January 30, 2024
Pam Harris
Episode 189

Ep 189: The Operator Meaning of Fractions

Math is Figure-Out-Able!

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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

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

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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

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

**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

Sorry, go ahead.

**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

I had to think about what you were asking me.

**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

Horrible. I'm bad. I'm horrible.

**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

Of moms and dads,

**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!

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