Pam  0:02  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:11  
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching,

Pam  0:19  
Because we know that algorithms are super cool achievements. But, y'all, they're terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems could be developing.

Kim  0:32  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships. 

Pam  0:38  
We invite you to join us to make math more figure-out-able.

Kim  0:43  
Hey. I'm just going to tell you what happened. 

Pam  0:45  
Oh, no. What happened? 

Kim  0:47  
You pushed the button to record. 

Pam  0:48  
I know to record, sorry.

Kim  0:49  
No, it's okay. And I started to tell you because I didn't realize you had pushed the button. My phone is on Do Not Disturb, but texts are getting through. So, then I go, "What's happening? What's going on?"

Pam  0:49  
Oh, it's your latest update. Remember, you told me that you did the latest update and you're not happy.

Kim  0:54  
Don't do it. Don't do it. 

Pam  1:04  
Yeah. Well, so I got the opportunity, and I was like, "Nope. Kim said, don't."

Kim  1:07  
I mean, you're going to have to at some point, but.

Pam  1:09  
I know.

Kim  1:10  
Not a fan. All right.

Pam  1:12  
I like to let it come out, and then let everybody work out all the bugs. And then once the  bugs are worked out, then I'll

Pam and Kim  1:19  
Dive in.

Pam  1:19  
Okay, cool

Kim  1:20  
Good plan. Alright. So, hey, we have had quite a few episodes where we haven't done a lot of math. Let's do some math today. 

Pam  1:27  
What?!

Kim  1:28  
I know, I know.

Pam  1:29  
Yeah, it's one of my favorite things to do is doing some math. Alright, cool. So, Kim, I have some math prepared for us to do today. 

Kim  1:37  
Okay, sure.

Pam  1:38  
The way we're going to do this is... I'm just going to sort of set the stage. So, rather than saying, "Hey, Kim. What are you thinking about?" I'm going to kind of do it... Kim, can you be like a sixth grade kid? Kind of channel.

Kim  1:50  
I'll try.

Pam  1:51  
Channel sixth grade.

Kim  1:53  
Okay.

Pam  1:53  
Which means, we've got kids that have lots of different backgrounds in that sixth grade class. But I'm going to kind of try to facilitate this string like I would with a sixth grade class. So, I'm... 

Kim  2:03  
Okay.

Pam  2:05  
Yeah. I think it'll become clear as we go. Alright here's our first problem.

Kim  2:04  
Okay. That sounds good. 

Pam  2:08  
Okay, everybody, you got $14.00. So...

Kim  2:11  
Okay.

Pam  2:11  
...picture $14.00. Yay, $14.00. Not a lot of money, I know, $14.00. You can buy lunch, right? $14.00?

Kim  2:16  
Mmhm.

Pam  2:17  
Though, lunch is getting more expensive. Anyway.

Kim  2:19  
It is.

Pam  2:19  
You got $14.00. It's extra from whatever. Whatever we were just doing, we got $14.00 extra. And there's 8 of us, and so this extra $14.00 we're going to split it among the 8 of us. I know it's not a lot of money. But $14.00 split among the 8 of us. Go and solve that, everybody. Everybody thinking. How much money would each of us get if we shared that $14.00 fairly among 8? Now, I'm going to let kids think a little bit. I might circle, might like see what's going on. But I am going to then say... Maybe if I found it, I might call on a kid. But I'm going to say, "Did anybody think about we got 8 people $14.00." Well, maybe I should say. So, if you listen to the podcast, maybe pause for a second. Think about that. How much money would each of the 8 kids get if $14.00 was shared evenly? I'm going to look and I'm going to say, "Did anybody like give everybody $1.00?" So, Kim. Let's say your eyebrows raise, or you've got your thumb up, or whatever. Could you be a kid who said, "Well, yeah. I can think about $14.00 shared among 8 kids by kicking it off, giving everybody $1.00." What? Say more.

Kim  3:18  
Yeah. So, each kid gets $1.00. Each person gets $1.00. And that would be 8 of the dollars.

Pam  3:25  
Okay, so I'm going to... Sorry, can I pause you for just a second?

Kim  3:28  
Yeah, sure.

Pam  3:28  
Because I'm writing as you're saying that.

Kim  3:29  
Okay.

Pam  3:30  
So, as the student's saying that... Well, maybe I'll listen to you first. Maybe I should listen. That just occurred to me that a better teacher move would be to listen to you first. So, go ahead.

Kim  3:38  
Okay, so I have $6.00 left.

Pam  3:41  
Okay.

Kim  3:42  
So, I'm going to then give $0.50 cents to everybody, and that uses up 4 of the dollars.

Pam  3:48  
See, I really do think I would be writing on the board at this point. I think.

Kim  3:51  
Okay, you could write on the board. 

Pam  3:53  
Okay, so I think I would have written $8.00 "fraction bar" 8. When I said, "We're giving everybody $1.00," I would have said, "So, these 8. You gave $8.00 to 8 people." And then under that, I would write $1.00."

Kim  4:07  
Exactly what I have.

Pam  4:08  
$8.00 divided by 8 and $1.00. And then you said there was $6.00 left? And what did you do with the $6.00?

Kim  4:12  
Yeah. And so, for 4 of the dollars...Yeah, 4 of the dollars, I gave $0.50 to each person. And that's $4.00 for 8 people.

Pam  4:21  
So, now I've written $4.00 "fraction bar" 8. And you're saying that's $0.50? Now, here, we're going to pause for a second, and I'm going to say, "Is that $0.50, everybody?" I'm going to look around. I'm going to call on somebody else. Like, "How do you know that $4.00 shared among 8 people is $0.50?" So, Kim, I don't know. What would be a justification of yours for that?

Kim  4:40  
So, $1.00 would cover 2 people. It's $0.50 and $0.50.

Pam  4:44  
Mmm, mmhm.

Kim  4:45  
So, each of the dollars would be for 2 people. So, $4.00 for 8 people.

Pam  4:50  
Nice. I could also imagine a kid saying, "Well, if $8.00 shared among 8 people was $1.00, now we have half the amount of money, shared among the same number of people, so they would get half the amount of money.  So, they had a $1.00 before, now they have $0.50." Do you have anything else you can think of? Just because never stop with only what you can think of.

Kim  5:13  
No.

Pam  5:13  
Cool. Okay. So, so far, you have $8.00 shared among 8 people, $4.00 shared among 8 people. But we needed a whole $14.00.

Kim  5:20  
Mmhm, that's only $12.00. So then I have $2.00 left among 8 people, and that's a quarter for each person. 

Pam  5:26  
And how do you know that $2.00 shared among 8 people is a quarter?

Kim  5:31  
You just said that $4.00 for each person would be $0.50, so $2.00 would be half as much.

Pam  5:38  
Nice. Then I might stand back and look at what I have on the board. So, right now I have $8.00 "fraction bar" 8 plus $4.00 "fraction bar" 8 plus $2.00 "fraction bar" 8. And I might say, "Do those look like fractions?" So, underneath that, I have $1.00 plus $0.50 plus $0.25. But I'm kind of curious if anybody goes, "Oh, yeah. 8 divided by 8 is 1, and 4 divided by 8 is one-half, and 2 divided by 8 is one-fourth." And those fractions are equivalent to that $1.00 and that $0.50 and that $0.25. Something I might try to pull that out. And so, how much money did each kid get if we were sharing the $14.00 among 8 kids? $1.00? 

Kim  6:17  
$1.75.

Pam  6:18  
$1.75 because we add those up, you get $1.75. And just for fun. Kim, what would that fraction be if we wrote that as a mixed number or an improper fraction? Either one. You could write 1 and 3/4. Okay. And the improper fraction?

Kim  6:36  
Fourteen-eighths. 

Pam  6:38  
Ah. Hey, there's our original fourteen-eighths. Nice. Could you also have fourths? 

Kim  6:42  
Yeah. You could have seven-fourths.

Pam  6:45  
And you could have seven-fourths. That might have been a fail on my part right there. Pretend I didn't ask that. Pretend I don't. Let's say the 1 and 3/4.

Kim  6:45  
I wanted to call it fourteen-eighths. 

Pam  6:45  
You did, didn't you? 

Kim  6:45  
I did.

Pam  6:45  
I'm going to stay at the 1 and 3/4. So, so far on the board, I've got 1.75 or $1.75 and 1 and 3/4. Okay. 

Kim  7:05  
Okay.

Pam  7:05  
Next problem. So, see teachers fail every once in a while. You just sort of smooth it over. Kim knows where I'm going, which is why she's cracking up with me. Okay, so great first problem. Second problem. This time, $7.00. This time, shared among 4 people.

Kim  7:23  
Mmhm.

Pam  7:23  
$7.00 shared among 4 people. So, new problem.

Kim  7:27  
Okay.

Pam  7:28  
$7.00. Got the extra $7.00. But this time we only have 4 people. 

Kim  7:30  
Mmhm.

Pam  7:30  
$7.00 shared among 4 people. I'm going to let people work. Kids are working. Sixth graders are working. I'm looking around. By the way, I could be doing this in fifth grade.

Kim  7:40  
Mmhm.

Pam  7:40  
Maybe even fourth.

Kim  7:41  
Mmhm.

Pam  7:43  
And for sure, up. But I'm thinking sixth grade because I think that will help everybody kind of whatever. Okay, $7.00 shared among 4 people. I'm going to do the same kind of looking I did before, and I'm going to ask the same question. "Did anybody give everybody $1.00? Did anybody start with..." And let's say Kim is the kid who I called on before. I'll probably say, "Did anybody use Kim's strategy of giving everybody $1.00?" Whoever that was. That kid I called on for the $1.00. So, Kim, can you run with that? What would it look like if a kid had given everybody $1.00?

Kim  8:11  
Yeah, so each kid is going to get $1.00. That's 4 of the $7.00. 

Pam  8:15  
Okay. 

Kim  8:16  
And then I have $3.00 left. So, I have $3.00 for 4 kids. Honestly, if I'm a sixth grader, I might go the same $2.00 shared among those 4 people is $0.50 again, and then I have 1 more dollar, and that's a quarter for everyone. So, I'd have $1.50 and $0.25.

Pam  8:38  
So, let me just tell you what I wrote while you were talking. And I would probably say, "Whoa, that was fantastic. Good strategy. Say it again while I represent it on the board." And so, when you said $4.00 shared among 4 people, I wrote $4.00 "fraction bar" 4. And you said that's because everybody gets $1.00, so underneath that, I wrote $1.00. And then you said $3.00 shared among 4 people. And you said, "But if I was sixth grader, I probably thought about that as..." So, I actually wrote $3.00 shared among 4 people. $3.00 "fraction bar" 4. And then when you said, "But I probably break that up," then I erased it, and I wrote $2.00 "fraction bar" 4. So, $2.00 shared among 4 people. And you said that was $0.50.

Kim  9:18  
Mmhm.

Pam  9:18  
And could you? How are you reasoning about $2.00 shared among 4 people?
How do you know it's $0.50?

Kim  9:24  
We just said it was $0.50 in the previous problem.

Pam  9:29  
$2.00 shared among 4 people? 

Kim  9:31  
Oh, no. Sorry, we just said the $4.00 for 4 people was $1.00. And the $2.00 would be half as much.

Pam  9:39  
Half as much.

Kim  9:40  
Yeah.

Pam  9:40  
Because it's... Yeah, nice. And then I might ask at that point, "Did anybody look at it like as a fraction kind of like we did above?" And somebody might say, "Oh, yeah. Two-fourths. That's like a half or 0.5."

Kim  9:51  
Mmhm.

Pam  9:51  
And then you said you had $1.00 left over, so I have $1.00 "fraction bar" 4, which looks like one-fourth.

Kim  9:59  
Mmhm.

Pam  9:59  
But we might also want to reason. You know like, if $2.00 divided among 4 people was $0.50, then half the amount of money, $1.00, shared among the same number of people would be half the amount of money. So, not $0.50 but would be $0.25. Cool. And then what is $1.00 plus $0.50 plus $0.25?

Kim  10:15  
$1.75.

Pam  10:17  
Hey, and then I'm looking to see if anybody's smiling. Kim, I know you are.

Kim  10:22  
Mmhm. 

Pam  10:22  
I might say, "Why are you smiling?" And do you want to read my mind?

Kim  10:26  
Yeah.

Pam  10:27  
Why do you think I'm...

Kim  10:28  
Yeah, go ahead.

Pam  10:28  
Yeah.

Kim  10:29  
Because each person gets the same amount of money as last time.

Pam  10:35  
And last time... Keep going. 

Kim  10:37  
Last time there were $14.00 for 8 people, and this time there's $7.00 for 4 people. And in both those cases, each individual person gets $1.75.

Pam  10:48  
Interesting. So, then we might go back and look at that fourteen-eighths, $14.00 share among 8 people as a fraction. And look at the $7.00 shared among 4 people as a fraction. And is fourteen-eighths... How does fourteen-eighths compare to seven-fourths? Oh, yeah. They're equivalent. Cool. So, we sort of scale down the numerator, scale down the denominator. Sort of cut both of them in half. But we could also look at it in the fair-sharing context. I had $14.00 shared among 8 people. I cut the money in half. Now, we only have $7.00. But I also cut the number of people in half.

Pam  11:19  
Is that (unclear).

Kim  11:19  
Not literally, though. Not literally.

Pam  11:19  
Ha! You don't want to cut the people? Cut the number of people in half. Is that better? 

Kim  11:24  
Yeah, there you go.

Pam  11:25  
Okay, cut the number of people in half. That was a good catch. So, if we have half as much money shared among half as many people, will they get the same amount of money? 

Kim  11:36  
Yeah. 

Pam  11:37  
That's interesting. And we could sort of talk about that. And we can... I think that would be a good thing. That's really good relational reasoning that we want to kind of help build in kids. Okay.

Kim  11:45  
And I can imagine it now is when you'd put seven-fourths next to your fourteen-eighths

Pam  11:50  
Well said. Yes.

Kim  11:51  
that you were talking about doing before.

Pam  11:53  
Yeah, and so up above where I said, "Oops, I forgot." Or I didn't mean to ask for seven-fourths. Exactly that. When we had that 1 and 3/4 for the $1.75 that is seven-fourths, which is then kind of that sort of next problem. Yeah, nicely done. Cool. Okay, so then I would give... So, those are the first two problems in the string. Third problem in the string. How about this time we have 12 extra dollars. $12.00 hanging around. And we had to share with those 8 kids. Those original 8 kids. So, we've got $12.00 shared among 8 kids. Now, I'm really curious what kids are doing. Walking around, looking at what they're doing. I'm going to try to pull out a couple of different ways of thinking. "Did anybody do the, let's give everybody $1.00, strategy." So, Kim, will you just be the kid that gave everybody $1.00?

Kim  12:38  
Yep. Everybody gets $1.00, and that's 8 of the dollars. 

Pam  12:38  
So, $8.00 divided by 8.

Kim  12:39  
Mmhm.

Pam  12:39  
That's $1.00. Okay. 

Kim  12:42  
Mmhm. And then we have $4.00 left to share among 8 people. And we said that that was $0.50.

Pam  12:51  
We've already done $4.00 divided by 8 people.

Kim  12:54  
Yeah.

Pam  12:54  
Up above. And I could circle that above where we've got that written.

Kim  12:57  
Mmhm.

Pam  12:57  
Like, that's $0.50. And then, oh, oh. But, Kim, shouldn't they've gotten the same amount of money as the other two problems? No, it's a whole new problem. 

Kim  13:01  
Yeah.

Pam  13:01  
Like, so I might make that joke. You know like, "Uh-oh. Did we goof? They should be getting the same amount of money."

Kim  13:05  
But it would be really great if a sixth grader says, "I'm going to go back and use the portion of the $14.00 where we broke it up from that $8.00 and $4.00. Yeah.

Pam  13:21  
Because yeah. And if you guys could see my paper right now, we had $8.00 divided by 8, and $4.00 divided by 8, and then we had $2.00 divided by 8 in the $14.00 divided by 8. But in that problem, we exactly had the $12.00 divided by 8. So, we sort of pull that out. Nice. Trying to think if I would go anywhere else. I'd probably encourage kids to just look at fractions. I might have had kids... No, that's probably good there. Okay, so then, in this case, we got $1.00 and we have $0.50, so each kid gets $1.50. I'm thinking. I'm thinking. Yeah, that's probably good for that one. Then I might say, "Okay, next problem. What if this time, we only have 3 extra dollars? But that's okay because we're only going to share it with 2 kids. So, there's 2 kids. They're going to share fairly. Fair sharely? They're going to share fairly $3.00. 2 kids sharing $3.00."

Kim  14:15  
Why are you laughing? If I'm in a sixth grade class, everybody right now is like, "That's me! That's me! I'm getting the 2! I'm the 2!"

Pam  14:22  
"I'm one of the 2! I'm one of the 2! Give me the money! I want the money!"

Kim  14:28  
Absolutely. 

Pam  14:28  
And at this point, I might be like, "Okay, only 2 kids. You're going to get more money, right? Nice, there's only 2 kids. Or are you? Did anybody do the, give everybody $1.00, strategy?" So, Kim, let's do that one first. Let's give everybody $1.00. Okay.

Kim  14:38  
Yeah, everybody gets $1.00. And then there's $1.00 left to share among the 2, so they get $0.50.

Pam  14:43  
Alright, so I wrote down $2.00 divided by 2 is $1.00. And then there's $1.00 leftover divided by 2. Ooh, and $1.00 divided by 2, that's $0.50. So, that is...Oh, and that is like the one that we had that was like the $12.00 divided among 8.

Kim  15:01  
Mmhm.

Pam  15:02  
And so, now I'm going to look. Is there a relationship between the $12.00 divided by 8 and the $3.00 divided by 2? I'm going to pause. And then, Kim, do you want to? What relationship do you see between the $12.00 divided by 8 and the $3.00 divided by 2 students?

Kim  15:12  
Yeah, they're... Each pair of students gets the same amount. So, in the 3 divided by 2, they get $1.50. But also the 12 divided by 8, they get $1.50 because you have a fourth as much money and a fourth of the number of kids.

Pam  15:14  
Nice. Like, 12 divided by 4, $12.00 divided by 4 is $3.00.

Kim  15:17  
8.

Pam  15:17  
Sorry? 

Kim  15:18  
Wait, 12 divided... Oh, sorry. I interrupted.

Pam  15:20  
$12.00 divided by 4 is $3.00.

Kim  15:23  
Yep, mmhm.

Pam  15:23  
And 8 kids divided by 4 is 2 kids. 

Kim  15:25  
Mmhm.

Pam  15:25  
And so, the problems are equivalent. So, hey, it's interesting. Of the four problems that we've just done, we have two sets of equivalent problems. Fourteen-eighths was equivalent to seven-fourths, and twelve-eighths is equivalent to three-halves. And so, that's interesting. I wonder, looking back, if you ever ran into fourteen-eights if you'd prefer to just solve seven-fourths. Solve, like find 7 divided by 4, rather than finding 14 divided by 8.

Kim  16:08  
Mmhm. 

Pam  16:09  
Could you?

Kim  16:09  
Mmhm.

Pam  16:09  
Like, you could choose at this point. You could say, "Hey, I could just scale up the numerator/denominator, or I could scale down the numerator/denominator. As long as I had an equivalent problem, I could choose the one that I wanted to solve. 

Kim  16:20  
Mmhm.

Pam  16:20  
In both of these cases, from the $14.00 divided by 8 and the $7.00 divided among 4 kids, we scaled the $14.00 down to $7.00, and the 8 kids, the number of 8 kids, down to 4 kids. And the $12.00 shared among 8 to $3.00 shared among 2, we scaled down both of those. I wonder if there'd ever be a time when you might want to scale up. Probably not. Probably not. That'd be weird. Next problem. How about... This time, I'm not really actually going to do money so much. I'm just going to say that I've got 5 and a quarter, so 5 and 1/4

Kim  16:56  
Mmhm.

Pam  16:56  
divided by... And I'm writing that on the board like a compound fraction. So, I've got 5 and a 1/4 "fraction bar" underneath it. 5 and 1/4, "fraction bar" one-half. So, 5 and 1/4 divided by one-half. Let me just say again because it's all not visual. I've got 5 and one quarter, 1/4, big "fraction bar" underneath it, and then one-half. So, compound fraction. Alright, what's your gut urge, Kim?

Kim  17:26  
I want the denominator to be one, so I'm going to scale up times 2.

Pam  17:34  
Because one-half times 2 is 1. Mmhm.

Kim  17:36  
Mmhm. And then I'll have 1 in the denominator, so I'm going to double the 5 and a 1/4 and make that 10 and a 1/2. So, my new problem I have is 10 and a 1/2 divided by 1, which is 10 and a 1/2.

Pam  17:49  
So, there might be a time when you just want to scale up both the numerator and the denominator to find an equivalent problem that's just divided by 1. Well, that seemed really helpful. 

Kim  17:59  
Yeah.

Pam  18:00  
Wow, 5 and a quarter times 2. That's not too bad. It's 10 and a 1/2. A half times 2, double that, that's just 1. Nice, nice. Last problem of the string. 

Kim  18:09  
Mmhm. 

Pam  18:09  
What if I gave you a problem like 3 and 1/5 all divided by, so big fraction, bar one-third.

Kim  18:17  
Mmhm.

Pam  18:18  
Hmm. What are you thinking about? Well, actually, so pause. Podcast listeners, pause. Might you want to scale down? Scale up? Both the numerator and the denominator, find the equivalent ratio, equivalent fraction that might, I don't know, be easier to solve. Alright, Kim. What do you got?

Kim  18:35  
So, I'm looking at the third in the denominator, and I'm wishing that it was also 1.

Pam  18:40  
Okay.

Kim  18:40  
So, I'm going to scale up times 3. So, I've written times 3. So, 3 and a 1/5 times 3 is 9 and 3/5. And a third times 3 is 1. So, I have 9 and 3/5 divided by 1. So, 9 and 3/5.

Pam  18:56  
You're saying 3 and a 1/5 divided by 1/3 is 9 and 3/5?

Kim  19:01  
Mmhm.

Pam  19:01  
Bam. Because 9 and 3/5 divided by 1 is equivalent to our original problem, 3 and 1/5 divided by one-third.. And you would rather, if I gave you the choice, you'd rather solve 9 and 3/5 divided by 1 than 3 and 1/5 divided by one-third.

Kim  19:18  
Oh, sure. Yeah. 

Pam  19:18  
And we could look back just a little bit. When I gave you 5 and a 1/4 divided by one-half. We could have asked, how many one-halves are in 5 and a quarter.

Kim  19:26  
Mmhm.

Pam  19:26  
We could make sense of that. Like, how many one-halves are in 5? Is that 10? We have ten 1/2s in 5?

Kim  19:33  
Mmhm.

Pam  19:34  
But then we'd have to ask how many one-halves are in one-fourth. Well, that's not too bad. How many one-halves are in fourth? Well, like  there's only a half. There's only a half of a half in a fourth. 

Kim  19:42  
Mmhm.

Pam  19:42  
And we would get our same 10 and a 1/2. So, that's a quotative look, how many one halves are in 5 and a quarter. And we could do that. We could also do the partitive look that you took by finding an equivalent fraction, equivalent ratio that was easier to sort of think about 10 and a 1/2 divided by 1.

Kim  20:04  
Yeah. 

Pam  20:04  
But in the second problem, I gave you, 3 and a 1/5 divided by one-third, it's a little more difficult to think about how many one-thirds are in 3 and 1/5. We could. But for this problem, I think I would just rather scale up like you did. Triple both of them. Get 9 and 3/5 divided by 1. Bam, that's just 9 and 3/5. Y'all, there's some fraction fun on this podcast day. Hope you enjoyed that one. What did you think, Kim?

Kim  20:28  
You know, I love this strategy, and I think we get a lot of questions about fractions and division, and a lot of people asking, you know, how can I help my students make sense of fractions? And I think often we leave out this partitive meaning.

Pam  20:41  
Mmhm.

Kim  20:41  
And there's so much power and so much value in helping students understand both quotitive and partitive meaning of division with fractions.

Pam  20:52  
And notice, that while we got there with this string, we also did a lot of equivalent fractions above. We did a lot of reasoning about division and fair sharing. 

Kim  21:03  
Mmhm.

Pam  21:04  
We did a lot of connection between fractions and decimals. Like, it's about developing mathematical reasoning. It's not just getting... Because somebody could be like, "Pam, for all those problems, we could have just like, invert and multiply. Like, we could have just done long division algorithm for the first bit." You know like, it could have been far quote unquote "easier" for us to solve those problems. But notice, how we use these problems to develop reasoning, not just for kids to get answers. Yeah, cool. Thanks for having fun doing some math today. 

Kim  21:33  
Yeah, for sure. It's our favorite. 

Pam  21:34  
Alright, y'all, thanks 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!