Ep 188: The Whole Matters

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!