right? You go in sometimes with a goal in mind. Especially if you're an interventionist, or coach...

Pam  15:25

A guest teacher.

Kim  15:25

...or a leader. Yeah, in any of those kinds of situations, you're walking in, sometimes having a conversation with the teacher ahead of time and having different understandings of what it might look like to say, "My kids know, unit fractions are..." But then, sometimes, you know, you just make assumptions about a class, and you get to take it into a different place. So, that's why, you know, it's super important that you know the terrain and you know all the important bits of mathematics around the thing that you're doing. Because I was able to then say, "Okay, we're not going to do this kind of Over thing. We're going to now simply work on what is two-thirds as it relates to one-third? What is four-fifths as it relates to one-fifth.

Pam  16:09

Nice, nice.

Kim  16:11

Yeah.

Pam  16:11

Yeah, and it reminds me of your mantra, "know your content, know your kids." In that instance where you're a guest teacher, and you cannot know the kids. At least, you know, your goal is to get to know them as much as you can in the conversation.

Kim  16:23

Yeah.

Pam  16:23

Then, it's so important to know the terrain, so that you can shuck and jive with them as you get to know them. Yeah.

Kim  16:29

It definitely makes it more challenging.

Pam  16:30

Yeah, cool. Alright, Kim, let's end with a different Problem String that has a different sort of slant on this idea of an operator meaning of fractions. What does it mean to treat fractions kind of as operators? A little bit of a different slant. So.

Kim  16:45

Okay.

Pam  16:46

First problem. If I said one-fifth of a number, some random number. One-fifth of that number is 8. One-fifth of the number is 8. If that's true, what's two-fifths of that number? Maybe pause just a second, so people can think. If one-fifth of a number is 8, what's two-fifths of that same number?

Kim  17:09

That would be 16.

Pam  17:10

How do you know?

Kim  17:11

Because I'm just doubling the amount that you first asked about. So, I'm going from one-fifth to two-fifths, two 1/5s. So, 8 times 2 is 16.

Pam  17:23

So, you just didn't even figure out the number. You just sort of like (unclear).

Kim  17:26

Oh, was I supposed to figure out the number?

Pam  17:28

No. No, no. I'm just acknowledging. I'm acknowledging.

Kim  17:31

Okay.

Pam  17:31

That you're like, "Well, if one-fifth of whatever it is, is 8. And you're just asking me for 2 of those?"

Both Pam and Kim  17:36

Yeah.

Pam  17:36

"Well, then it's just 2 of 8 to 16." Is that kind of how you were thinking?

Kim  17:40

Yes.

Pam  17:40

Cool. What if I said, a fifth of a number. Could be different number, could be the same number. What if a fifth of a number is 8? What's four-fifths of that number?

Kim  17:50

We'll, it's going to be the same number. But I'm still going to scale. So from one-fifth to four-fifths is going to be times 4. So, 8 times 4 is 32.

Pam  18:02

Cool. I'm a little curious. The first problem we said two-fifths was 16. Could you scale from that?

Kim  18:07

Oh, yeah, yeah. Okay. Yep.

Pam  18:09

So, you could double the 16 to get two-fifths to four-fifths. Cool. If one-fifth of a number is 8, what's one-tenth of the number?

Kim  18:19

It's going to be 4 because a tenth is one-half of one-fifth. We know a fifth is 8.

Pam  18:26

We know a fifth is 8. And a tenth is a half of a fifth, then half of that 8 is 4. So, you're saying a tenth of the number is 4?

Kim  18:35

Yes.

Pam  18:36

Yes. Okay.

Kim  18:36

Sorry.

Pam  18:37

It's alright.

Kim  18:37

Pam  18:38

I just kind of said it backwards. Okay, so next question. If a fifth of the number is 8...same scenario...what's nine-tenths of the number?

Kim  18:48

Nine-tenths. Okay. So, you just asked me one-tenth, and that was 4. So, nine-tenths is going to be 36.

Pam  18:59

Because?

Kim  19:03

Because I could scale up times 9. But also I'm thinking about if one-tenth is 4, then ten-tenths would be 40. I think it's the first time that I thought about what. Because I wanted to go Over here. So, it was the first time I considered what the whole amount was, so that then I could back up from it.

Pam  19:21

And so, if the whole is 40.

Kim  19:24

Yeah.

Pam  19:24

One-tenth of that's 4.

Kim  19:27

Mmhm.

Pam  19:27

Nine-tenths is 4 back from 40.

Kim  19:30

Mmhm.

Pam  19:31

And that's another way of getting the 36.

Kim  19:33

Yeah.

Pam  19:33

Let's back up actually, if you don't mind, now that you think the whole is 40.

Kim  19:38

Yep.

Pam  19:39

I had asked you if one-fifth of a number is 8, what's two-fifths of the number? Does knowing that the whole is 40? Like finding two-fifths of 40? Does that make you?

Kim  19:50

No, I don't love that. I would still scale up.

Pam  19:52

Still want to just scale from the one-fifths to the two-fifths?

Kim  19:54

Yeah.

Pam  19:55

How about the four-fifths? When I said four-fifths?

Kim  19:58

No, I like scaling better.

Pam  20:00

Just scaling. Cool. I wonder if the four-fifths for me might have been. If I know five-fifths is 40.

Kim  20:08

Yeah.

Pam  20:08

And we started with one-fifth was 8, then I could back up the 8 from the 40 to get the 32. (unclear)

Kim  20:14

I just don't think I... Yeah, I think for whatever reason I didn't necessarily think of going five-fifths and back a fifth to get to four-fifths.

Pam  20:22

Yeah. But (unclear).

Kim  20:22

I don't know why. Yeah.

Pam  20:24

The nine-tenths kind of. Yeah.

Kim  20:25

Yeah, I don't know why.

Pam  20:25

And so is it bad or?

Kim  20:28

Pam  20:30

Or are we just are we just being flexible? We're just like, "What are we thinking about?"

Kim  20:34

Yeah.

Pam  20:34

And we kind of go from that. Alright, so the last question of the string is if one-fifth of a number is 8, what's the number?

Kim  20:41

So, yeah, 40.

Pam  20:42

And you already had found out that it was 40. Cool. So, that's different Problem String that kind of gets...

Kim  20:46

Yeah, I don't know that we've done one of those.

Pam  20:48

Yeah. I have a bunch of those in Lessons & Activities for Building Powerful Numeracy.

Kim  20:52

Oh, good to know.

Pam  20:53

Yeah, it's a different way of kind of working on both operator meaning and the scaling idea that if you've got one-fifth, then how can you find two-fifths? And if you got one-fifth, how can you find one-tenth? What's the relationship between a fifth and a tenth?

Kim  21:07

Yeah.

Pam  21:08

Yeah. Very nice. Cool.

Kim  21:09

Very nice. Alright, so you know, I wasn't going to share a review, but I think we have a little bit of time. So.

Pam  21:14

Oh, okay.

Kim  21:14

This is super short. And it says, "Great for experienced educators who love teaching math." That's the title. I like when the title is like also something fun.

Pam  21:25

That's cool. That's a great title. Yeah.

Kim  21:26

I can't even read the jumble of the letters, so I'm sorry.

Pam  21:32

You mean the person's handle is crazy.

Kim  21:33

Yes, sorry, sorry. But this says, "Awesome content presented in ways that inspire and clarify." I like the clarifying, right? So, I sure hope we've done that today. I hope everybody enjoyed listening in as we talk more about fractions.

Pam  21:49

Hey, and let me just maybe say thank you so much for the review, person with lots of letters and stuff in your name. It says, "Great for experienced educators who love teaching math." I mean, I think it's good...

Kim  22:00

For anybody.

Pam  22:01

...inexperienced educators too.

Kim  22:02

And not educators. We've had lots...

Pam  22:04

Oh, yeah.

Kim  22:04

Pam  22:05

(unclear) and friends.

Kim  22:06

For anybody. Math's for everybody.

Pam  22:07

But we sure appreciate the five stars in the review. That helps other people find the podcast because we are trying to spread that word. Ya'll, 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. Let's keep spreading the word that Math is Figure-Out-Able!