am  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather. 

Kim  0:09  
And

I'm Kim, 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  0:16  
Because we know that algorithms are amazing historic achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:30  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:37  
We invite you to join us to make math more figure-out-able. 

Kim  0:41  
Hi.

Pam  0:42  
Hey, there. How's it going? It's good. It's good. Let's record a podcast. What do you say?

Kim  0:47  
Okay, let's do it. Hey, you know, every once in a while I like to share a review. So... 

Pam  0:52  
Oh, nice. 

Kim  0:53  
Yeah, you haven't seen this one yet, but this one the title of it or the subject line is Mind Blown! 

Pam  0:59  
Oh!

Kim  1:00  
Which is super fun, super fun. And my eyes are old, so honestly, I cannot see the name of who I grabbed it from. I'm going to let you see it, so you can maybe try to see the name. 

Pam  1:10  
Oh, okay, I'm looking.

Kim  1:11  
Don't read.

Pam  1:11  
I'm just

reading the name. It's I-H-MUSNN... I think it's initials.

Kim  1:22  
Wow, you're eyes are better than mine.

Pam  1:22  
H-M-U-S-N-N. Thank you for telling us your mind was blown and for the five stars. 

Kim  1:23  
Yeah, yeah.

Okay, so let me read. "I just finished listening to the second podcast about Mean, Median, Mode." So, this is a while back.

Pam  1:31  
Okay. So, we did a couple in a row, and this was the second one. Mmhm.

Kim  1:34  
Yeah, yeah.

"I just have to say, I cannot believe how I didn't understand how..."

Pam  1:38  
You're doing great. You're doing great. 

Kim  1:42  
"I just have to say, I can't believe I didn't understand how I could just view the leftovers as the fractional difference to determine Mean. Not only is more visual than to add it all up and divide, but so so much easier. Pam and Kim, you continue to blow my mind with how much more figure-out-able you make math for this 50 year old math teacher."

Pam  2:03  
Sweet!

Kim  2:04  
Listen, I think we got several comments about that. The idea that mean is

evened out. 

Pam  2:10  
It's not just a process. It's not just a procedure. You add them all up and divide, but you can actually think about what a

mean is. 

Kim  2:16  
Mmhm.

Pam  2:16  
Yeah, yeah. 

Kim  2:17  
I think that we got several comments about that. That people were like, "Wait a second. Are you serious? You can just kind of move some numbers around and make it more fair?" Which is lovely. I'm so glad that people... I'm glad we did that episode. I'm glad people got something out of it. 

Pam  2:32  
And if

you haven't listened to it, go check that episode out. We'll put the episode number in the show notes.

Kim  2:38  
Yeah. 

Pam  2:38  
So, yeah, you can get have a better sense. Data is important, as Jo Boaler is telling us. To get a data view on... We should have a data flare in our teaching. Thanks to Jo Boaler and Kathy Williams and their new book that's not probably out yet. It's coming out soon. Yeah,

cool. 

Kim  2:55  
I'm jealous (unclear).

Pam  2:56  
Alright, hey. So, Kim, for today... Wait, what did you say? You're jealous what?

Kim  3:00  
That you get to read it.

Pam  3:01  
Oh, yeah, it was super. It's been fun to get some advanced copies to (unclear).,,

Kim  3:05  
Yeah.

Pam  3:05  
...endorsements. Yeah. 

Kim  3:06  
Very cool. 

Pam  3:07  
Very cool. Okay, so we've been planning to film our Building Powerful Fractions 2 workshop. Because we already have Building Powerful Fractions 1, which is a workshop that you can take from us right now.

Kim  3:18  
Yeah.

Pam  3:19  
Which I think is one of our best. It's super good. And as we've been planning to film Building Powerful Fractions 2, we've been talking a bunch about code switching. 

Kim  3:30  
Yeah.

Pam  3:31  
And Kim, you kind of brought up the term "code switching".

Kim  3:34  
Oh, I don't think it was me. I think we talked about the idea, but I think it might have actually been

Sue that named it.

Pam  3:40  
(unclear) some credit then.

Kim  3:41  
Yeah.

Pam  3:42  
Sue Simmons works with us. She's amazing. We love her. 

Kim  3:44  
Yeah, yeah. 

Pam  3:45  
Yeah. 

Kim  3:45  
Yeah. 

Pam  3:46  
You know when you say someone's not a jack of all trades, they're a master of all trades. That's kind of Sue.

Kim  3:50  
Absolutely. 

Pam  3:51  
If we ever need anything that we can't, then it's like, "Hey, Sue..." 

Kim  3:51  
She's a Macgyver. That's an old person reference. 

Pam  3:51  
Oh, Macgyver. That's a really good comparison.

Kim  3:52  
(unclear). She absolutely is. 

Pam  3:55  
I like it.

Kim  4:02  
Yeah.

Pam  4:02  
Anyway, so she brought up this idea of code switching. So, Kim, tell everybody. What do we mean by code switching?

Kim  4:07  
Yeah. So, it's the idea that at least in this particular circumstance that we were talking about, the idea of flowing back and forth between fractions, decimals, and percents so seamlessly that you choose what you want in the moment. So, like, if you're solving a problem, if you're thinking about describing information that you choose which representation you want because there are some times where some are better, easier, more familiar than others. It's kind of like, you know, how if you're ever speaking to somebody who's bilingual. That, you know, I find it fascinating that somebody who is bilingual will be having a conversation with somebody else, and you can hear them flow in and out of English and Spanish or French and Spanish or whatever, and they just... It's beautiful to watch them as they choose words. Kind of go in and out of the languages. 

Pam  4:59  
Sort

of has the best meaning for the situation. And they're like, "Oh, it's like this word really fits here." 

Kim  5:04  
Yeah.

Pam  5:05  
Yeah, yeah. That is really cool. And I'm a fan of languages. Really appreciate that reference, yeah. So, Kim, if I were to ask you a specific question, maybe we could give everybody an example of how you might use a different interpretation or, like you said, representation of numbers. So, if I were to just give you something random like three-fourths of two-fifths. Three-fourths of two-fifths.

Kim  5:30  
Yeah.

Pam  5:30  
Tell me what's your first gut? 

Kim  5:33  
Like how I think about it? 

Pam  5:34  
Yeah, because I actually remember. I gave you this problem. It was a few years ago. And I had some things in mind, and everything you said was not what I had. I was like (unclear).

Kim  5:45  
Whoops.

Pam  5:45  
Yeah. No, it definitely on the oops. It was like, "Oh, that's so interesting." So, what do you think about when I say three-fourths of two-fifths?

Kim  5:53  
Yeah, so I think the way that I picture that in my head is 75% of four-tenths. 0.4. So, like, if you gave me that and it was a problem, I would say 75% of point 0.4.

Pam  6:09  
Yeah. So, it was three-fourths that you turned into 75%.

Kim  6:13  
Mmhm.

Pam  6:14  
And it was two-fifths that you turned into four-tenths. 

Mmhm.

And so...

Kim  6:19  
But

I think... So, the reason I said 0.4 is to give people a visual of that I do it in a decimal. Like I'm picturing a decimal not (unclear).

Pam  6:27  
Oh, not a fraction. Sure, enough. Yeah, I hadn't even thought of that. Yeah.

Pam and Kim  6:30  
Yeah.

Pam  6:31  
Yeah. And why? Why three-fourths turning into 75% here? Why not (unclear).

Kim  6:36  
Because

if I'm thinking about 0.4, then 75% of that is easily 0.3. Which I would say maybe three-tenths, but like in a decimal. Like, $0.30,

$0.30.

Pam  6:47  
So, sort of like you're thinking about 75% of $0.40. 

Kim  6:51  
Yeah. 

Pam  6:52  
And you just like. You don't even think about one-fourth or 25%?

Kim  6:58  
No. 

Pam  6:58  
To get $0.10, and then scale it up to get $0.30? 

Kim  7:00  
No.

because those fractions, I own the decimal and percent equivalencies without thinking about it. That works well for me because I know that two-fourths... I mean, two-fifths is $0.40 or 40%. Like, I've traveled that path in my mind so much that I don't really think about it. 

Pam  7:19  
Yeah, so

if anybody's kind of like still grinding on those numbers a little bit. If you had $0.40, you could think about one-fourth of $0.40 is $0.10. Two-fourths of $0.40 would be $0.20 cents. Three-fourths of $0.40 is $0.30. And so, you're saying that three-fourths of two-fifths is $0.30? And how would you write that?

Kim  7:20  
Decimal 3. 

Pam  7:24  
So,

even though the question was fractions, three-fourths of two-fifths, you would like (unclear).

Kim  7:48  
Who am I talking to? Am I talking to somebody who is okay with the equivalencies? Then I'm going to write 0.3. 

Pam  7:54  
Okay, okay. 

Kim  7:55  
If it's somebody who's like, "You have to put it in fraction form, then I would write three-fifths." 

Pam  8:00  
Maybe.

If you're in a high school class, and you're like, "If the problem started in fractions, you have to end in fractions."

Kim  8:06  
Yeah, yeah, yeah. 

Pam  8:04  
Yeah. So we might. We're kind of joking. Or joking. We're kind of chuckling about that a little bit because I think sometimes we get a little anal. Is that the word? Like, a little picky?

Kim  8:06  
We get stuck. 

Pam  8:10  
We get a little picky. You know, if the problem was in fractions, then we have to write the answer in fractions. Okay? Is that just because it's more gradable? How many times have we talked to teachers lately? In fact, when I was just... Golly, I think it was when I was in Hawaii. They were talking about some sort of electronic homework thing that the kids do, and they were they were saying how frustrated they were that when they're creating the answer keys, they feel like they have to put in so many different choices because otherwise it gets marked wrong when the kids aren't. It isn't actually wrong. And I think that's one thing you've complained about in the past. 

Kim  8:52  
Yeah.

Pam  8:52  
The kid could actually answer the question correctly, but if the teacher hasn't entered in enough of the correct different. (unclear).

Kim  8:56  
Good heavens, yes. When my... It was Covid. My older son had to put in a ton. And it was... I can't remember what he was taking at the time. Geometry, Algebra 2, I don't know. And it was like, holy cow! If you typed anything wrong. (unclear) very long answers. And I was like, "(unclear)"

Pam  9:16  
Wrong. Or even just

an extra space. Or, you know like, there were things that didn't even make the answer wrong. It's like we're saying we're code switching. They could enter in an equivalent expression, but if the teacher hadn't thought of that one, then eh. 

Kim  9:17  
Yeah.

Pam  9:17  
So, that's kind of a bummer for everybody, right? It's a bummer for the kid, but it's also a bummer for teachers that are trying to enter. That was the conversation I'd had in Hawaii when they were saying, "Yeah, it's just so hard to try to think of all of the different kind of possibilities." And you want to give the kid credit if they got it. Yeah, blah, blah, blah. Okay, so you're saying that when I give you a fraction problem, like three-fourths of two-fifths, you would turn it into percent and decimal. Like, I'll just note. 75% of 0.4. So, it's not like you went to percents or you went to decimals. You went to both. And then you left your answer as a decimal. But you could have switched it back to like three-tenths.

Kim  10:03  
Yeah.

Pam  10:04  
As a fraction. 

Kim  10:05  
Sure. 

Pam  10:05  
Cool. So, I'm wondering, Kim, if I were to ask you, you know, the exact same kind of question like two-thirds of three-fourths or two-thirds of two-fifths.

Kim  10:17  
Yeah.

Pam  10:17  
Would you do the same thing? 

Kim  10:19  
No.

Pam  10:20  
Because?

Kim  10:21  
Because

two-thirds is funky in decimals. So, yeah.

Pam  10:27  
With all decimals?

Kim  10:29  
That particular decimal. Like, I don't want to find 66.6666%

of 0.75.

Pam  10:38  
Okay.

Kim  10:39  
Yeah. 

Pam  10:40  
What would you rather do? 

Kim  10:41  
Oh,

that's a good question. What would I rather do? Two-thirds of three-fourths. What I want to do? Huh. I feel like I might want to think about three-fourths of two-thirds. 

Pam  10:54  
I wonder

if that's because you're thinking about three-fourths of 2. 

Yeah, maybe.

Thirds. So, if you think about three-fourths of two-thirds.

Kim  11:04  
But

then even still, like I want to go 1.75

in the numerator. People are going to love that. 

Pam  11:10  
1.75/3.

Kim  11:12  
Mmhm. 

Pam  11:13  
Yeah. I want to stay in in thirds.

Kim  11:16  
Mmhm. 

Pam  11:16  
So, what if I asked you about one-third of three-fourths. 

Kim  11:22  
I

don't like it. Why would you want that? 

Pam  11:26  
I don't know. One-third of 3 anythings. 

Kim  11:28  
Yeah. 

Pam  11:29  
Wait,

what do you mean "yeah"?

Kim  11:33  
1/3 of 3 anythings.

Pam  11:34  
Yeah,

what's 1/3 of 3? So, this is fourths. 

Kim  11:35  
Yeah.

Pam  11:35  
One-third of three-fourths, but I wonder if it was one-third of three-fifths, or one-third of three-sevenths, or one-third of three-elevenths. 

Kim  11:44  
Yeah,

so then it would just be a fourth.

Pam  11:46  
So, one-third

of 3 things. One-third of three-fourths would be one-fourth. And then two-thirds.

Kim  11:52  
Two-fourths.

Pam  11:52  
Two-fourths. So, in this case, you actually stayed in thirds.

Kim  11:57  
Mmhm. 

Pam  11:57  
Once I forced you to. So, when I said three-fourths of two-fifths, you were like, "Ah, going to percents and decimals." But we could think about two-thirds or three-fourths. You flipped it around and was like, "Well, I think I could think about three-fourths of 2 (unclear).

Kim  12:12  
Know why I think I did? I think because we were just talking about 75.

Pam  12:16  
Yeah. And when I asked you about a third of 75, that didn't flow.

Kim  12:20  
Mmhm.

Pam  12:20  
At least as well. 

Kim  12:21  
Yeah.

Pam  12:22  
Does it flow better now that we've kind of thought about? Like, could you think about a third, 1/3 of 75.

Kim  12:28  
Yeah, I don't think that it's that I can't. I think it's like because we're constantly like remembering and using what we've recently tinkered with. 

Pam  12:37  
Totally.

Yeah, that's what's pinging in your brain. 

Pam and Kim  12:39  
Yeah. 

Pam  12:40  
Yeah, interesting. So, I find it interesting that when I deal with thirds, I very rarely, if ever, go to decimals and percents. 

Kim  12:49  
Sure.

Pam  12:49  
I almost always stay in fractions and think about one-third and then scale it to two-thirds.

Mmhm. 

So, that's kind of interesting. Let me give you another example or  question. What if I asked you for 0.5 times 16? What are you thinking about? 

Kim  13:05  
Hang on a second. 

Pam  13:06  
Oh, sorry. Yeah?

Kim  13:07  
I just realized that I wrote down on my paper 1.75. Somebody's screaming in their car right now. They're like, "It's not three-fourths!" I think that's why I was stuck on 7-5. I said 1.75 because that's what I wrote down. It's 1.5. Thirds. So, sorry.

Pam  13:23  
Oh, I didn't even hear you. Okay. 

Kim  13:26  
I looked

down at my paper, and I was like, "Why does it say 1.75?

Pam  13:29  
So, when you said...

Kim  13:30  
(unclear) 75.

Pam  13:31  
Okay,

so you're saying that three-fourths of 2...

Kim  13:35  
Yeah.

Pam  13:36  
...is 1.5. 

Kim  13:37  
Yeah.

Pam  13:37  
So, three-fourths of two-thirds would be 1.5/3. 

Kim  13:39  
Yeah.

Pam  13:41  
Which then we could scale to get an equivalent. So, that would be 3. Like, if I double 1.5, that's 3. 

Kim  13:49  
It's still half, yeah. 

Pam  13:51  
If I double 3, it's 6. Now it's 3 out of 6. Or 1.5 out of 3 is still one-half.

Kim  13:55  
Yeah.

Pam  13:55  
Right? Okay. 

Kim  13:56  
Sorry. Okay, ask you a question

again. 

Pam  13:59  
Alright, so the next one for us all to think about.

Kim  14:01  
Yeah.

Pam  14:01  
Hope everybody's hanging on. Hang on, everybody. 

Kim  14:04  
Gosh. 

Pam  14:04  
So if I were just to ask you a completely non-related problem. 1... Oh, sorry. 0.5 times 16.

Kim  14:11  
Yeah.

Pam  14:12  
Do you stay in decimals? 

Kim  14:13  
I do not. 

Pam  14:14  
What do you

do?

Kim  14:15  
That's half. In that circumstance, it's fraction one-half. It's one-half of 16.

Pam  14:20  
So, instead of thinking about 0.5 times 16, and writing it down, and drawing the line, and then doing the turtle, and the magic 0, and all the...

Kim  14:28  
Stop! 

Pam  14:28  
Well, because you could see that if a kid had not had this. You know, "Oh, it's decimal multiplication. I know how to do that. I'm going to write these down. But you're saying, "No, no, no. Like, think about 0.5 is one-half. Oh, it's one-half of 16." Okay, so you can just think about half of 16 is 8. Cool. How about if I asked you 1.5 times 16?

Kim  14:28  
Yeah, that's one 16 and another half of 16.

Pam  14:29  
And we really want kids to be able to do that, right? We want them to be able to think about 1.5 as 1 and a 1/2, so that they don't, again, write everything down, do the whole algorithm, butt cheek the decimal at the end. Oh, or we could just think about 1 and a 1/2. How do you think about one and a half 16s? What is that?

Kim  15:13  
I think about one 16, which is 16. And half of 16, which is 8. So, that's 24.

Pam  15:15  
So, 1 and a 1/2 times 16 is 24. Cool. What about something like 0.95 times 88. Or 95/100 times 88. What do you think

about that? 

Kim  15:28  
That is percent to me. 

Pam  15:31  
Okay. Say more. 

Kim  15:32  
So, 100% would be 88, and then I'm going to go back 5%. Because it's so close to 100, so you know I like Over. So, I'm probably going to think 10% is 8.8. 8 and 8/10. And then 5% would be... See, I'm saying points specifically because I want people to... (unclear). 

Pam  15:56  
Not picture fractions.

Kim  15:57  
Yeah, yeah. 

Pam  15:58  
Yeah. 

Kim  15:58  
So, 5% would be 4.4.

Pam  16:01  
Mmhm.

Kim  16:02  
So, then I'm going to go 88 minus 4.4.

Pam  16:04  
It's that 100% minus that 5%. What is 88 minus 4.4?

Kim  16:09  
88 minus 4 is 84, so 83.6.

Pam  16:15  
Cool. So, you're thinking about 0.95, and I could have said 95/100, but we're trying to give everybody a vision of, what, a vision. 

Kim  16:22  
Man...

Pam  16:23  
A visual is what I... (unclear).

Kim  16:26  
I would be so unhappy to do an algorithm for that. Like, to write down 88 times. Ugh.

Pam  16:32  
What would kids do, right? If it's 0.95 times 88, they'd probably write the 0.95 on the top, and then 88 underneath that, and then they would do a ton o' multiplications, and then butt cheek the decimal. 

Kim  16:43  
Yeah. 

Pam  16:43  
Or we can help kids realize the equivalencies involved and do this code switching thing. Like, how do you want to think about what's happening here? And then use the interpretation of these rational numbers that fits the situation best. And maybe even compare. You know, like how kids are thinking about it. Cool. 

Kim  17:02  
Yeah. 

Pam  17:05  
How about... You ready for one more? 

Kim  17:06  
I feel like

I'm having a test right now.

Alright, (unclear).

Pam  17:11  
Alright, what about 120% of 5? 

Kim  17:15  
Okay.

Pam  17:16  
Because I could algorithm this, right? I could say, "Let's see. I'm supposed to turn that percent into a decimal."

Kim  17:21  
Oh, yeah. 

Pam  17:22  
And then I could, again, do the long multiplication and butt cheek at the end. 

Kim  17:27  
Yeah.

Pam  17:27  
But how are you thinking about 120% of

5? 

Kim  17:30  
I like that it's 5 because I'm going to say 100% is 5. And then 20%, I think I could do with percents, but I also could think about it in terms of fractions. So, 20% is a fifth. So, a fifth of 5 is 1. 

Pam  17:46  
Oh, nice. 

Kim  17:47  
So, then that would be 6. 

Pam  17:48  
So,

5 plus 1.

Kim  17:50  
Is the answer.

Pam  17:50  
So, you did 100% plus 20%.

Kim  17:53  
Well, I did a fifth. So, I did 100% and a fifth of 5.

Pam  17:57  
Fifth of 5.

Sure enough. Alright, cool. I like it. 

Kim  18:02  
Yeah. 

Pam  18:02  
So, there's this idea of code switching that's highly involved in working with rational numbers, fractions, decimals, and percents. Yeah.

Kim  18:10  
And I think what's important to consider here is that when... You know, I think often we say, "Okay, I'm in the decimal unit as an extension of maybe some whole numbers. And then like in some other point of the year, I'm going to talk about fractions. And then as kids get older, we're going to talk about percents."

And there's such value... 

Pam  18:17  
Can I pause you for just (unclear). 

Kim  18:29  
Yeah. 

Pam  18:32  
In that... What was the first one you said? Decimal unit. In the decimal unit, we might actually say, "How do I turn this into a fraction?" and give kids kind of a procedure for doing that.

Kim  18:44  
Mmhm. 

Pam  18:44  
And then when we get into fractions, we could say, "So, how do you turn this fraction into a decimal?" and we could give them another procedure to do that. 

Kim  18:53  
Mmhm.

Pam  18:54  
Or here's an alternative. Instead of doing that, what if we did more equivalence to start with. Less procedures and more equivalence, and then engage them in the types of tasks that we're just doing and ask kids to like how could you use what you are reasoning about fractions, and decimals, and percents to do this code switching? So, I don't... Sorry, can you remember what you

were? 

Kim  19:15  
Yeah, I think. You know, I do think that there are teachers who are trying to make the connection between problems. So, I've seen...

Pam  19:21  
Of course.

Kim  19:21  
Somebody just sent me a worksheet that's like, you know, "Hey, I'm going to give it to you in fractions. You write the decimal or the percent equivalent." And I think people sometimes try to connect that there equivalent, but not in the way that we were just describing, like actually using what you know in a way that is meaningful and that you see the value of flowing between. And I do think that there are certain times where I gravitate towards percent or I gravitate towards decimals or fractions. There are. There's some general ideas that I've I've come to generalize about when I don't want one form or another. And I think you develop that through experiences over and over again. I mean, we could make a list and say, "You don't want decimals for these fractions," but that's meaningless. It doesn't really help anybody.

Pam  20:10  
It's another thing to memorize. 

Kim  20:12  
Yeah. I think flowing back and forth between decimals and percents is really nice. I think it's, you know, they may be a bit easier. But sometimes the fractions are super nice, and sometimes not at all. Like, the third that we were just talking about.

Pam  20:29  
Yeah,

like if I'm dealing in... If there's a problem and there's something with one-third or two-thirds, I'm rarely, if ever, going to think about decimals or percents.

Kim  20:38  
Yeah. 

Pam  20:38  
I'm almost always going to think about divided by 3. 

Kim  20:41  
Yeah. 

Pam  20:42  
Like, one-third, to me, is going to like symbolize divided by 3. But if I'm dealing with fourths or fifths, I might think about 25%. I might think about 20%. 

Kim  20:52  
Mmhm.

Pam  20:52  
Well, I might even think about decimals, if I'm finding the fraction of that decimal. Like, in other words, if it's the sort of second thing I'm thinking about. Because I might think about like you did earlier, 0.4. Like, I could think about 40%, but I could also think about $0.40 and then find chunks of that. 

Kim  21:11  
Yeah. 

Pam  21:11  
So, there are definitely times where third, sixth, ninth, those are all almost always going to stay in fractions for me.

Kim  21:21  
Yeah.

Pam  21:21  
Like, very rarely, if ever, are they going to turn. Is that true for you?

Kim  21:26  
Yeah, I think so. Except when I just want a general like estimate. Which I think there are times in life. You know, maybe not on a classroom worksheet. But in life, you know, 33% is good enough.

Pam  21:38  
Yeah, so... 

You'll just realize whatever you just estimate, it's going to be a little less because it should have been 33.33333%

Kim  21:48  
Yeah.

Pam  21:48  
Yeah. Okay, cool.

Kim  21:49  
Yeah. I think what's important to know is that the more experience that you have, and you create kind of this comfortable, like natural ability to go back and forth, the more that you will use it. So, I remember. Quite some time ago, you were like, "Man, why do you do that? Why do you kind of go back and forth? Or how do you decide when? And you do it all the time now because it's a thing. And I think once students, once adults, become aware that it's a thing to do, then the more they do it, the more you. I would encourage people to give it a try. Like, when you see a form that you're like, "I don't know. What do I know about this?" You know, stop for a second and think about do I know the fractional equivalent? Do I know the decimal equivalent? Do I know

the percent? 

Pam  22:34  
And then see what pings for you at that point.

Pam and Kim  22:34  
Yeah. 

Pam  22:34  
And I think one of maybe the most important things we want to bring out in this episode is rather than the worksheet you mentioned earlier, where it is if I give you this form, give me the other form. We're suggesting, let's actually solve problems. 

Kim  22:48  
Yeah.

Pam  22:49  
And suggest. So, we could do Problem Strings that kind of suggest, "Hey, here's one form. How about this form? How about that one? Now, let's compare." Maybe we'll do that in a future episode where we actually do a string like that. To end this episode, I want to end with just maybe one more example. 

Kim  23:04  
Okay.

Pam  23:04  
What if I were to give you the problem 76 times 36. Now, off the cuff, just kind of naked, you would look at that maybe and say it's whole number multiplication. You're going to do some whole number multiplication strategy. But I'm kind of kind of curious. What do you think about when I say...you're chuckling...76 times 36. Well, maybe, listeners.

Kim  23:27  
Yeah (unclear).

Pam  23:27  
Yeah, what comes to mind? Like, if I say 76 and 36, are there some other forms of rational numbers that are pinging for you right now? And if there's not, that's okay. Let's travel the path and see if we can make that path and more well traveled path. 

Kim  23:41  
Yeah.

Pam  23:41  
Alright, Kim, what are you thinking about? 

Kim  23:43  
Okay,

I'm going to think about 75% of 36.

Pam  23:48  
Okay.

Kim  23:48  
And I know that 75% of 36 is 27. So, 25% is 9. So, scale up. 75% would be 27. 

Pam  23:59  
And

I might have to say 25% is of 36 is 9 because a fourth of 36 is 9. My brain might have to go there. 

Pam and Kim  24:06  
Okay.

Kim  24:07  
So, once

you know that 75% of 36 is 27, you can scale up to say that I know 75 times 36 is 2700 because you're going to scale up by 100 to get from 75% to 75,

Pam  24:29  
Mmhm. 

Kim  24:29  
So, now we're at 2700.

Pam  24:29  
And that's 75 times 36.

Kim  24:29  
Yeah, so you just need one more 36.

Pam  24:31  
To get 76

times 36.

Kim  24:32  
Mmhm. So, that's 2736.

Pam  24:35  
Bam. Which is one of my favorite problems ever.

Kim  24:38  
Is it really? 

Pam  24:38  
It is. It is.

Kim  24:40  
Okay.

Pam  24:40  
I think we told the story once on the on the podcast that Craig came home. My third son came home, and his teacher had said, "No more baby strategies. You guys are going to have to learn the multiplication algorithm." And hand drawn. I don't blame this teacher, by the way. I blame the district because they hadn't trained her. And it was a hand drawn worksheet with problems that you, "Don't use your baby strategies because you'll be too inefficient." And the first problem was 76 times 36. And I literally said to my, then, sixth grade son, I said, "Hey, Craig, when you hear 76, what do you think about?" And he said, "75, three-quarters." So, he didn't go percent. He went fractions. 

Kim  25:18  
Yeah. 

Pam  25:18  
Okay, alright. He's like, "Three-quarters". And I was like, "Do you know three-quarters of 36?" And he's like, "Well, I can one-quarter." And then he scaled it up to get three-quarters. And I was like, "Okay, so then can that help you with..." Exactly what you did.

Kim  25:27  
Yeah. 

Pam  25:28  
...help you with 75? Okay, just scale that." And I was like, "Okay, here's the hard part. Once you know 75 times 36 is 2700, but you need seventy-six 36s." He looked at me, and he was like, "Are you kidding me?" And I was like, "Why are you mad right now?" Craig, who never gets mad. 

Kim  25:46  
Yeah.

Pam  25:46  
Craig is one of the most even keeled, low key, just doesn't get riled. He was hot. And he goes, "Baby strategies? She's trying to tell us not to use baby strategies but to use the algorithm?" And this is the particular teacher that had drawn the turtle over the whole numbers to help the kids know. And then when the magic 0, she said, "The turtle lays an egg." And he's like, "Baby strategies! We could have been learning! All year long, we've been memorizing stuff about decimals, percents, and fractions. We could have been doing problems like this, getting better at multiplication!" And now, I would give him these words, "and code switching." 

Kim  26:23  
Yeah.

Pam  26:23  
Yeah. So, it was a very meaningful moment for me to go, you know like. You had already, I had already been doing some code switching at that point, but now I knew we could do it purposely with kids. 

Pam and Kim  26:34  
Yeah.

Kim  26:34  
Wow. 

Pam  26:35  
Yeah. And we could do purposely with kids. Alright. Kim, that was fun. Thanks.

Kim  26:38  
Yeah, mmhm. 

Pam  26:39  
Alright, thanks for tuning in everybody and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. And thanks for spreading the word that Math is Figure-Out-Able!