January 23, 2024
Pam Harris

Ep 188: Fractions - The Whole Matters

Math is Figure-Out-Able!

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Math is Figure-Out-Able!

Ep 188: Fractions - The Whole Matters

Jan 23, 2024

Pam Harris

Does it really matter what we call the whole? In this episode Pam and Kim take a closer look at fraction problems where the unit matters.

Talking Points:

- Kim and the beach
- How can there be 2 possible answers to the question "What is 1/2?"
- Focusing on the unit
- Naming the unit
- What happens when the unit changes?
- Help students focus on the unit, changing the unit, and finding a fractional amount of that new unit
- Even test writers struggle with fractions
- Re-unitizing will support students to reason through fraction multiplication

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

Listen on

Share Episode

Does it really matter what we call the whole? In this episode Pam and Kim take a closer look at fraction problems where the unit matters.

Talking Points:

- Kim and the beach
- How can there be 2 possible answers to the question "What is 1/2?"
- Focusing on the unit
- Naming the unit
- What happens when the unit changes?
- Help students focus on the unit, changing the unit, and finding a fractional amount of that new unit
- Even test writers struggle with fractions
- Re-unitizing will support students to reason through fraction multiplication

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 **00:07

And I'm Kim.

**Pam **00:08

And you found a place where math is not about memorizing and mimicking, where you're waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We know we can mentor students to think and reasons like... Reasons? To think and reason like mathematicians did when they were students. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. Because, ya'll, it kept me from being the mathematician I could be. And now, I am a better one than ever before. Whoo!

**Kim **00:44

I have another review, but this one's mostly because I aspire to be this person. The title says "Retired and still a fan". Nice! I want to be retired.

**Pam **00:54

You want to be retired? You're younger than I am.

**Kim **00:55

Way too early. It's way too early. I want to retire and lay on a beach. You know, I never used to love beaches. (unclear).

**Pam **01:02

I was going to say. You were not a beach person when I met you.

**Kim **01:05

Um, yeah. Yeah, not at all. Neither am I. Because I'm mountains. Like, climbing and hiking and...

**Pam **01:11

It's because you can't sit still. But it sounds like you now maybe are more chill than you have been.

**Kim **01:15

I just like to sit there. Yeah. Okay. Anyway. So, SquirrelGirl.

**Pam **01:20

SquirrelGirl.

**Kim **01:21

Yeah.

**Pam **01:21

I like it.

**Kim **01:21

The names are my favorite.

**Pam **01:23

That is awesome.

**Kim **01:24

So, she says I love listening to both of you, even though I'm no longer teaching.

**Pam **01:28

Huh.

**Kim **01:29

Interesting, right? Will I listen to podcasts when I'm done?

**Pam **01:32

Yes. Yes, you will.

**Kim **01:33

Okay. "(unclear) learning through you. Some people think I'm crazy that I still listen to a teaching podcast. But some habits are really hard to break and you two are one of those habits." Are we bad habits?

**Pam **01:45

No! We're awesome habits. That's fun.

**Kim **01:48

That's super fun. Okay.

**Pam **01:48

Thanks, SquirrelGirl. That's fun. I like being a good habit. Okay. I've never been called a good habit before.

**Kim **01:55

Well, actually.

**Pam **01:56

Oh.

**Kim **01:56

Fractions. You just said fun. Fractions are fun.

**Pam **01:58

Fractions are fun.

**Kim **01:59

That's what we're going to talk about.

**Pam **02:00

Fractions are our friends.

**Kim **02:01

Yeah. Some people find them troubling, right? And we think it's because of some of these really important things that we're chatting about in this short series that often get overlooked or maybe misunderstood.

**Pam **02:14

Well, or didn't even know they exist, right? That was me. I mean, I... Yeah, I didn't. I did fractions well. Quote, unquote, "did well". Which meant, I memorized all the rules, and I knew when to apply them.

**Kim **02:24

Yeah.

**Pam **02:24

But reasoning? Yeah, no.

**Both Pam and Kim **02:27

Yeah.

**Pam **02:27

Some of the funniest things I did was while I was writing Lessons & Activities for Building Powerful Numeracy. There's a whole section in there on fractions. You should check it out. It's pretty good. While I was developing the materials for that, I would ask my personal kids. I would say, "How do you think about this problem?" And they would blow my mind. I'd be like, "Wait, say that again? Like, half what?"

**Kim **02:46

Yeah.

**Pam **02:46

Yeah, it was awesome. And then, I would call you, and you would usually... You and Craig were usually on the same wavelength. (unclear).

**Kim **02:51

That's because I like Craig so much.

**Pam **02:53

I mean, he's pretty likable. Yeah.

**Kim **02:55

Was it because he was in my class? Probably not.

**Pam **02:58

(unclear).

**Kim **02:58

Definitely not. Craig is super sharp.

**Pam **03:00

So was Matt. I don't know.

**Kim **03:01

Yeah.

**Pam **03:01

They both had you. Yeah, yeah.

**Kim **03:02

Well, it was true. It was earlier. It was earlier in my numeracy when I had Matt. Not earlier in my numeracy, but my teaching.

**Pam **03:10

Which is different, right? Like, you might you might do some things in your head... Not in your...

**Kim **03:16

Right.

**Pam **03:16

It's not all in your head.

**Kim **03:16

Yes, I didn't model. I didn't (unclear).

**Pam **03:18

Yeah. And, listeners, you might be like, "Yeah, of course, I think that way," but have not really had a way to help your students develop the same kind of thinking. And we're so excited to be able to do that. Teach all of us to think and to teach it. Here we go. Let's do some of that today. So, one of those very important things that is often not emphasized or taught at all, maybe because we don't know it or we don't realize the importance of it, is how important the unit is. "What do you mean, Pam? What do you mean the unit?

**Kim **03:49

Can I interrupt for a second?

**Pam **03:50

Of course.

**Kim **03:50

Because I think people might think it's important, but they don't know the kinds of things that they might be saying or doing that...

**Pam **03:59

We might not recognize things that we're saying or doing that are...

**Kim **04:02

Yeah, that are...

**Pam **04:03

...less helpful.

**Kim **04:03

Yes. Yes, yes, yes.

**Pam **04:05

Okay. Okay.

**Kim **04:06

Maybe,

**Pam **04:07

And, you know, and I don't know that we can see that all the time until somebody helps us kind of see our perspective. I know for sure it's been really helpful... Many of you know I'm a grandma now. Whoo!

**Kim **04:19

Yay!

**Pam **04:19

And I have a dear friend who has heard me give some advice to my daughter-in-law, and then tackled me and said, "Are you kidding me?! Shut your mouth!" And so, you know, you can't always see outside yourself, until someone kind of mirrors, "Hey, this is what you just said," and I'm like, "Oh! No, I didn't really sound like I was that judgmental, did I?!" And they're like, "Yes." I'm like, "Oh, let me take that back. No, you're perfect in every way daughter-in-law!" And she really is. And he's so good. And they're so amazing. Yes. So, my point is sometimes you can't always see what you're doing until maybe you hear it from a different perspective. So, let's see if we can do some of that today.

**Kim **04:57

Yep.

**Pam **04:58

Alright, so what do I mean that the unit is important when we're talking about fractions? Well, Kim, if I were to say to you, hey, here are 2 lovely candy bars. Which, see, that would be for me. For you, it would be, what? Like, two boxes of gummies?

**Kim **05:12

2 pizzas. Whole pizzas.

**Pam **05:14

Pizzas? No! We're talking about... Stay rectangular.

**Kim **05:17

Oh. You can get square pizza.

**Pam **05:19

Oh, Kim.

**Kim **05:20

Okay, fine. (unclear).

**Pam **05:21

I was trying to do the candy thing.

**Kim **05:23

Can I have a sandwich?

**Pam **05:24

If I'm doing the candy thing, then I would do candy bars, and you would do? What's a rectangular.

**Kim **05:30

Oh, actually, I just bought a candy bar. Which is so rare. It was a toffee, pretzels, sea salt.

**Pam **05:37

(laughs).

**Kim **05:38

Oh, I mean.

**Pam **05:39

And you liked it?

**Kim **05:41

Well, mostly I ate one piece and my family ate the rest. But yes, I loved it. So good.

**Pam **05:45

Oh, nice. So, we're just joking a little bit because Kim typically likes gummy.

**Kim **05:50

Sour.

**Pam **05:50

Sour, salty stuff

**Kim **05:54

I also bought peppermint toffee.

**Pam **05:56

Ew.

**Kim **05:56

It was so good. Mmm.

**Pam **05:57

No! Never!

**Kim **05:59

Anyway. Candy bar.

**Pam **06:00

I can maybe do dark chocolate...

**Kim **06:02

No!

**Pam **06:02

...peppermint. But only a little bit. Yeah, no, no, no. Okay, anyway. Kim, I have two candy bars. Okay. What if I asked you what is one-fourth?

**Kim **06:13

I'm going to go with half a candy bar (unclear).

**Pam **06:15

Okay, why?

**Kim **06:16

Because you said you have two of them. Two candy bars. And a fourth of that amount of original candy bar is one-half a candy bar.

**Pam **06:25

So, if I said what's one-fourth, you inferred that I meant one-fourth of 2 candy bars. One-fourth of all of it. Everything that I have sitting there?

**Kim **06:32

All of it. Correct.

**Pam **06:35

And if that was true, I met one-fourth of all of it, then a half of one of those candy bars would be a fourth of the whole shebang.

**Kim **06:42

Yeah. So, a fourth is a half. They're equal.

**Pam **06:45

Ah! Nice!

**Kim **06:46

No!

**Pam **06:47

No! So, one-fourth of 2 candy bars is one-half of a candy bar.

**Kim **06:54

Correct.

**Pam **06:55

And the way I just said that was trying to emphasize what the unit was every time I said a fraction.

**Kim **06:59

Yes.

**Pam **07:00

So, what if I had those two candy bars, but I said, hey, Kim, what's one-fourth of a candy bar? Even though I have 2 sitting here, I only like to have one-half of a candy bar. What would you say?

**Kim **07:08

One-fourth of a candy bar?

**Pam **07:09

Yeah.

**Kim **07:10

You just said half. Okay.

**Pam **07:11

Oh, did I?

**Kim **07:12

Yeah, you did.

**Pam **07:13

Oh, for Heaven sakes.

**Kim **07:13

It's okay. A fourth of a candy bar is if I were to ignore the second candy bar and split the first candy bar into 4 piece, and it would just be one portion of the fourth. It would be a fourth of 1 candy bar.

**Pam **07:26

So, one-fourth of a candy bar.

**Kim **07:27

Yep.

**Pam **07:28

So, I could literally ask you the same question. What's a fourth?

**Kim **07:32

Yep.

**Pam **07:32

And you could have said it's a half a candy bar. You could have said it's a fourth of a candy bar. And both of them would have been correct, depending on what you were thinking of as the unit.

**Kim **07:42

Correct.

**Pam **07:42

Can we agree on that?

**Kim **07:43

Yeah.

**Pam **07:44

So, that's tricky. There's two possible answers, depending on what you were focused on as the unit. And you might say, "Psh. Pam, obviously, there were 2 candy bars sitting there. Obviously, the kid's going to know we meant all of it." And we're going to say, "Um, maybe not." Like, let's actually just be careful, be purposeful, about identifying the unit.

**Kim **08:06

Sure.

**Pam **08:07

Alright, so here's another example. What if I showed you 6 dots? And they're not really arranged any way. There's just 6 of them? Can you picture 6 dots

**Kim **08:17

I actually just drew. Is that okay?

**Pam **08:19

Yeah, absolutely.

**Kim **08:19

Drew circles.

**Pam **08:20

Okay. So, Kim, what's one-half?

**Kim **08:25

3 dots is one-half,

**Pam **08:27

Okay. Of?

**Kim **08:30

The entire set of 6. So 3 is half of 6.

**Pam **08:33

So, if I mean the entire set of 6 dots, then you would say 3 dots is a half of the 6 dots. Cool. Can you think of... If I were just to have those 6 dots sitting there, can you think of another unit you could focus on, and that you could find half of that?

**Kim **08:48

Yeah, I can have one whole dot is half of 2 of the dots.

**Pam **08:54

Whoa. So it's almost like... Did you ever eat Twinkies as a kid? Not Twinkies. Ding Dongs. Did you ever eat Ding Dongs?

**Kim **09:00

No! I ate Twinkies.

**Pam **09:01

Well, okay. Can you make sure Ding Dongs? You know the ones?

**Kim **09:04

No, I don't know which ones. Is that (unclear).

**Pam **09:06

(unclear) Now, I'm not sure. Well, I'm thinking they're the chocolate ones. The chocolate cupcake things.

**Kim **09:11

Yeah.

**Pam **09:11

Okay, so can you picture those? Came 2 to a pack.

**Kim **09:15

Mmhm.

**Pam **09:16

So, I'm kind of picturing that when you said, "Well, if there's 2..."

**Kim **09:19

2 pack.

**Pam **09:20

Yeah, it's a 2 pack. So, if it's a 2 pack of those chocolate frosted things, and I said, "What's one-half, then you said, "Well, you're just going to have one of them." Is that right?"

**Kim **09:32

Mmhm.

**Pam **09:33

So, if we had these 6 frosted things here. But you're picturing them in 2 packs. And I said, "What's a half?" You could literally say it's just one of the frosted things."

**Kim **09:43

You totally making me think about if I am as good as we're going to suggest people need to be. Because, like, what if something came in a box of 6 and my kid said, "Can I have half?" And I assume like a half of a doughnut. And he's like, "No, 3 of the donuts."

**Pam **10:00

Well, so you just said... Okay, so I have written down on my paper that you could have found half of a 2 pack. That was the last one you did.

**Kim **10:07

Yeah.

**Pam **10:07

You could have found half of the whole thing, which is 1 out of the.... Sorry, which is 3 out of the 6. I wrote that down wrong. 3 out of 6. But then, you just said... Yeah, the 1 donut, right? If you had 6 doughnuts sitting there, and the kid said, "Can I have a half?" you're thinking in your head that they would do what?

**Kim **10:24

Eat one-half of 1 donut.

**Pam **10:27

Like, there's 6 of them sitting there. But surely the kid didn't mean 3 of the 6. (unclear)

**Kim **10:32

My kid did.

**Pam **10:34

(laughs). But they could have been focused on just 1 of the doughnuts, and you're just going to have half of that doughnut. So, we have sort of 3 correct answers, even though there is 6 sitting there, depending on what you're focusing on as the unit.

**Kim **10:46

Mmhm.

**Pam **10:46

That's kind of important that we need to be able to help kids, we need to do things with them, and let that unit change. So, actually throw up 6. Throw up. That sounds terrible.

**Kim **10:57

Oh.

**Pam **10:57

I know, sorry.

**Kim **10:58

(unclear).

**Both Pam and Kim **10:58

(laughs).

**Pam **11:00

Put up 6 doughnuts on the board.

**Kim **11:03

Sure.

**Pam **11:04

And then, ask them if I'm talking about all of them, what's one-half? What if I'm talking about this 2 pack, what's a half? What if we had a 3 pack? Kim, what if we had a 3 pack?

**Kim **11:14

Of doughnuts?

**Pam **11:15

Yeah.

**Kim **11:15

Half of it would be 1 and 1/2 doughnuts

**Pam **11:18

1 and a 1/2 of a donut, right? So, notice how even in the answer we're referring to the unit.

**Kim **11:24

Yeah.

**Pam **11:24

One-half of the 3 donut pack.

**Kim **11:27

Oh, and I said 1 and a 1/2 doughnuts because I was thinking about you would have more than 1 donut, so I call it donuts.

**Pam **11:34

Mmm. Mmhm. (unclear).

**Kim **11:35

But you said 1 and a 1/2 of a donut.

**Pam **11:38

Yeah, yeah.

**Kim **11:39

Okay.

**Pam **11:40

So, different ways that we're going to kind of make sense of fractions by letting that unit change. So, we're not only suggesting that we need to be purposeful about naming the unit, but we need to actually do exercises with students where we have multiple things, and we change the unit and discuss what the fractional part is as that unit changes. "What if this is the unit?" "What if that's the unit?" "What if this is the unit?" And let kids sort of... They almost have to kind of close one eye and tip their head a little bit.

**Kim **12:10

Yeah.

**Pam **12:11

It's almost like an optical illusion where you could see you the one, and then see the other, and you have to kind of zoom in and out a little bit. Does that makes sense when I zoom in and out? Sort of zoom in where 1 donut is the unit. Zoom out where all 6 is the unit. Zoom in not quite as much where we had a 2 pack or out a little bit where we had a 3 pack. All of those different units, then, can help kids re-unitize. And that is a skill that we need. It's a path we want their brains to travel often. We want their brains to travel that path of thinking, "What is the unit? Ooh, now that I know what the unit is, let me think about the fractional part of that unit. Ooh, let me shift. Now, what if this other thing, bigger, smaller, is the unit? Now, let me find the fractional part of that unit." It's super important.

**Kim **13:00

Yeah, people ask all the time like, "What do you do with, you know, high fliers?" Or whatever. "How do we extend kids?" And I'm feeling like this would be a really nice...

**Pam **13:10

We both just took a breath, by the way, because we don't like to label kids. Keep going.

**Kim **13:13

I air quoted with my fingers (unclear) can't see. But I'm telling you, this would be a super fun like warm up as you're coming into school, as you're transitioning classes, whatever, to have some images on like on the on the wall or whatever. And it's literally like what's one-half? What's one-third? And, you know, it's something open enough that many people can enter in. Everybody can enter in with an answer. But for kids who are not... Not where we think they need extension, but who are actually interested in thinking. Like, I think Luke would super love that. Where he gets to stare and think about different ways to re-unitize. I'm going to give you one.

**Pam **13:53

Okay. Go.

**Kim **13:55

Okay, you might have to draw

**Pam **13:57

I got my pen.

**Kim **13:58

Okay.

**Pam **13:58

Pen. Pen. Pen. Not a pencil.

**Kim **14:00

You've got fruit. So, you got 3 strawberries Draw fast.

**Pam **14:06

Strawberries?

**Kim **14:07

Yep.

**Pam **14:07

How do you even draw a strawberry? Okay.

**Kim **14:09

You could put a circle with a S. Quick draw, people. Quick draw.

**Pam **14:13

I have 3 squirrely looking triangle things on my paper. Okay.

**Kim **14:16

If you were in my class, I would have already moved on because we quick draw. We're not drawing the actual thing. We got an apple. Circle with an A. We got an orange.

**Pam **14:26

Okay.

**Kim **14:27

And you get a bunch of 5 bananas. So, bunch of them together.

**Pam **14:31

5 of them?

**Kim **14:32

Yep.

**Pam **14:33

Okay. Alright. I literally have fruity looking things on my paper just so you know.

**Kim **14:37

That is supposed to be fruit, so that's good.

**Pam **14:39

Okay.

**Kim **14:39

Okay, now what's a half?

**Pam **14:43

Okay. I'm going to say if my unit is the apple, then it's a half an apple.

**Kim **14:51

Okay.

**Pam **14:51

If the unit is the strawberries, so it's a half of the strawberries, then it's 1 and a 1/2 strawberries.

**Kim **14:59

Okay.

**Pam **15:00

Is a half of the 3 strawberries. If it's of the round fruit because I have an orange and an apple, then it would be 1 of those fruits, would be half of those 2 round fruits. If it's the 5 bananas, then it would be 2 and a 1/2 bananas, would be half of the 5 bananas. If it is the 10 total fruits. Right because I have 10 total fruits?

**Kim **15:27

Mmhm.

**Pam **15:28

Then, I'm going to do the easy thing and say it's the bananas because there's 5 of them. So, that 5 of the bananas would be half of the 10 fruits. However, I could cut them all up and make a fruit salad.

**Kim **15:40

There you go!

**Pam **15:41

And I could divvy up half of the fruit salad, weight it out, and give you half of all of the fruit cut up. And that would be half of the fruit salad. What do you think?

**Kim **15:50

Yeah, I like.

**Pam **15:51

Lots of different ways.

**Kim **15:52

What about if we thought about an individual type of fruit? We have 4 kinds of fruits.

**Pam **15:58

Mmm.

**Kim **15:59

Half of the kind of fruit

**Pam **16:01

Half of the kinds of fruits. So, I'd have 2 kinds of. It's almost like you said to make your fruit salad, you can have half of the kinds of fruits. And I would say, mmm, okay, I'm going to have the strawberries and bananas, please.

**Kim **16:15

Okay. There you go. Make you a smoothie.

**Pam **16:17

2 of the two of the kinds of fruits. When I was at the NIH, they had a strawberry banana smoothie that was super yummy. And the beginning of the week, I had to have lots of calcium, so I was like, "Sweet!" So, every meal, I was having a strawberry banana smoothie. It was awesome. And then, they were like, "No, way to much calcium." And so, then, they cut me down to none, and then I had no more.

**Kim **16:34

Aw. Sadness.

**Pam **16:36

(unclear).

**Kim **16:36

Moderation, moderation. I don't really love bananas in a smoothie. It's overpowering. Kind of (unclear).

**Pam **16:41

You know, these were bad. There was just a little bit of banana. Yeah, I'm kind of actually with you on that. You know what I really can't take? Bananas and chocolate. No! No! No! No! Like, if anybody ever wants to.

**Kim **16:50

I have frozen banana chocolate pieces in my freezer.

**Pam **16:54

Oh! Bleh!

**Kim **16:55

So good!

**Pam **16:56

Bleh! Bleh! No, not my favorite. Okay, Kim, I get another one for you.

**Kim **17:00

Yeah.

**Pam **17:01

Picture a rectangle.

**Kim **17:03

Okay.

**Pam **17:04

And it's kind of a... How do I even describe it? It's a horizontal oriented rectangle. So, it's like, wider than is tall.

**Kim **17:13

Okay.

**Pam **17:13

Does that make sense?

**Kim **17:14

Mmhm.

**Pam **17:14

And cut that rectangle into 7 equal sections. So, it's kind of like a candy bar that's got 7 sections.

**Kim **17:24

Okay.

**Pam **17:25

Did you draw vertical lines?

**Kim **17:27

I did.

**Pam **17:28

Okay, cool. And 5 of those sections... This is terrible that I just can't show it to you, and I have to tell you all about it. But 5 of those sections are shaded.

**Kim **17:37

Okay.

**Pam **17:37

So, let's see if we can do that again. (unclear)

**Kim **17:40

(unclear). Like I'm picturing.

**Pam **17:41

Mmhm.

**Kim **17:41

Like, if we were making an area model, I'm picturing like a 2 by 7.

**Pam **17:48

Okay. Mmhm.

**Kim **17:51

And then, I cut horizontally 6 cuts to make 7 pieces.

**Pam **17:57

Vertical lines when you say cut horizontally. You drew vertical lines.

**Kim **18:00

Yeah, yeah, yeah.

**Pam **18:01

Yeah. And 5 of those are shaded, right?

**Kim **18:03

Correct.

**Pam **18:04

Okay, cool. So, with that image. So, hopefully everybody can kind of see that image. With that image, can you see five-sevenths? The fraction five-sevenths. If you can, what's the unit? Five-sevenths of what?

**Kim **18:20

The area model. Candy bar.

**Pam **18:22

The whole thing.

**Kim **18:23

Is the unit. And I have 5 shaded of the 7 pieces. So, that's five-sevenths of the candy bar.

**Pam **18:30

Yeah, so it's almost too easy because I had to tell you how to draw it, right? So, cut into 7. You're all, "Duh, Pam. It's already sevenths. So, we shaded 5 of them. This isn't hard."

**Kim **18:39

Yep.

**Pam **18:39

Five-sevenths of the whole thing is shaded. Okay, cool. Can you see... Same picture. Same picture. Same picture. Can you see seven-fifths?

**Kim **18:51

Seven-fifths.

**Pam **18:55

Of something. Seven-fifths of something.

**Kim **18:57

Someone's screaming at me right now. "Kim!"

**Pam **19:01

"Kim it's right there!"

**Kim **19:03

Mmhm, mmhm.

**Pam **19:03

So, close one eye and tip your head to the side.

**Kim **19:06

Old lady, young lady.

**Pam **19:07

Yeah.

**Kim **19:07

Okay, so I'm thinking seven-fifths means I need more than the whole, so I'm going to call the 5 shaded pieces the whole because then I'd have 1. Just five-fifths. And then, I have the 2 leftover as 2 more fifths. And five-fifths and two-fifths make my seven-fifths. So, the whole is the 5 shaded pieces.

**Pam **19:39

The unit is the 5 shaded pieces.

**Kim **19:41

Mmhm.

**Pam **19:42

So, it's almost like you've got?

**Kim **19:45

It's almost like if they were separated it might be easier to see, but because they're connected, the five-fifths and the two-fifths are connected, that made it a little bit harder for me to see.

**Pam **19:58

So, if I thought about the five-fifths, the shaded part, as the whole candy bar, it's almost kind of like a had 2 candy bars sitting... No, I don't know how. I'm trying to think of a actual scenario.

**Kim **20:09

It would be like I had another five-fifths candy bar next to it, and I chopped off the three-fifths and got rid of them. So, I really only had five-fifths, which is the whole, and then two-fifths.

**Pam **20:20

Of a different candy bar.

**Kim **20:21

Of a different candy bar. But only considered the one candy bar. It's like my kid tried to sneak in those 2 pieces and said, "No."

**Pam **20:31

You got a whole candy bar cut into 5 sections. That's the whole. And you've got these 2 extras. Yay! Bonus! It's like seven-fifths. The whole five-fifths and 2 more fifths. So, the same picture can represent five-sevenths of the entire rectangle that we drew. And with shaded 5 out of those 7 equal parts. But if we focus in, zoom in, on just the five-fifths, just the 5 shaded parts and call that the whole, then we have extra. We've got two-fifths extra in that, and we could call that 1 and 2/5 or seven-fifths.

**Kim **21:05

Mmhm.

**Pam **21:06

Alright, so hopefully everybody... We have no idea how this is going to work. Like, let us know when you hear us describe stuff like this if you're like, "What are you even..." Like, it would be helpful for us to get some feedback from you, podcast listeners, if this works. You're like, "No, no, no. It's good enough. Like, you described it well enough." We're over describing in the hopes that you can kind of see what what we're suggesting. We would never do this out loud with students. With students, we would show these images. Yeah. We wouldn't say, "Hey, I'm going to describe it, see how well you can transcribe my..." No. None of that. So, how are we helping build students ideas of fractions? Well, in one way, we are really helping them focus on the unit, and then changing the unit and helping them find a fractional amount of that new unit. So, it's like we identify a unit and find a fractional amount, change the unit, find that same fractional amount, change the unit again. And sort of in and out, so kids brains get used to traveling that road of like, "Wait, wait. If this is the unit, then what's the fractional amount?" And that can be super, super helpful for kids. Okay, so, Kim.

**Kim **22:24

Yep.

**Pam **22:24

There's a poster that we've dealt with.

**Kim **22:26

Yeah.

**Pam **22:28

Do you want to describe it or me?

**Kim **22:29

Sure. I'll give it a go, and you can tell me if I'm... I do shortcut describe, and you often slow me down.

**Pam **22:35

Let's see how you do.

**Kim **22:37

Re-say what I said.

**Pam **22:37

So, this is another example of where we would just show you this poster.

**Kim **22:40

Absolutely.

**Pam **22:41

And hey, we should probably say this comes from Susan Lemon's wonderful book called Teaching Fractions and Ratios for Understanding.

**Kim **22:49

Yep.

**Pam **22:49

I'd love to meet Susan Lemon someday. She's brilliant. And she has this poster in her book. And we've used it in professional development before. And go ahead, Kim.

**Kim **22:57

Okay.

**Pam **22:58

Okay.

**Kim **22:58

So, picture a poster. And there are three columns, three pictures. So, I'm going to describe each one. So, on the left side, there are two squares sitting on top of each other. So, I'm picturing almost like a block. Like of kids playing with blocks. And there are two blocks on top of each other.

**Pam **23:17

Okay.

**Kim **23:18

And the bottom one is shaded, and the top one is not shaded. So, two on top of each other. Bottom's shaded. Top not. And it says one-half below it.

**Pam **23:30

And that makes sense. Because you got two blocks, one shaded. So, 1 out of 2, one-half.

**Kim **23:35

Yep.

**Pam **23:35

Clear. Okay.

**Kim **23:37

Next to that, to the right, there are the same squares, 3 on top of each other.

**Pam **23:43

Same size squares. Not the exact same squares. Same squares. Mmhm.

**Kim **23:47

The bottom one is shaded. The one above it and the one above that. So, the second and third one are not shaded. And that column is labeled as one-third.

**Pam **24:01

And how many did you say were shaded?

**Kim **24:03

1 is shaded out of the 3 squares.

**Pam **24:06

So, that makes sense. 1 out of the 3 is shaded, and so that column is labeled as one-third.

**Kim **24:10

Yep.

**Pam **24:11

Okay.

**Kim **24:11

And then, the final column is very similar. 4 squares on top of each other. The bottom one is shaded, and the other three, the second, third, and fourth one are not shaded. So, overall, when you look at it, the bottom squares are shaded, but all the ones above are not shaded.

**Pam **24:31

And I if I remember correctly, I think is that actually...

**Kim **24:33

That final one, sorry, was called one-fourth.

**Pam **24:36

And I think there was a column to the right of that, that was one-fifth. It was the same idea. It was five squares and one shaded on the bottom. So, let me describe the poster in maybe a slightly different way. You described it brilliantly. Along the bottom we've got one-half one-third, one-fourth, one-fifth. The fractions written out. The numeral. The numerals written in fraction notation. And then, we have the same size square across the bottom is shaded. In each column, one square is shaded. Same size square. And then, above that is a bunch of unshaded. And so, in the one-half, there's one unshaded. In the one-third, there's two unshaded. In the 1/4, there's three unshaded. And in the one-fifth, there's four unshaded. Is that right?

**Kim **25:26

Yeah.

**Pam **25:26

Yeah. Okay, so Susan Lemon and we asked the question, what do you think about this poster? So, I kind of hope that you've drawn it a little bit, so that you can maybe talk about what you think about the poster. Maybe you're visioning it. But yeah. Kim, we're always a little surprised when...

**Kim **25:48

Yes.

**Pam **25:49

Especially...

**Kim **25:51

Very surprised

**Pam **25:51

...maybe the last time that we put this up in front of a group of teachers. Teachers said, "Oh, this is fantastic because kids really get a feel for what these fractions mean. And it will be up on the board. And they'll be able to..."

**Kim **26:02

"I wish I had one of these."

**Pam **26:04

Yeah.

**Kim **26:05

A few people said.

**Pam **26:06

Like, "Create one with Math is Figure-Out-Able on the bottom, and we'd buy it. We could stick it on our wall." And, Kim, why does Susan Lemon put it in her book and we put it up for teachers to consider. What what amiss?

**Kim **26:21

Mmm. So, I think what the point is to cause a little discourse and conversation because if any of these columns were in isolation, the one-half, or the one-third, or the one-fourth, it would be a nice representation of that fraction.

**Pam **26:38

Mmhm.

**Kim **26:38

But as soon as you put them together on a poster, they are no longer accurate representations of those fractions. (unclear).

**Pam **26:49

At least if they're referring to the same whole.

**Kim **26:52

Right.

**Pam **26:53

So, like, the one-half is 1 out of 2 equal pieces, but then right next to it, we have a different unit. Now, there's 3 of the same size squares. It's not just that they're the same size squares for the 1 out of 3. They're the same size squares as the 1 out of 2. So, it's as if I said to Kim, "Here's a candy bar with 3 pieces. And I've shaded this one. You get one-third." Now, here's... I'm not even sure. How do I say that? Here's two-thirds of that candy bar? Like, that might not be the best representation. Let me think for just a second. From the one-half to the one-fourth. The one-half has two squares, one of them shaded. The one-fourth has four squares, one of them shaded. It's as if I gave Kim the candy bar with the four squares and said, "Hey, you get a fourth." And then I said, "Now here's a different candy bar, and you get a half." The point is it's a whole different candy bar. And that candy bar is is much smaller. The one that is labeled one-half is much smaller than the one that was labeled one-fourth. I don't know that I'm saying this very well. It's almost like I said, "Which candy bar would you prefer? Do you want the biggest candy bar there that had the five squares, and you get a fifth? Or do you want the littlest candy bar there that had two squares. Do you want a half?" And that's okay, as long as we're super clear that it's different candy bars we're talking about. But the poster does not make that clear. The poster sort of seems like it doesn't matter what the unit is. And we're saying it does. Well, it's not just us. Unit matters.

**Kim **28:34

Yeah.

**Pam **28:35

Yeah. Got anything else to say about that one?

**Kim **28:39

I think that it's a bit of a challenge. Because if you have a fourth, one-fourth, and you're also saying that two-fourths is equivalent to a half, then if you get two-fourths in that column, you're eating more than one-half in the first column.

**Pam **28:58

Oh, that's nicely said. Yeah.

**Kim **29:01

So, I want the two-fourths not the one-half because I'm going to end up eating more chocolate.

**Pam **29:08

Yeah, nice. That was probably super helpful. Cool. Okay, so what? Like, you might be like, "Pam, really does this matter?" Well, let me give you an example that we can't find any more. I wish I would have taken a picture of it. I'm sure it's somewhere in the ether of the internet. But there was a high stakes test question in a state which shall not be named, where they gave the students two pizza looking things, two circles, that were cut into some equal sized pieces. And I don't even remember what they were cut into. And then, the question asked them either to shade in or what fraction is represented. It must have been what fraction was represented. So, they said what fraction is represented? And here's the problem. It was a multiple guess question, and the answer was there if the unit was one pizza and the answer was there if the unit was both pizzas. And nowhere did the question say, "Of both pizzas, what unit is shown? What unit is shaded? Or what fraction shaded?" is probably what it said. It didn't say that. It didn't identify what the unit was. So, basically there were two correct answers.

**Kim **30:19

Yeah.

**Pam **30:19

That got taken off and taken down. And we can't find it anymore. Rightly so because you have to identify the unit if you're going to be able to correctly identify the fractional parts.

**Kim **30:29

Yeah. It's like the test writers made assumptions that they wanted the students to also make assumptions about.

**Pam **30:36

Yeah. And depending on which assumption you made, you could have gotten either of those answers correct. Mmhm.

**Kim **30:40

Yeah. Yeah.

**Pam **30:41

Yeah.

**Kim **30:41

So, re-unitizing is super important in fraction multiplication and proportional reasoning problems, too. We haven't really talked about that quite a bit. So, if I were to say a half of two-thirds of something. It's like if I asked you what's a half of two-thirds, we're also assuming that people understand that the two-thirds is of something.

**Pam **31:07

Mmm. Mmhm.

**Kim **31:08

So, if we write down one-half times two-thirds, and we think about it as one-half of two-thirds, then the first thing you might think about is two-thirds of what?

**Pam **31:18

Absolutely.

**Kim **31:20

And if we don't help students, we don't do work to help them think about the unit, they could be thinking about... They need to be thinking about two 1/3s as 1/3 is the unit, and we need 2 of them.

**Pam **31:32

Whoo! And you just did so much re-unitizing in that one little bit.

**Kim **31:38

Yeah.

**Pam **31:38

We're finding a half of something. There's a unit. We're finding a half of two-thirds. Whoa, that's two of something.

**Kim **31:48

(unclear). Mmhm.

**Pam **31:48

Yeah. That's one-thirds. And that's one-third of something.

**Kim **31:51

Right.

**Pam **31:52

Yeah, so lots and lots of re-unitizing that students need to be able to do in order to reason through fraction multiplication. We will do more work on that soon. Bam! 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. And thank you for spreading the word that Math is Figure-Out-Able!

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