Ep 267: Connecting Strategies, Properties, and Models in Multiplication

If your only focus is answer getting, students may never experience and grapple with and understand multiplicative properties. In this episode Pam and Kim discuss how to use models to help students progress to more sophisticated strategies using multiplicative properties.

Talking Points:

• Modelling different strategies on area models, ratio tables, and equations
• Using models to visualize properties
• Strategies that use the associative property are more multiplicative than the those that use the distributive property 

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Transcript

Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather, who needs to scroll faster to the beginning of the podcast. Oh, gosh. You think I'd have this memorized by now? 

Kim  0:13  
Yeah,

Kim  0:14  
I tried. 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  0:23  
Oh, my gosh, that was so loud. I could just hear our editor. He's like, "And Kim!" Sorry, bud. We know that algorithms are amazing human achievements, and they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than they should be developing and using.

Kim  0:42  
Yeah, in this podcast, we help you teach, mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:49  
Did I kind of sound ornery when I just said that? 

Kim  0:51  
What that I was loud? 

Pam  0:52  
No. Well, that, yeah, sorry too.

Kim  0:54  
I don't care. 

Pam  0:54  
Just at the "using less sophisticated reasoning that should be!" Yeah, okay. Alright, ya'll, we invite you to join us to make math more figure-out-able. It's hot here. Hey, let's give a shout out to Sherry. I'm pretty sure that was Sherry. I think that was your name. Who I met in Florida at FCTM. I had a great, amazing time at FCTM. 

Kim  1:15  
Aw, that's fun. 

Pam  1:16  
Got to hang out with some amazing people. That was... So, all ya'll that we hung out with, that was a blast. Sherry was like, "Pam, I got to tell you. When I listen to your podcast, can you hear me?" And I was like, "Sorry, what?" She's like, "Oh, I talk right back to you and Kim. I just tell you what I think." And I was like, "Um, you just keep on talking back to us." 

Pam and Kim  1:32  
(unclear). 

Pam  1:32  
Anyways, for Sherry, we hear you. We hear you. You keep talking. Keep talking because we hear you. 

Kim  1:37  
Love it. And love it.

Pam  1:38  
You know what? Kim and I talk back to each other all the time. Sometimes less pleasantly than others. But most of the time, it's pretty

Pam  1:46  
pleasant. 

Kim  1:46  
Stop. We're good. We're good.

Pam  1:48  
Most of the times. 

Kim  1:49  
So, today, we're going to... Listen, I found this message that I sent to you. I don't have a date on it, but it was 7:24 in the morning. And I believe it was like Saturday morning. So, honestly. 

Pam  2:00  
Saturday morning.

Kim  2:01  
I think. 

Kim  2:02  
I'm pretty sure it was. So, I sent you this message, and it was something like we're talking about modeling multiplication on both area models and ratio tables. So, we've talked before about how these models are important, and that one of the reasons why we think an area model is important is for Spatial Reasoning. And so, then one morning I was thinking. And I think I asked you have you ever considered that we need both because of the properties that you can...

Pam  2:26  
Wait, both what? 

Kim  2:28  
Both area model and ratio table. 

Pam  2:33  
Okay.

Kim  2:33  
We need both because of the distributive and the associative property. And one of the things I said was as kids become more multiplicatively. They mostly land in ratio table because the associative property can be modeled there, and that it's maybe trickier to see the associative property on an area model. So, I think it just went on to say like, "Ah, properties." And so, we decided to dig that message out and spend some time talking about properties, and strategies, and models.

Pam  3:04  
I love it. Let's do it. But I really quick have to just poke fun a tiny bit that you said "As kids become more multiplicatively."

Kim  3:12  
Yes! It's 7:24 on a Saturday morning. Probably true.

Pam  3:16  
Love it. So, ya'll, you might have heard me say that the ratio table is our model of choice as students get older. But there is a caveat. We actually need both area models or open arrays and ratio tables. But we also need to know which one when, and for what purpose, and all the things. 

Kim  3:38  
Yeah.

Pam  3:38  
Different models. Different purposes. Yeah.

Kim  3:40  
Yeah. So, can we focus on the properties for multiplication that we that we have kids meth with. Meth. Oh, my Heavens.

Kim  3:48  
Mess with. 

Pam  3:49  
We're not going to meth with anything.

Kim  3:50  
No, we're not. So, Fosnot would say that kids group the groups. I think that's the language that she uses. And kids typically start with the distributive property, right? So, that's what kids do first. (unclear).

Pam  4:05  
Yeah, maybe let's (unclear).

Kim  4:06  
(unclear).

Pam  4:06  
Yeah. So, if we had an example of a multiplication problem. They're thinking about 7 times 8.

Kim  4:11  
Mmhm. 

Pam  4:11  
And the very first thing they do is very additive. They need seven 8s, and so they're adding those eights together. But then they start to, like you said, group the groups. And so, they might say, "Well, do I know some eights? Well, I know two 8s, I could double those. Or I know five 8s, and then I could add the leftover eights." So, that's an example of grouping the groups. And we're suggesting, Cathy Fosnot know as well, that that's kind of a first move kids make. First, they're thinking additively, adding all of the product, the factors together.

Kim  4:38  
Mmhm. 

Pam  4:38  
But then they start to group the groups. And we can represent that grouping the groups you're saying with the distributive property. 

Kim  4:44  
Yeah, and on an area model.

Pam  4:46  
Yes. Nice, nice.

Kim  4:47  
So, if you take a look at the strategies that we suggest, are the major strategies for multiplication.

Pam  4:54  
Mmhm.

Kim  4:54  
One of the things that kids typically start with is splitting two by two digits into Place Value Partial Products.

Pam  5:02  
Mmhm. 

Kim  5:02  
So, you can represent that on an area model. So, for example, for a problem like 18 times 25, kids could break up the 18 into a 10 and 8, and the 25 into a 20 and 5. And you can represent those chunks on an area model. Hopefully, it's proportional. 

Pam  5:18  
So,

Pam  5:19  
I've just done that. I've just draw a long, skinny rectangle because 18... Well, it's not too. 18 is shorter than 25, so this is not a square. So, the 18 is shorter. The 25 is longer. And I've cut, like you said, the 18 into 10 and 8.

Kim  5:33  
Mmhm. 

Pam  5:33  
And on the 25 side, I've gone way... So, the 10 and 8 are not... It's not... I haven't cut it in the middle.

Kim  5:39  
Mmhm. 

Pam  5:39  
The 10 side is longer than the 8 side. Though, I really should fix it  because my 10 is way longer than my 8. So, let me actually...

Kim  5:51  
You go ahead and fix that. 

Pam  5:51  
10 is only sort of longer than 8. Okay, so now the 10 is longer than the eight, but only by a little bit. And then the 20 and the 5, the 5 is smaller than the 8, but now it's over here on the vertical. I'm cutting that down. That five is smaller than the 8, and so now I've cut the other side into 20 and 5. So, I have a rectangle cut into 4 rectangles. Although, the bottom right rectangle looks kind of square-ish. It's an

Pam  6:06  
8 by 5. 

Kim  6:07  
Yeah. And so, on the area model, you can see what the traditional algorithm is generally built from. Karen Camp would say that this is double distribution.

Pam  6:19  
Mmhm.

Kim  6:19  
As kids are distributing those multiplications.

Pam  6:22  
Ooh, and so can I represent that with... I did it on an array. But now, I'm going to do it with... 

Pam and Kim  6:26  
Equations.

Kim  6:27  
Mmhm. 

Pam  6:27  
Yeah. So, I've got 18 times 25 equals parentheses 18 is 10 plus 8. So, I've got parentheses 10 plus 8 times parentheses 20 plus 5. And then that double distribution. Or I'm distributing everything. Then I would have 10 times 20. That's the big rectangle, like you said.

Kim  6:45  
Mmhm.

Pam  6:46  
Then I've got... Now, here I could go different orders, but I'm going to go ahead and do 8 times 20. And that is the rectangle below the 10 times 20. Then I've got 8... Sorry. Yeah, 8 times 20. And then I've got the 10 times 5. That's over to the right. And then I've got the little, tiny one, and I'm adding that little chunk down there that is the 8 times 5. 

Kim  7:05  
Mmhm.

Pam  7:06  
And so I'm ending up with those four chunks. Now, I've written them in in an expression. 

Kim  7:12  
Mmhm.

Pam  7:12  
Okay.

Kim  7:13  
So, what's really nice about this using an area model to represent this distribution, this double distribution, is that kids can see the parts that they're multiplying as they're becoming early multipliers of double-digit numbers.

Pam  7:27  
Mmhm, mmhm. 

Kim  7:28  
And this distribution can't be represented on a ratio table when you split things into Place Value Partial Product. So, here's a (unclear)

Pam  7:36  
Well, when you split both numbers, right? 

Kim  7:38  
Yeah, when you split both. So, here's a moment where we need the area model to do this Place Value Partial Product splitting.

Pam  7:45  
Nice. 

Kim  7:46  
And then as students further develop, then we spend time helping them not split both of the factors. We want them to keep one factor whole. And they may choose to split the other factor. And in this moment, we have some choices that we can make where kids can represent this still on an area model. But now that we've kept one factor whole, we can start to represent this thinking on a ratio table. 

Pam  8:13  
Mmhm, mmhm. So, let's

Pam  8:16  
stay with our 18 times 25.

Kim  8:18  
Mmhm. 

Pam  8:18  
What would be a way that you might split up either the 18 or the 25. Which one do you want to keep whole? 

Kim  8:24  
Let's keep the 25 whole. 

Pam  8:25  
Okay, so I'm drawing an area model. Same rectangle I had before, the 18 by 25.

Kim  8:33  
Mmhm. 

Pam  8:33  
It's a little too long. I shouldn't have made it that long. Anyway. So, you said keep the 25 whole?

Kim  8:34  
Mmhm. 

Pam  8:34  
Okay, so then I'm probably going to break it up into 10 by 25 and 8 by 25.

Kim  8:42  
Okay.

Pam  8:42  
But I'm not going to break the 25 up, right? 

Kim  8:44  
Yeah, right.

Pam  8:45  
So, it looks similar to the one I had before.

Kim  8:47  
Mmhm. 

Pam  8:48  
But I only am cutting it in half. I'm not cutting into four chunks. So, now I have to think about 10 times 25. I can do that. It's 250. 8 times 25. I can think about quarters. So, that's 200. 

Kim  8:56  
Mmhm. 

Pam  8:56  
And then I could add those together to get the the total. And bam, that's a whole lot more efficient than the all four chunks that I just did.

Kim  9:04  
Right. 

Pam  9:04  
But I could also represent that... Do you want me to ratio table or (unclear)?

Kim  9:08  
Well, I just did that on my paper. As you were drawing, I put the same partials.

Pam  9:13  
Okay.

Kim  9:13  
On a ratio table. So, I have a vertical ratio table that I started one to 25.

Pam  9:18  
Because you kept the 25 whole, so you're thinking about 25s, mmhm.

Kim  9:22  
Mmhm. So, then I have 10 and 250. So, ten 25s is 250. So, you have that same partial that you have on your area model, I have in my ratio table.

Pam  9:31  
Sure enough. 

Kim  9:31  
And then I have 8 to 200. So, eight 25s is 200.

Pam  9:36  
Nice. And then you can add those together to get 18. 

Kim  9:38  
And

Kim  9:39  
what a beautiful moment to compare the same strategy on two different models.

Pam  9:44  
Yeah, nice. Yes. And we can cement, get clinch. Clinch? Cinch. Cinch that.. What in the world? That strategy better if we can see the relationships into... And we can actually do a third model.

Kim  9:57  
Mmhm. 

Pam  9:57  
Because now we can write that 18 times 25. So, I'm writing an expression here. 18 times 25. I'm going to then equals. We're keeping a 25 whole, so that's going to be 10 plus 8 times 25. And then distribute 10 times 25 plus 8 times 25.

Kim  10:14  
Yeah. 

Pam  10:14  
And then I could solve each of those. And now, we have three models that are representing that same strategy.

Kim  10:20  
Mmhm. 

Pam  10:20  
Again, all using the distributive property. 

Kim  10:22  
Mmhm.

Pam  10:23  
So, smart. So, Place Value Partial Products. And Smart Partial Products all use the distributive property.

Kim  10:29  
Mmhm. 

Pam  10:29  
Okay?

Kim  10:31  
And then we often see kids. We nudge them towards Over strategy.

Pam  10:36  
Mmhm.

Kim  10:37  
So, students might look at a problem like 18 times 25 and think, "Oh, that's really close to 20 times 25."

Pam  10:44  
Sure enough. 

Kim  10:44  
"So, I'm going to solve 20 times 25. And you could draw. Let's have you draw that. 

Pam  10:50  
Okay. 

Kim  10:50  
Sketch out for us a 20 times 25. 

Pam  10:52  
So, now,

Pam  10:53  
the rectangle I'm drawing is close to the two that I had before, but it's actually deeper.

Kim  10:59  
Mmhm.

Pam  11:00  
Because now it's 20 by 25 not 18 by 25. And then I cut off the bottom 2, so the inside I have an 18. So, I've got a whole rectangle that's 20 by 25. And golly, that would be 500 is the area of that whole thing.

Kim  11:13  
Mmhm. 

Pam  11:13  
And then get rid of the two 25s which is 50. And so, 500 minus 50 is that 450 that we've gotten for all the other ones.

Kim  11:20  
Mmhm. 

Pam  11:20  
What would the ratio table look like for that strategy?

Kim  11:23  
Same. So, I have the same parts that you have. I put a 1 to 25.

Pam  11:28  
Mmhm. 

Kim  11:29  
And then I actually put 20 to 25. Kids might think about 2 to 25. 

Pam  11:35  
To scale up the 20.

Kim  11:37  
Then scale up to 10. 

Kim  11:38  
So, I just went straight to 20 times 25 is the 500.

Pam  11:39  
Mmhm. 

Kim  11:40  
And then back up the 2 to get to 450. Back up to two 25s is 50. (unclear). 

Pam  11:47  
So,

Pam  11:47  
your ratio table... Oh, sorry. 

Kim  11:49  
No, go ahead. 

Pam  11:50  
Your ratio table might look 1 to 25. 20... Yours would look like 1 to 25, 20 to 500, 2 to 50, 18 to 450. 

Kim  12:00  
Mmhm.

Pam  12:00  
But you could see a kid that might say 1 to 25, 2 to 50, 20 to 500, and then the 18.

Kim  12:00  
And they have the two already there. Which is really cool. 

Pam  12:00  
Yeah, once you got the 2 to get to 20, bam, it's there to subtract from the 20. 

Kim  12:07  
Yeah.

Pam  12:08  
Yeah (unclear).

Kim  12:09  
So, they're thinking a lot about the same relationships but can represent it a little differently.

Pam  12:17  
And so, then the distributive property kind of becomes like glaring if I write it with an expression. So, I've got 18 times 25. And this time I'm writing that as 20 minus 2. That's in parentheses. So, the quantity of 20 minus 2 all times 25. When I distribute that's 20 times 25 minus 2 times 25.

Kim  12:37  
Mmhm. 

Pam  12:37  
And then those are the same values as that 500 minus 50. Which is 450.

Kim  12:43  
So, at this point, we've used the area model as a starting place to talk about Place Value Partial Products. And we can represent both Smart Partial Products and Over strategy with either an area model or ratio table. And then we get to Five is Half of Ten.

Kim  12:58  
Mmhm. 

Kim  12:58  
And so, Five is Half a Ten is another brilliant strategy that could be on either an area model or on a ratio table. So, 18 times 25 on an area model. What are you sketching, Pam?

Pam  13:13  
I'm going to sketch the same one that I had in the first two. So, it is an 18 by 25.

Kim  13:19  
Mmhm. 

Pam  13:19  
And I'm going to break that into... I'm going to keep the 18 whole.

Kim  13:21  
Mmhm. 

Pam  13:21  
And I'm going to break it vertically into 20 and 5. So, I've only cut the area, the rectangle, in two chunks. One of them is a rather large chunk. It's 18 by 20. And then a small chunk that's 18 by 5. Then you might be like, "Pam kids don't know 18 times five?" Well, that's where the Five is Half of Ten strategy comes in. Do I know... And this is actually hard to put on an area model because now this is probably happening either in my head or with an expression next door. But I'm actually thinking about 18 times 10 is 180, and so half of that would be 90. So, the 18 by 5 is 90. And that's where the Five is Half of Ten comes through. So, I'm not actually loving the area model to show that I used Five is Half of Ten in this problem. 

Kim  14:07  
Because you

Kim  14:08  
didn't show the 10 in your area model. 

Pam  14:10  
Yeah, yeah. 

Kim  14:11  
So, I think I'm thinking about really young students as they're thinking about Five is Half of Ten for the first time.

Pam  14:16  
Mmhm.

Kim  14:16  
And they might think about 10 and 10 and 5. I think that's my third (unclear).

Pam  14:21  
Ah, okay, okay, okay.

Kim  14:24  
But

Kim  14:24  
yeah. But here's a beautiful moment where, as an early multiplier, the area model might make sense because you're breaking up that 25 into 10, 10, 5. 

Kim  14:34  
Yep. 

Kim  14:34  
And... 

Pam  14:34  
Okay, okay. 

Kim  14:34  
...you might, as students become more sophisticated, it might make more sense to put it on a ratio table. 

Pam  14:35  
So, what does your ratio table look like for this one? 

Kim  14:41  
I did 1 to 18.

Pam  14:44  
Ah, so now you're keeping the 18 whole. That seems... Like, I wish everybody could see my iPad right now. It's actually a work of art.

Kim  14:52  
Is it just enough for you? Or are you communicating to everyone?

Pam  14:55  
I think I would be communicating to everybody actually at this point. Yeah. So, I've got the problem to the left. The area models are all lined up in a column. And the ratio tables are all lined up in a column. And the equations or expressions are all lined up in a column. I didn't write, I didn't draw column marks, but they're just kind of neatly, so we can compare now. And you can clearly see the distributive property happening in all of those. Yeah. In the strategies that we've done before. So, keep going. You have 1 to 18. All the ratio... The reason I said that was all the ratio tables before have 1 to 25.

Kim  15:17  
Mmhm. 

Pam  15:17  
But this one's got 1 to 18 because you're keeping the 18 whole.

Kim  15:32  
Yeah. 

Pam  15:32  
Yeah. 

Kim  15:32  
And then I did twenty 18s is 360.

Pam  15:36  
Mmhm. 

Kim  15:36  
And then 5 is 90. And in that moment, when I broke down the 5, I went 10 would be 180, 5 would be 90.

Pam  15:45  
So, you could have put 10 to 180.

Kim  15:47  
I could have, yeah.

Pam  15:47  
And 5 to 90 and just have the extra. It's okay to have extra in the ratio table, right?

Kim  15:51  
Mmhm

Pam and Kim  15:52  
Yeah. 

Kim  15:52  
Okay. So, so far, we've been talking about area models and ratio tables.

Pam  15:58  
Mmhm. 

Kim  15:58  
And then we get to using

Kim  15:59  
quarters.

Pam  16:00  
Before... Well, actually, before we go there, if you don't mind, let me do the expression for this one. 

Kim  16:04  
Mmhm.

Pam  16:05  
So, this is going to be 18 times 25 is going to be equal to 18 times the quantity 20 plus 5.

Kim  16:13  
Mmhm.

Pam  16:13  
Which is 18 times 20 plus 18 times 5. And then we could keep going.

Kim  16:19  
Mmhm.

Pam  16:19  
But I do think we probably should mention that to get the 5, we actually use the associative property. 

Kim  16:27  
Oh, good. I'm glad you said that. Yeah, yeah. 

Pam  16:28  
So, you really can see the distributive property when I write 18 times the quantity 20 plus 5. And so, then 18 times 20 plus 18 times 5. But to find the 18 times 5, here's the one little kind of glitch in my perfect world. We actually use the associative property because it's more about saying if I know 10 times... Well, how do I say this? If I'm looking for 5 times 18.

Kim  16:52  
Mmhm. 

Pam  16:52  
I can write that as 10 times one-half times 18. Maybe I should say I'm going to actually write that as one-half times 10.

Kim  17:02  
Mmhm. 

Pam  17:03  
One-half times 10 times 18. And that's... Now I'm going to... Because one-half times 10 is 5.

Kim  17:09  
Mmhm. 

Pam  17:09  
But I'm going to now ignore the half for a minute and just do 10 times 18. But I still have that half. And so, what's half of 180? So, on my paper, I have 5 times 18. Underneath that, I have parentheses one-half times 10. That's the 5. Times 18. Then, I've moved the parentheses, so now it's one-half is outside the parentheses times 10 times 18. That's  where the associative property is coming in to do the Five is Half of Ten part.

Kim  17:36  
Mmhm. 

Pam  17:36  
But so, we've had long conversations, Kim, about, do we put Five is Half of Ten in in the associated property. Or do we put it in the distributive property? I tend to put it more distributive property. It definitely uses the associative property. But it's because we rarely just find times 5. 

Kim  17:54  
Right? 

Pam  17:54  
We almost always use that to get, in this case, 25 times something. Or 15 times something. Or 50. Or 49 times something. 

Kim  18:00  
Right, right.

Pam  18:01  
We're almost always using it. The reason we did it was to get a chunk to use in the

Pam  18:05  
distributive property. 

Kim  18:06  
Yeah. Yeah, I think that's important to note because the properties matter, right? And we want to name what's actually happening. But because we're getting 5 for another reason, I think it's this interesting mix of

Kim  18:21  
both.

Pam  18:21  
Mmhm, mmhm. Okay, so you were about to go to another strategy. 

Kim  18:24  
Quarters. 

Pam  18:25  
Woohoo! 

Kim  18:25  
Yeah. Yeah, yeah.

Pam  18:27  
Alright, 18 times 25. Which model are you going to use for that one?

Kim  18:32  
I'm going to use... Ooh. Well, I almost said I was going to use equations, but... 

Pam  18:36  
I'll let you. 

Kim  18:36  
Can I use? I can use equations? 

Pam  18:39  
Mmhm.

Kim  18:41  
Well, and so we're talking about area models and ratio tables, so I'm going to use a ratio table.

Pam  18:46  
Okay. 

Kim  18:46  
Because factoring on an array or a  area models really tricky. 

Pam  18:51  
I think it's hard to show. 

Kim  18:53  
Yeah. Yeah. 

Pam  18:55  
Yeah, it's not natural.

Kim  18:56  
(unclear). Yeah, so I'm going to think about... Well, no, I think I'm going to write equations. Have we talked about equations being an appropriate model? 

Pam  19:05  
Yes.

Kim  19:06  
Okay. 

Pam  19:06  
Yeah, go for it. 

Kim  19:07  
Alright, so I'm going to think about 18 times 25 as 2 times 9 times 25.

Pam  19:14  
Okay.

Kim  19:14  
And I'm going to say that I'm going to reassociate the 2 with the 25. I'm going to make it be 9 times 50. 

Pam  19:22  
Okay.

Kim  19:22  
So, I'm going to call that 450.

Pam  19:25  
And I might call that more of a Doubling and Halving strategy than using quarters. Would you be okay with that?

Kim  19:34  
If you want to.

Pam  19:36  
Just

Pam  19:36  
because you pulled out a 2. 

Kim  19:37  
Okay. 

Pam  19:38  
So, I might. I get that you were thinking about 2 quarters.

Kim  19:44  
Mmhm. 

Pam  19:44  
And so, that totally makes sense. I'm going to sort of... The names aren't important. The relationships are more important. So, I would let you call that I was thinking about quarters. But it wouldn't be the strategy I refer to as using quarters and scaling.

Kim  19:59  
Oh, that's so interesting, because when you talk about using quarters, you normally think of like fourth of stuff. 

Pam  20:04  
I do.

Kim  20:05  
And I think you and I have beat this out before that I think it's either or.

Pam  20:09  
You and Sue are on the same page on this thing. Too bad. So, if I were to use the... So, let's do with Doubling and Halving. Would you agree that you just Doubled the 25 and you Halved the 18.

Kim  20:23  
Sure, yeah.

Pam  20:24  
Okay. And a way that you did that was you were factoring. You were like, "Hey, I can factor that 18 and grab that nice 2 out of there." So, we have a strategy called Flexible Factoring. And the way that you just did that, I might actually say was more Flexible Factoring than anything. Because it's kind of like you factored the 18, and you said yourself, "Mmm, I can do that in a nice way to take advantage this quarter over here." So, an example of Flexible Factoring, I think, is when you grab a factor like you just did with the 2, and it results in Doubling one number, and you grabbed it, so it Halved the number that you grabbed it from. 

Kim  21:02  
Would you 

Kim  21:03  
agree with me that Doubling and Halving is a smaller, more specific version of Flexible Factoring that both use the associative property?

Pam  21:14  
Absolutely.

Kim  21:14  
You just won't call mine using quarters. You have to say more about that.

Pam  21:19  
I think you use quarters inside of a Flexible Factoring strategy.

Kim  21:23  
Okay, alright, I can go with that. 

Pam  21:26  
Okay, (unclear).

Kim  21:26  
(unclear). What's your using

Kim  21:27  
quarters? 

Pam  21:28  
Okay, so my using quarter strategy would be to think about 18 times 25 as 25 times 18.

Kim  21:34  
Mmhm. 

Pam  21:35  
And then say to myself I'm going to find a fourth of 18. And when I said a fourth of 18, I wrote down 0.25 times 18. 

Kim  21:44  
Okay.

Pam  21:45  
But I'm thinking about a one-fourth of 18. And so, I might say to myself, "Well, half of 18 is 9, and so a half of that is 4.5." That's how I found a fourth of 18. I could have just divided 18 by 4 but whatever. So, I've got a fourth of 18 is 4.5. But I don't want one-fourth of 18. I don't want 0.25 times 18. I want 25 times 18. So, that's times 100. So, if I scale that times 100, then I scale the 4.5 times 100, and that gives me the 450. So, to me, using quarters and scaling is very much thinking about 25 times things and 75 times things, as can I find a quarter of it, and then scale up. Or can I find three-quarters of it and scale up. And then I could either add or subtract 1 or 2 to find 24 times something, 26 times something.

Kim  22:31  
Mmhm. 

Pam  22:31  
74, 76 times something. That's the strategy that I call Using Quarters and

Pam  22:31  
Scaling. 

Kim  22:31  
Yeah. So, we just kind of talked about using quarters Doubling and Halving and Flexible Factoring. So, I don't know how much you want to dive into those more deeply, but (unclear).

Pam  22:51  
I

Kim  22:51  
think we should talk about models. Oh, go ahead. (unclear).

Kim  22:54  
Well, I was going to say. All of those strategies rely on or are based on the associative property. So, the more sophisticated strategies, towards the end of what students mess with, are all associative.

Pam  23:04  
Which also means they're all more multiplicative. 

Kim  23:08  
Yeah.

Pam  23:08  
Because if we look at all the ones that we did in the distributive property starting with like Place Value Partial Products, we split the factors additively. So, like we split 18 into 10 and 8. And 25 into 20 and 5. That was the Place Value Partial Products. If we did Smart Partial Products, we split the 18 into 10 and 8. We left the 25 whole. But notice, we split the 18 into 10 plus 8. So, we like partitioned the factors. Partition is a way to talk about splitting the numbers kind of additively. Same when we did the Over strategy. We kind of made 18 into 20 minus 2. That's an additive relationship. But as soon as we start doing Doubling and Halving or Fexible Factoring, or Using Quarters and Scaling, now it's all based on factoring the factors.

Kim  23:54  
Right.

Pam  23:54  
We're not... We're splitting up the factors, but we're splitting them up multiplicatively not additively.

Kim  23:59  
Mmhm. 

Pam  23:59  
And that's more sophisticated. I have to be reasoning multiplicatively for that to happen. Ya'll, this is one of the reasons why I push so hard against the multiplication algorithm. Because if the only thing you ever do is rote, memorize a bunch of single-digit facts, and then apply them in the multiplication algorithm, you never develop either the distributive property. Maybe a little bit. But not with Over strategy for sure. But then for sure, you never get to these associative property multiplicative strategies where you're really factoring the factors in order to make an equivalent problem that's easier to solve. That's a big deal. Like, you haven't filled out your multiplicative reasoning at all. Like, sometimes people will say, "Pam, my kid's in the multiplication algorithm. They're thinking multiplicatively about all those single-digit facts." Okay, but that's like the most multiplicative they ever get. And there's a whole lot the area? The area? Can I use that word? The density. The part of the developing mathematical reasoning domain that is multiplicative reasoning is dense. It's not just reasoning with single-digit facts and getting a bunch of answers. There's more distributive property, kind of partitioning the factors thinking. And then there's the more associative property, factoring the factors thinking. And we want to build that in kids, and we can. The answer is not only can we, but it's fun. 

Kim  25:25  
Yeah.

Pam  25:25  
Bam. Okay.

Kim  25:28  
(unclear) important to note that it's important teacher knowledge to know how the properties and the models, the strategies are all connected.

Pam  25:36  
Mmm, nice.

Kim  25:36  
But kids don't need to name this stuff. This is not a conversation where you say, "Today we're going to work on using quarters, and that's associative, so we're going to only record it on a ratio table. This is you to know, so that when you're modeling students thinking, you can grab for the model that is

Kim  25:55  
actually useful. 

Pam  25:56  
Yeah, that's nicely said. So, that's why in our Problem String books, in Developing Mathematical Reasoning - Avoiding the Trap of Algorithms, in our Hand to Mind - Foundations for Strategies kits, you're going to see that we emphasize open arrays, ratio tables, and equations, but very specifically in specific orders, and at specific times, and with specific strategies. 

Kim  26:18  
Yeah.

Pam  26:18  
Ya'll, if you haven't downloaded our major strategies ebook yet, we invite you to download that sucker at www.mathisfigureoutable.com/big because it is a big addition to what we know about the major models and strategies, and that will help you see how we model the different strategies with which models and the order of sophistication that we teach those strategies in. Cool. Ya'll, 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. And keep spreading the word that Math is Figure-Out-Able.