Math is Figure-Out-Able with Pam Harris

Ep 174: The Most Sophisticated Multiplication Strategy

October 17, 2023 Pam Harris Episode 173
Math is Figure-Out-Able with Pam Harris
Ep 174: The Most Sophisticated Multiplication Strategy
Show Notes Transcript

Our students can learn to approach math problems like mathematicians! In this episode Pam and Kim look at one of the most sophisticated multiplication strategies that many mathematicians use.  
Talking Points:

  • Creating an equivalent problem by Doubling and Halving.
  • What makes a problem reasonable to solve without a calculator?
  • Double/Halve; Triple/Find a Third; Quadruple/Find a fourth
  • What makes Doubling and Halving one of the most sophisticated multiplication strategies
  • Flexible Factoring -  for more information see Ep 153
  • Pitfalls of direct teaching and strategy

Check out the Instructional Routine called Relational Thinking at:

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Pam  00:01

Hey, mathematicians! Or fellow mathematicians. Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.


Kim  00:11

And I'm happy because Pam messed it up today. Hi, I'm Kim.


Pam  00:16

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. Ya'll, we can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. There, I got that part of down fully.


Kim  00:42

Welcome back, everybody.


Pam  00:45

Hey, Kim.


Kim  00:45

So, a special good morning to Susan Smith. I know you're listening because she sent a review, and she said, "I listened to multiple math podcasts, but Math is Figure-Out-Able is one I NEVER..." all capitals "...miss. I have binged listened to many of the past podcasts I missed before I discovered it. And now, I never skip my weekly listen to the latest episode. I always walk away with a new insight or deeper understanding that will make me a better math teacher and a better mathematician. They're even powerful episodes I've listened to more than once. Don't miss out! You'll always be glad that you tuned in." So,  thanks, Susan.


Pam  01:22

Susan, that's super nice. Kim, this is super fun when I don't hear them before you.


Kim  01:25

I know, I told you. 


Pam  01:27

Yeah, that's cool. Alright, thanks (unclear). 


Kim  01:28

So, hey, if you're a new listener, I would encourage you to go back and check out some of the old episodes that sound interesting to you, right? So, we're on episode 174 already.


Pam  01:37



Kim  01:37

Can you believe that, Pam? 174!


Pam  01:38



Kim  01:39

And some are in a series, but you don't necessarily have to go in order for all the other ones. You can pick and choose what sounds interesting to you. And if you're like, "No, really, Kim, Pam, where do I start? You can check out episode 156. And that title is "Start here" And that will give you some big ideas about the Math is Figure-Out-Able movement. 


Pam  01:59



Kim  01:59

Alright, let's get going. We've been long-winded last couple weeks. 


Pam  02:02

Alright, nope, today we're down and dirty.


Kim  02:05

We're being serious today.


Pam  02:06

Getting her done, getting her done. 


Kim  02:08

Alright. So, Pam, I know I've told you this.


Pam  02:10

If you believe that.


Kim  02:14

I know. And I'm telling a story today, so maybe that will speed things.


Pam  02:18

Oh, you're saying you tell stories faster than I do is what you're saying. That's what you're saying right there. 


Kim  02:22

I can be wordy, but I feel like you might be a little wordier than me.


Pam  02:26

I don't know. Would I say you're wordy? I don't know if I would say you're wordy.


Kim  02:30

Tell my family that. Okay, so you actually know this story, so I'm just going to tell it for the benefit of listeners. So, I was in a fourth grade class doing a Problem String. I don't know, it's been a year or so, We were doing some filming. I was getting a couple of different strings. And we did like the first two problems of a string for this particular strategy that I was going for. And I was asking students to describe the model that I was using. And one kid said, "Oh, you're doing the blah, blah, blah, strategy." And then he said, "But I don't know why you would ever do that." And I was like, "Oh, okay." He was kind of a little... Sweet kid, but he's kind of a little arrogant about it. Like, you know, "That not for me." And, frankly, it didn't bother me. I kind of lean into that thing. And we kept going.


Pam  03:23

Yeah, you don't mind snark. You don't mind snark at all.


Kim  03:25

No, not at all.


Pam  03:26

You know, it's interesting, Kim. I remember watching the video because I wasn't there on the filming day. And I remember watching the video and going, "Oh, man. He already owns the strategy. Like, this is going to fail." And so, when you kept going, for just a second, I was like, "Why didn't you just stop and do something else? It's not going to work." 


Kim  03:43



Pam  03:44

Yeah, so let's talk about how that went maybe a little bit later in the podcast.


Kim  03:50

Okay. Alright. We'll finish up. 


Pam  03:51

So, let's work on that strategy that you were working on with that particular class and have some fun with that. Oh, Kim, I just realized I don't... Hang on, I don't have enough paper. There's no way this particular Problem String is going to work with that little square space I have. Surely I have a blank piece of paper in this pile of paper. Do you know what I started doing? You don't even know this.


Kim  04:13

What's that?


Pam  04:13

I've started just taking copy paper.


Kim  04:16

Oh, that's what I use.


Pam  04:17

Well, I know see.


Kim  04:18



Pam  04:18

And so, I didn't ever used to do that. I used to use... What are those called? Those 9 by 11 legal pads.


Kim  04:25

Those little yellow pads? 


Pam  04:26



Kim  04:27

I hate those.


Pam  04:29

And I ran out of them, and I had copy paper, and I was like, "I'll be Kim. I'll just be Kim." Now, I put mine on a clipboard. I think yours is just loose. Is that right? 


Kim  04:37

Yeah, it is. It is.


Pam  04:38

Yeah, mine's on a clipboard. Okay. Alright, first problem. 15 times 18. Ha! Have fun with that one because there's so many nice things you can do with that one.


Kim  04:45

Super. Oh, my gosh. You don't have anything in mind? I can do what I want?


Pam  04:49

Yeah, do what you want. 


Kim  04:50

Okay, I'm going to go with 10 times 18, which is 180. And then, I still need 5 times 18 And I know 5 is half a 10. So, if ten 18s was 180, then five 18s is going to be 90. So, together that is 270.


Pam  05:11

Bam! Nice smart partial product. Nice using Five is Half of Ten. Cool. So, 15 times 18, you're saying is 270 


Kim  05:19



Pam  05:19

Next problem. 


Kim  05:21



Pam  05:21

What is 9 times 30?


Kim  05:24

9 times 3 is 27, times 10 is 270.


Pam  05:29

So, 9 times 30 is 270?


Kim  05:31



Pam  05:32

So, we now have two problems that have the same product. And I know you guys are listening, so I'll just say it again. We had 15 times 18 was 270. And you just said 9 times 30 is also 270.


Kim  05:42



Pam  05:43

Kim, are those two problems related? If I were to draw... So, you mentioned.


Kim  05:48

I'm literally sketching as you say, "If I were to draw..."


Pam  05:51

Yeah, so you mentioned you asked that kid in class to talk about the model. 


Kim  05:54



Pam  05:55

So, I have a 15 by 18 on my paper. If I were to draw a 9 by 30... 


Kim  05:59



Pam  05:59

...based on that 15 by 18. Could you help me? What would the 9 by 30? look like?


Kim  06:03

Sure. So, I drew... It's like kind of squarish, but a little bit longer on the 18 side. So, 15 down and 18 across is what I have. Is that what you have? 


Pam  06:13



Kim  06:14

So, across the top, I have 18. But now, I only want 9 across the top, so I'm cutting that 18 in half. I just kind of drew a dotted line down mine cutting that 18 in half. So, it's like a 9 and a 9 by 15. And I'm going to move that second half of that squarish one. It's not square, but you know what I mean. And I'm moving it down. So, I'm making it twice as long and half as wide. My 9 by 30 is half as wide and twice as long as the 15 by 18.


Pam  06:48

So, you actually just created a 30 by 9. 


Kim  06:51



Pam  06:52

Yeah. Even though I said 9 by 30. So, maybe I shouldn't have asked you. I should have just told you. Let me tell you what I would have put on the board. So, I loved your explanation. It makes a lot of sense. You cut the 15 by 18 in half. You moved half of it down, created a long, skinny rectangle. A tall, skinny rectangle (unclear).


Kim  07:10

You want me to make it be a 9 by 30. Is that what you want?. 


Pam  07:13



Kim  07:14



Pam  07:15

If I was running the Problem String, I would have drawn a 9 by 30, and I would have said, "So, how does the 9 relate to the 15?" And you would have said well, a little bit more than half. I don't know. And then, the 30 would have been, what? Not quite twice as long as 18.


Kim  07:29



Pam  07:30

So, you would have had this long, skinny, but horizontal 9 by 30. Is that right? 


Kim  07:35



Pam  07:36

And then, I would have said, "Alright, so we're agreeing that they both have the same area of 270. Can you make that fit?" 


Kim  07:43



Pam  07:45

So, would you mind describing from the 9 by 30, that horizontal flat-ish rectangle? How would you move from it to the 15 by 18?


Kim  07:54

So, in that case, I would rotate it.


Pam  07:58

To create the one that you ended up with? 


Kim  08:00



Pam  08:00

Yeah. And then, you would have kind of that 30 by 9, and then you could have done the same kind of cut in half thing and everything. So, that seems important that if you cut one of the dimensions in half, and slide that bit of area down to make a long skinny one, which then in effect doubles the other dimension, you didn't lose any area. Huh. And we ended up with the same product. "I wonder if that happens all the time?" is what I would say to students, yeah?


Kim  08:29



Pam  08:29

Next problem. How about if I asked you 4.5, 4 and a 1/2, times 60.


Kim  08:36

So, I'm looking at my previous problem, and I'm noticing that a 9 is twice as much as 4 and a 1/2. So, I'm sketching. And so, right below my 9 by 30 that you had me draw, I'm drawing 4 and a 1/2, but it's only half as long on the 4 and a 1/2 side. So, I'm going to actually make it be 60 long. And it kind of is off my paper, so I drew some dots. But a 4 and a 1/2 by 60 is skinnier than the 9 and longer than the 30.


Pam  09:09

And skinnier by how much? 


Kim  09:10



Pam  09:11

And longer by?


Kim  09:12



Pam  09:13

Double. Cool. So, you kind of halve one dimension, double the other. What is? Did you ever find 4 and a 1/2 times 60? 


Kim  09:20

Oh, I didn't, but it's also 270. 


Pam  09:24

And how do you know it's 270?


Kim  09:27

Because if I cut that 9 by 30 into two pieces to make it be like a 4 and 1/2 by 30 and a 4 and a 1/2 by 30, I could take that bottom 4 and a 1/2 by 30 and slide it to create a 4 and a 1/2 by 60. So, I didn't lose any area.


Pam  09:44

So, we have an equivalent problem. Huh.


Kim  09:46



Pam  09:46

It seems like you've created equivalent problems all the way along. You've got 15 times 18 is equivalent to 9 times 30 is equivalent to 4 and a 1/2 times 60 is equivalent to 270. So, three equivalent problems, where you're kind of talking about cutting one dimension in half, doubling the other dimension, and the area stayed the same, therefore the product stayed the same. That seems kind of helpful. I wonder if you could do something like that for a problem, like 24 times 3.5?


Kim  10:18

Yeah. I was trying to make it relate for just a second.


Pam  10:22

I mean, it might. It might. It might not. Right? 


Kim  10:24

Yeah. I'm... Okay, I don't really love the 3 and a 1/2, so I'm going to double that one to make it 7. 


Pam  10:31



Kim  10:32

And so, that I can maintain the same area, I'm going to halve the 24 to make that 12. 


Pam  10:39



Kim  10:41

And so, that is 84. 


Pam  10:44

Do you just know 12 times 7? 


Kim  10:45

Um... Yeah, I do. 


Pam  10:47

That's okay. 


Kim  10:49

Why do I say that like it's a disappointment? "Yeah, sorry."


Pam  10:53

Because you want to be able to talk about how you think about it, and you're like, "Darn it! I just know that one." Hey, ya'll, just take a minute. Just take a minute to appreciate that that's mathing. It's mathing to say to yourself, "Ah, crud! I wish I had a way. I wish I had a reason right now to tell you how I'm thinking about that. And darn it! I just know that one." 


Kim  11:13



Pam  11:13

Wouldn't you love it if that was the result in your classroom? Where kids, instead of go... How often do we have them go, "I just know that one." And you're like, "Yeah, but tell me how you're thinking about it." "No, I just know that one." Instead, then we're like, "Oh, crud! I just know that one. Oh, man! That stinks! Golly, I wish I didn't just know that!" I think that's awesome. Okay, cool. So, you could double one of the dimensions, halve the other. That you're saying that you could like take that area, slide it over, create that new rectangle that would be a 12 by 7. If you didn't know 12 by 7, which I'll just be honest, I have to refigure every time. 


Kim  11:48

Oh, okay. 


Pam  11:49

I know. Yeah. So, I often just think about 10 times 7. And as soon as I have... So, like 12 times 7 is like 10 times 7 plus 2 times 7. But I don't do that much work. I literally think about 10 times 7, and I go 70, 84. 


Kim  12:02



Pam  12:02

Like, I don't even think about that I need two more or whatever. It tells me I know that I'm going to be 60 for 12. Then, I'm going to be 60, 72, 84, 96. And I can just never remember which ones which.


Kim  12:14



Pam  12:14

And so...


Kim  12:15



Pam  12:15

Oh, yeah?


Kim  12:15

That's me for 8 plus 5. Like, I think we all have... I, for whatever reason, 8 plus 5 I'm like, "Is that 12? 13? 14? I don't..." But I go back to the fact that I know 8 and 4 is 12. For whatever reason, that was my favorite fact as a kid. So, then, I'm like, "Oh, it's 12, 13." Isn't that weird? Yeah, so.


Pam  12:35

Ah, so then 8 plus 5 has to be different. No, you're reminding me actually. So, I was just in Oklahoma a couple weeks ago. Few weeks ago? I don't know how long ago. Anyway, super group of people. And Kathy, the calculus teacher. When I said, "Does anybody not know 8 times 7?" And, "You know, you can admit it." Whatever. And I always love it when it's the calculus teacher that raises their hand and goes, "Yeah, I don't know it." And she goes, "Yeah, I just can never remember if it's 54 or 56." 


Kim  12:59



Pam  12:59

So, she refigures. I can't remember exactly what she does. But I think she figures 9 times 6, and then says, "Oh, it's not that." 


Kim  13:10

That's the 54, so it can't be that.


Pam  13:12

Yeah, she goes, "Oh, that's 54, so therefore it's 56." That was her... I think Kathy. Hopefully, I just gave you credit there.


Kim  13:20

I want lots of adults to admit that. 


Pam  13:22

I mean...


Kim  13:22

I want lots of adults to say, "You know what? Like, I have to refigure one in a mathy way, and it's okay." 


Pam  13:28

And it's more than okay because what we're saying is, don't fill your brain with that trivia. Fill your brain with relationships... 


Kim  13:35



Pam  13:36 that you can refigure it in a efficient enough manner that it doesn't bog you down in the work you're doing. And at the same time, because you are using relationships, your brain is more developed and you can do more. And all the things Yeah, okay, cool. 


Kim  13:47

Back to the Problem String. 


Pam  13:48

Next problem. How about 32 times 12.5? I mean, Kim. Now, I'm giving you a 2 by 3 multiplication with decimals. 


Kim  13:59

Yep. So, I, again, don't love the 12.5. So, I want that to be 25, so I'm going to create the problem 16 by 25 by halving the 32 and doubling the 12 by 5.


Pam  14:18

Okay. 16 times 25. Yeah. 


Kim  14:20

And I actually know that one. I have done enough work with quarters, but...


Pam  14:27

"Sure wish I knew something about quarters." I'm quoting Kim, by the way, there. We have a video of Kim working with fourth grade kids. And this kid says something about 25s, so she goes, "Sure wish I knew something..." Or, no. You said something like, "What are you thinking about?" And he goes, "Quarters." And you go, "Sure wish I knew something about quarters." Anyway. Sorry, keep going.


Kim  14:44

So, if at that point I didn't know something about 16 by 25, that's still a really nice problem because if you double and half again, then you get 8 by 50. And you could even do it again and get 4 by 100.


Pam  15:00

So, you're saying 4 times 100 is 400, and 8 times 50 is 400, and 16 times 25 is 400. Therefore, 32 times 12.5 is also 400? 


Kim  15:11

Yeah. Yep. 


Pam  15:12

So, you created lots of equivalent problems.


Kim  15:14

I did.


Pam  15:14

To solve that sucker. Even though you probably could have only created one.


Kim  15:17

Well, and it's funny that that we're having this moment right now because one of my boys is absolutely obsessed with Double, Halve. He wants to like. It's kind of like Over for him. Like, he seeks for that everywhere, and I have to say to him like, "Stop when you get to one that you know." Because he wants to just like, "And I know this and I know this all the way down to like times 1 or whatever. Like, "Hey, okay. You can stop when you know."


Pam  15:45

Kim, let him play.


Kim  15:47

I know. 


Pam  15:48

So, you're not really commanding. 


Kim  15:49

Oh, gosh no.


Pam  15:50

You're not slapping his hand and saying "STOP!"


Kim  15:51

No, I'm like, "Do you know that one?" He's like, "Yes." Okay. Alright.


Pam  15:55

Hey, Kim, this is reminding me that in a recent Q&A for our online workshops. So, we have online workshops. We have live Q&As Participants get on. They can ask questions. One of the questions that a participant asked was, "What are problems that are reasonable to solve without a calculator?" Because I will often say, if you help develop the major strategies that lead to the major relationships, then you can solve any problem that's reasonable to solve without a calculator. And she said, "I think it might have less to do with the number of digits, like how big the numbers are, and more to do with the numbers themselves. Am I right?" And my answer to her is absolutely.


Kim  16:32

Yeah, that's right.


Pam  16:33

Because we have standards that say things like... Like, for example, often in fifth grade the standard is... No, sorry, fourth grade is what I'm thinking of. Where the quotient is four digits and the divisor is one digit. So, it's like don't go above those. You know, four digits divided by one digit. And I would say, it absolutely depends on the numbers. For example, 32 times 12.5 is a 2 by 3 multiplication that I think we could do in fourth grade, once kids learn to Double and Halve. 


Kim  17:02



Pam  17:02

So, it's way out of their standards because they're not doing decimal multiplication in fourth grade. But it's doable because of the numbers. So, yeah.


Kim  17:09



Pam  17:10

It's more important to think about the numbers, and less important to just randomly the size of numbers. Yeah, cool. Next problem. Could you do a similar thing? By the way, it's the last problem, Kim. You can do it. You can do it.


Kim  17:22



Pam  17:22

1.75 times 36.


Kim  17:28

Okay, so I'm going to go with doubling the 1.75. And I think about that like $1.75. 


Pam  17:37



Kim  17:38

So, double $1.75 is $3.50.


Pam  17:43

Do you want to? Like, do you just know that one? I don't know (unclear)


Kim  17:46

$1.50 and $1.50 is $3.00. And I have an extra quarter. So, I thought about the $1.75 as $1.50 and $0.25. 


Pam  17:55

And you doubled both of those.


Kim  17:55

So, double both of those to get $3.50. Which is 3.50.


Pam  17:59

Cool. Thanks for doing that. 


Kim  18:00

And what was the other? What was the other? I don't even know.


Pam  18:03



Kim  18:03

I think you said 36. Okay.


Pam  18:05

Aren't you glad? Aren't you glad?


Both Pam and Kim  18:06



Kim  18:07

So, then, I would think about that like doubling that again. Double, Halve again, I think. So, then, I would get...


Pam  18:14

Wait, sorry. So, hang on. Forgive me. Go ahead. Carry on. I'm interrupting.


Kim  18:20

So, I currently have 1.75 times 36. And I don't think I said what the other number would be. I Double, Halve to get 3.5 times 18.


Pam  18:31



Kim  18:31

I don't think I said the 18. That's not helpful. And then, I Double, Halve again to get 7 times 9, which is a fact that I know. 63.


Pam  18:40

Bam! So, you're saying 1.75 times 36 is equivalent to 3.5 times 18 is equivalent to 7 times 9, and they're all equivalent to 63. 


Kim  18:49



Pam  18:50

Ya'll, that is mathing. And we call that the Doubling, Halving strategy.


Kim  18:58

Which is really a not...


Pam  18:59

Terrible name. 


Kim  19:00

I mean, yeah.


Pam  19:01

Because notice that Kim didn't just Double, Halve. Like, I mean, kind of. You kind of kept going. But you could have quadruple, quartered.


Kim  19:10



Pam  19:10

We could bring in other problems where you might want to triple, third.


Kim  19:14



Pam  19:15

I don't know if you can say that. Is third a verb? Triple and find a third. So, Doubling and Halving is a great strategy to do early with kids because kids double. Doubling is a kind of a young thing that we can do early in developing multiplication, and then that can lead to Doubling and Halving. Let's talk about why Doubling and Halving is one of... I'm not going to say the most, but one of the most sophisticated multiplication strategies. Kim, give me one. What's one? What do you got?


Kim  19:45

So, it's definitely in the family of creating equivalent problems, which is a big idea. So, you're thinking about a problem that would be equivalent. In this case, we talked about it on an area model. So, it's maintaining area. You're having to think about both factors at the same time when you double one of them, what happens to the other one?


Pam  20:10



Kim  20:11



Pam  20:12

Yeah. And that requires simultaneous-ness, right? (unclear) 


Kim  20:15

Yeah, I was going to say. So, one of the things you think about. People ask us like how do kids know which number to double and which number to halve. I don't know that we've mentioned that at all. And so, my response to that is let them go at it, and they get to a point where they start to generalize the kinds of numbers that makes sense to double or to halve. In the beginning, when students are tinkering, they are not amazing all the time at it. And so, rather than tell them these are the kinds of numbers, we would prefer that they use Problem Strings, right, to help them develop the idea. But then, let them Double and Halve or quadruple and find the fourth of four numbers, and then they get to a point where they go, "Oh, that was not helpful. Let me back out of that and attempt something else."


Pam  21:06

Yeah, like pause and don't dive in to save too early. Give them a chance to go down a rabbit hole a little bit, back out, try something else. And that is mathematical behavior. Mathematicians try things that makes sense to them. And when it doesn't work out so slick, then they back up and try something else. And that's a mathy kind of thing to do. Very nice. So, main things that are key indicators of sophistication are that we have to anticipate. And so, like you said, in order to think about finding an equivalent problem, I think that even has to be an idea, right?


Kim  21:42



Pam  21:42

Like, think about students who were taught, "Here's how you multiply. Do these steps." All they're doing is like, "Oh, it's multiplication, I must now do these steps." And they just start doing the steps. What they don't do is say to themselves, "Hmm, what do I know? And could I think about this anticipation? Could I anticipate creating an equivalent problem? Can I think of a way to create an equivalent problem that would be easier to solve?" That very idea takes bandwidth. How do we create that bandwidth? It's by developing multiplicative reasoning, developing the kind of reasoning, doing all of the strategies before, so that kids can start saying, "Oh, I can actually anticipate. Can I create an equivalent problem that would be easier to solve?" So anticipatory thinking, equivalence, like you said, and the idea of simultaneously considering both factors. "If I do this to one, how will it affect the other one? Oh, that'll be nice. Let me try to go that way." Yeah. Very cool. Let's see. The model that we used for this one was an area model. We've talked about how the the dimensions and the area. We will also, at some point, start using equations. So, I'll be honest with you.


Kim  22:49

Mine kind of shifted along the way. 


Pam  22:50

Yeah, by the end of it, I was writing equations in between numbers. So, for example, on the 1.75 times 36, I have a double. I have kind of an arrow with a times 2 from the 1.75 to 3.5. And from the 36 to 18, I have a kind of an arrow and a divided by 2. So, if I Doubled the one, then I Halved the other. Every once in a while, I'll have high school teacher say, "But wait, wait. If you do something to one side, shouldn't you do the same thing to the other?" So, then, we have to discuss what sides mean. Sides of an equation, yes. But this is not an equation, right? This is a, I have an expression 1.75 times 36, and then I've drawn arrows between expressions with the times 2 and the divided by 2. So, you might find that interesting.


Kim  23:40



Pam  23:41

Yep. Cool. Alright, what else do we want to say about Doubling and Halving, Kim? 


Kim  23:46

Well, it's an equivalence strategy, right? So, we've been recommending that it's great to do some Relational Thinking.


Pam  23:53

Oh, yeah. The instructional routine Relational Thinking. Fantastic. Okay, so I've got one for you, Kim.


Kim  23:59

Oh, goodness. Okay.


Pam  23:59

Thanks for reminding me. What if we had a Relational Thinking that looked like this? 16 times 3.5. So, I'm writing a horizontal equation. 16 times 3.5 equals 8 times blank. 16 times 3.5 equals 8 times blank.


Kim  24:17

I'm going to say this in a way that I don't think we've said. So, you might have to clean up my language. 


Pam  24:21



Kim  24:22

If we have sixteen 3.5s, and I know that I don't want to deal with the 3.5s. So, I'm thinking about it in terms of like I have 16 of the 3.5s. 


Pam  24:36



Kim  24:36

So, I want to have the size of my group being twice as big, so I need half as many of them. So, what I'm thinking about is I don't want to deal with 3.5s. I want to deal with 7s. And because the size of that group, the size of that number is twice as much, then I only need half as many of them. I don't know if we've ever if we've said that here this way. So, I'm thinking about 8 times 7.


Pam  25:00

Yeah, we didn't say it here. I almost wonder if you kind of thought a little from the answer when you said that. Because the blank was the 7. And you kind of talked about it.


Kim  25:09

Oh. So, yeah. I wrote it down, and then I forgot which one was the blank. Sorry, I just wrote 8 times 7, and then I forgot which one you asked me about. So, scratch that. I need... Which one was the blank? The 7 was the answer?


Pam  25:22

7 is the blank.


Kim  25:23



Pam  25:23

7 is what you don't know.


Kim  25:24

Okay, so then I want half as many groups of whatever, then they need to be twice as big.


Pam  25:33

Yeah. And, ya'll, you might want to write that down to be able to follow that. If you have 16 times 5 equals (unclear).


Kim  25:39

3.5. 3.5. 


Pam  25:39

Oh, sorry.


Kim  25:40



Pam  25:41

If you have 16 times 3.5 equals 8 times something. The 16 to the 8, you're saying, "I started with 16 of something, 16 groups. Now, I only have 8 groups. If we're going to stay equivalent, the groups have got to get twice as big. 


Kim  25:57



Pam  25:58

It almost doesn't matter what the 3.5 is in this reasoning, right? If I've got 16 groups of something, and I've got to have an equivalent total, and now I only have 8 groups, I got to have twice as much stuff in each of those 8 groups in order to stay equivalent. So, twice 3.5 is 7. Is that? 


Kim  26:15



Pam  26:15

(unclear) That was funny. You didn't remember what the question was. Good job, Kim. Alright, ya'll, you can actually find more of these Relational Thinking types. We put down on social media all the time. We gather them, stick them on the website. So, you can go to Or just Google it. Google Math is Figure-Out-Able and Relational Thinking. And you can find examples of lots of these guys that you can use with your students. So, you might be interested to know that when we did the addition one we said most sophisticated addition strategy, Give and Take. When we did subtraction, we said most sophisticated subtraction strategy, Constant Difference. When we're doing this Doubling and Halving strategy today, we're saying it's one of the most sophisticated strategies. Why? Because it has those hallmarks. It's an equivalent strategy. It requires anticipatory thinking. There is simultaneity involved. However, there's one that is even more multiplicative. So, as multiplicative as this strategy, Doubling Halving is, it is really the precursor to the most sophisticated multiplication strategy we think kids need to learn, which is Flexible Factoring.


Kim  27:28

Yeah, super fun. And, you know, we're already kind of long, so let's just let them know that we've talked about Flexible Factoring on episode 153, so we highly recommend that you check that episode out to maybe tackle a strategy that might be new for you or to strengthen that strategy if you've already heard of it.


Pam  27:45

Yeah. And when you do, think about does it have those three things? Does it require simultaneity, simultaneous thought, anticipatory thinking, and is it about finding equivalent problems that are easier to solve? There you go. So, in this series, we're talking all about equivalence problems, finding equivalent problems are easier to solve using equivalence. So, there was a time when I was working early. Hey, I made a lot of mistakes when we did this early stuff. I learned so much. When I was beginning to work with teachers, and we were developing, and I was thinking, and all the things. One day I said, "Wouldn't it be cool if we have video of teachers doing these things with kids, so that other teachers who are new could go, 'Oh that's what it looks like.'" 


Kim  28:26



Pam  28:27

Well, I mean, we've now created the Problem String Hub in our membership site called Journey where you could do exactly that. But in this moment, we had a few teachers who were kind of leaders. They've been on the cutting edge, and so we said, "Would you mind, you handful of teachers? I'll come in with a camera, and I'll video you, and then we'll show everybody else." And the teachers are like, "Oh, we're a little nervous. But yeah, we'll give it a try." One of these teachers. I walked in the door, I set up the camera. And she said, "Okay, is everybody ready? By the way, I will never share this name because your name is safe with me, and so if there's something that we're going to kind of poke out a little bit, I won't tell you who it is. But one of those teachers. "Are you ready?" I was ready. The cameras all ready. She goes, "Okay. Alright. Here we go." She took a deep breath. She goes, "Students, today, you're going to learn the Doubling and Halving strategy. Everybody write that in your notebooks. Doubling and Halving. Okay, that's what we're going to do today. Ready? Here is the problem. 24 times 12. 24 times 12. Okay, is everybody?" Oh, sorry. Sorry, actually, it was 12 times 24. My bad. It was 12 times 24. She goes, "Okay, everybody write that down. Here's how you do the problem. Everybody ready? Ready? You Double the first number. Alright, what is double 12, everybody?" I'm in the back of the room going, "Is she just telling them steps to solve the problem?" Is she doing a non-example? Like, I was like "What is happening right now?" Because she was literally like telling them what to do. That's not how you do a Problem String, right? You do a Problem String, you launch the problem. You see how kids are doing it. She's like step-by-step telling kids to do. "Ya'll, double the first number." So, they dutifully did what she said. She goes, "Everybody double the first number." So, Double 12 that's 24. And then she said, "Alright, now, everybody and now you halve the second number. Okay, everybody do it. What's half of 24? I know that's going to be hard. It's going to be hard to halve 24. Everybody work on that." I was like dying in the back. Not laughing. I was like, "What is happening right now? I finally just shut the camera off. I was like, "Let's just... Ya'll, 12 times 24? Double and Halve that, you get 24 times 12." Then, they kept going. Then, they got 48 times 6. And then, they got 96 times 3. And then, I looked at her face. She's like, "Oh, no. What do we do now?" Because it didn't turn into this nice. She goes, "Let's practice on another one." And then, she gave another problem. So, then, she walked around. And she was like, "Nope. Double that one. Halve that one." There was no discussion about which number she should double. Which number you should halve. There was no letting kids. So, Problem Strings, Doubling and Halving, all of these strategies, are not about direct teaching steps in the strategy. It's about developing relationships in kids heads, so that the strategies become natural outcomes. 


Kim  29:39

Yeah. Yeah. Yeah, and I'll add. We'll go back to the story at the beginning. This is a perfect time. Because the kid in the class that I was filming had heard Double, Halve, right? Somebody along his journey had said, "Double, Halve, this is what you do." So, if direct teaching is what you do, you might get the response like this kid at the beginning, where he's heard a thing, and he's like, "I don't know why you would do that." But after, towards the end of the Problem String. And remember, we have this on film. We have video of him when I posed a problem. And we can hear him gasp a little bit and go, "Oh, so the reason why you'd want to do that is..." And then, he talks about creating an equivalent problem that was easier to solve. He had had an experience that forced him to think about why you would want to do that when it might be useful. And it turned into a lovely video because you could see the learning occurring right in that moment. Yeah (unclear).


Pam  32:06

Yeah, it was so cool.


Kim  32:08



Pam  32:08

It's such a neat moment for him to like first be kind of like, "Yeah, somebody's taught me Doubling and Halving, but I don't know why you would ever want to use that...OH!!" And the gasp was just brilliant. The look on his face. And yeah. So, we're after that, right? Where after like we own those things. It's about reasoning, thinking, and in this case, creating an equivalent problem that's easier to solve. Yeah, super. Alright, 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 Thanks for spreading the word that Math is Figure-Out-Able!