Pam Harris  00:00

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

Kim Montague  00:07

And I'm Kim. 

Pam Harris  00:08

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. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim Montague  00:35

Okay, so Pam, quite a while back, you did an episode about the Doubling and Halving strategy for multiplication, right? It's one of our episodes. 

Pam Harris  00:42

Which we totally love, right? 

Kim Montague  00:43

Yes. 

Pam Harris  00:43

We love, love, love. Uh huh.

Kim Montague  00:45

And I think that in that episode, we talked about how the name itself that we call it, Doubling and Halving, is a little bit too constraining. Let me give the listeners just an example. Let's say that you're given a problem like 50 times 16, you can create an equivalent problem that might be easier to solve by doubling one of the factors and having the other factor. So in this example, you could double the 50 to make 100 and half the 16 to make eight and then you have a new equivalent problem of 100 times eight, which is really pretty slick, right? 

Pam Harris  01:16

Yeah. Very nice.

Kim Montague  01:16

But the problem is, that's not just true for doubling and halving. You can also triple one dimension and find a third of the other or quadruple one dimension and find a fourth of the other and on and on. And so for years now... 

Pam Harris  01:30

In fact, can I just do that with that problem? 

Kim Montague  01:32

Sure. 

Pam Harris  01:33

Because you got to 100 times eight, but we could keep going. We could go down 200 times four. 

Kim Montague  01:38

Yep. 

Pam Harris  01:38

We could keep going to, I could do 400 times two. And a verily, we could even keep going to 800 times one, which means we could have gone from the 50. We could have quadrupled the 50 to 200 divided the 16 by four to four. 

Kim Montague  01:55

Yep. 

Pam Harris  01:55

And so on. That's an example of quadrupling, quartering, or times eight divided by eight, like 50 times eight to get the 400 and 16 divided by eight to get the two. So that, yeah, there's an example. Keep going.

Kim Montague  02:10

So for years, you've kind of lamented a little bit about how the name that even you and I give it is a little bit troublesome because it's maybe not all encompassing enough. And you have promised that if anyone can come up with a name that's better than Doubling and Having maybe more, it's mathematical, but it's slick, that I'm pretty sure that I've heard you say a number of times that you would send them a free shirt.

Pam Harris  02:36

A free Math is Figure-Out-Able shirt. Wow. Yeah.

Kim Montague  02:39

So recently, we have a listener who sent in a suggestion. And this is from Alexandra Ferrell, and she writes, "I'm currently doing your multiplication workshop." Yay.

Pam Harris  02:50

Yeah. Thanks for taking the workshop. Glad you're loving it.

Kim Montague  02:53

She does it in the evenings while she's cleaning the kitchen and making funny teacher T-shirts. And she says, "I'm a huge podcast fan. Don't stop those. They bring light to my Tuesdays." Thanks, Alex. "I've been thinking about the Doubling and Halving name, especially when I walk my dog. I think more of switching the factors out." And she gives an example of 28 times five, which is seven times four times five, which then could be seven times 20. And she says, "Maybe..." 

Pam Harris  03:23

Wow, that was kind of fast. Can I say that again? 

Kim Montague  03:25

Sure. 

Pam Harris  03:25

Twenty eight times five. The 28 is seven times four. So still times five, and then she's gonna pull the four times five together to get 20. 

Kim Montague  03:33

Yep. 

Pam Harris  03:34

Which leaves the seven hanging over. So she's thinking about 28 times five as the equivalent problem seven times 20. Got it. 

Kim Montague  03:41

Sure. Yep. 

Pam Harris  03:41

Yep. Thanks. 

Kim Montague  03:42

And she says, "Maybe you could call it the Factor Switch." Factor Switch. So Pam, what say you? What do you think of Alexandra's name, Factor Switch?

Pam Harris  03:53

Well, So Kim, I'm also kind of curious to know what you think about it. Do you wanna start?

Kim Montague  03:59

Sure. Um, so I have to say that I like it a lot more than many other names that we have had suggested over the years.

Pam Harris  04:08

Yes, yes. We have lots of people suggest things that we don't bring on the podcast right? So that says something here that we're actually talking about it.

Kim Montague  04:15

So I don't think I love Factor Switch. But in the same way that I don't love Doubling and Halving. It just, uh, hmm. How do we say this? It doesn't sit quite right with me. And I'm not sure - Factor Switch. Switch in what way? It's moving from one factor to another, it's decomposing and then re-composing. 

Pam Harris  04:42

Yeah. 

Kim Montague  04:43

I don't hate it. You know, I have struggled a little bit about this because I almost want to just say it's the associative property at work, but that's a little too broad. It doesn't.

Pam Harris  04:55

I believe Cathy Fosnot says that. She's like, she does. 

Kim Montague  04:57

Me and Cathy. It's too broad.

Pam Harris  05:00

She's like, it's - well, it's interesting, because when I was, when my numeracy was bad, which was for a very long time, and I was reading her work, and she said, "See, it's the associative property." I was like, "Huh?" 

Kim Montague  05:10

Yeah.

Pam Harris  05:12

And to be clear, she didn't, at least with that part that I was reading, there wasn't a lot of let me write out the factors. And then - 

Kim Montague  05:21

Yes, unpack it. 

Pam Harris  05:22

Unpackit, so I could see the parentheses shift. 

Kim Montague  05:24

Yes.

Pam Harris  05:24

So I can actually see the associative property. But when she would say see Doubling and Having is the associative property. I had no, I could not, like, nope. 

Kim Montague  05:33

Yeah. Well, I feel like we would have to make some assumptions about kids knowing that name, that is a little bit too, umm, yeah. 

Pam Harris  05:44

It's a long big word. 

Kim Montague  05:45

Yeah, you and I are not super big on that kids must know the name associative property. We want them to use it. Right? 

Pam Harris  05:50

Exactly. And so that's probably where, oh go ahead.

Kim Montague  05:54

Yeah, I was just gonna say that's kind of where I am with the factor switch thing. Like if they don't look at a problem, like her example, I don't remember what it was now. But 28 times five, they don't look at 28 and automatically go, "Oh, that's seven times four."

Pam Harris  06:10

Well, or we could even... 

Kim Montague  06:11

It almost feels like...

Pam Harris  06:12

We could change your example slightly, because she's doing really a factor switch. But if we kept that 28 times five, and made it be 14 times two times five, so 14 times two is the 28. And for listeners, what I meant by the parentheses in the associative property, so I've written 28 times five, and then under it, I've written 14 times two in parentheses, times five, because the 14 times two is at 28. And then in the next line, I would write 14 times two times five, like again, but I would put the parentheses now around the two times five.

Kim Montague  06:43

Yes. 

Pam Harris  06:43

I've got 14 times two times five, that becomes 14 times 10. That would be a true Double/Half. Right? 

Kim Montague  06:51

Yeah. 

Pam Harris  06:51

Because we're doubling and halving. 

Kim Montague  06:52

Quadrupling.

Pam Harris  06:53

She was quadrupling and quartering, right? No, yes? Yeah, I had to think back what she did, looking at my double up. So she was doing much more the kind of advanced example, the more general, that's what I wanted, the more general example, where, "Oh, I could pull the nicest factor out and reassociate it to with a different factor. I could switch the nicest factor to make, you know, like a really slick problem." That is definitely a higher, more complicated, more sophisticated strategy. So part of what Kim and I are looking for when I've given this or what I'm looking for when I've given this challenge to say, "Hey, come up with a better name." So it's kind of not fair, Alexandra, because what we're looking, what I'm looking for is something that fits when kids start doubling and halving. And when kids start, well, I want it to fit then, and then I want it to grow with the student. And that's part of why I don't like Doubling and Halving, Doubling and Halving fits really well with the young students. When, I mean, I was working with young students who were just beginning to think about multiplication, we were doing a Problem String, and it was something like two times eight. And I was doing on an open number line. So I had two jumps of eight with 16. And then the next problem was four times eight. And there were kids that were saying, "Well, I can just think about two of those six teens." And I'm like, "How can you, why can you think about two 16s? The problem is four times eight." And they're like, "Well, because those two eights are in that 16. And so I can, if the group is twice as big, I only need half as many of them." Bam! I mean, they're really messing right in that moment. Now, they don't have down, they're not like constructed completely Doubling and Halving. But it's a brilliant beginning. And we can build on that as they sort of group the groups together. And they're thinking about, "Well, I can think about if I have a group that's twice as big, I only need half as many of those groups." We would obviously want to build that with an open array as well, with an area model. But anyway, that's why, I still might send Alex's shirt. 

Kim Montague  08:58

That would be lovely. 

Pam Harris  08:59

But I probably, so yay, Alex gets a shirt, but I'm probably not going to switch names, yet. 

Kim Montague  09:05

Yeah. 

Pam Harris  09:06

What we might do, what I'm starting to consider is that maybe I'm going to start encouraging teachers to let the name Doubling and Halving kind of grow up with the students. So maybe Doubling and Halving becomes something like using the associative property or the factor switch, or what I've called Flexible Factoring. So actually, if you think about our major strategies, free ebook download, the last multiplication strategy that we would list, the most sophisticated, the most multiplicative of the multiplication strategies, is what I call Flexible Factoring, which I think is what Alexander was calling factor switch. 

Kim Montague  09:44

Yep.

Pam Harris  09:45

Yep? 

Kim Montague  09:45

Yep. Yeah,

Pam Harris  09:46

There you go. 

Kim Montague  09:47

All right. So for the last few weeks, we've been sharing or thinking about the Over strategy, a totally different strategy.

Pam Harris  09:55

I'm just totally laughing, because we just went on and on about that. And that's the topic of today's podcast. So hold on tight everybody. This one might be a little longer. Alright.

Kim Montague  10:04

So two weeks ago, we talked about addition and last week with subtraction. And so this week, let's tackle a little bit of multiplication. 

Pam Harris  10:12

Let's go over, let's go over a multiplication. 

Kim Montague  10:15

Okay, so this week, Pam, I'm gonna give you some problems. But before we dive into a Problem String, let's give the listeners a chance to think about a problem. And this problem is 99 times 79. So 99 times 79. Go ahead and pause the podcast and think about that problem. And then when you're ready, come on back, and we will have Pam talk about her thinking. (pause) All right, Pamela, you ready?

Pam Harris  10:44

I'm so ready. Bring it on, Kimberly. 

Kim Montague  10:45

Okay. All right, your first problem is eight times eight.

Pam Harris  10:51

Eight times eight, eight times eight is actually a fact a lot of kids know. Eight times eight is 64.

Kim Montague  10:57

Okay, cool. 64. All right, your next problem is eighty times eight.

Pam Harris  11:03

So we have this nice times 10 thing. So if I scale up eight times to 80, there's times 10, I can scale up to 64 times 10, which is 64 tens, or 640. But I'm starting to think a little spatially, so I've actually just drawn an eight by eight, and I've labeled the area as 64, an eight by eight rectangle. And then next to it, or underneath it, actually, I've drawn a really tall eighty by eight, so it has the same width, but it's 10 times deep, 10 times tall. Then 80 by eight and that area is 640. Okay?

Kim Montague  11:39

Okay, nice. Your next problem is 79 times eight.

Pam Harris  11:44

So I only need 79 eights. So before I had drawn that 80 by eight, I had drawn 80 eights, but only need 79 eights. So I'm going to (click) hack off, (chush, chush) just at the very end there. I'm gonna cut off one of those groups of eight. And so 640 minus eight is 632.

Kim Montague  12:03

Okay, next one.

Pam Harris  12:05

Okay. 

Kim Montague  12:05

What about 100 times 79? 

Pam Harris  12:09

One hundred times 79, I can also think about as 79 hundreds. And so 79 hundreds is the name 79, or is the number 7900 or 7900. Okay. Ninety-nine times 79. So if I've got 100 seventy-nines, I might draw that as like, really tall, 100 by not as wide, 79. A hundred by 79, I've got 100 groups of 79. So it's almost like, if I'm drawing the groups, on that rectangular array, on that area model, I would draw a line across the top to kind of cut off a group of 79, there's one group of 79 and then underneath that, I would do another group of 79. So I'm kind of drawing a bunch of horizontal lines now to kind of chunk off these groups, one group of 79, two groups of 79, three groups of 79. 

Kim Montague  13:02

Stacked on top of each. 

Pam Harris  13:03

All stacked on top of each other, making this rectangle, completed rectangle. And if I were to keep doing that, but I'm gonna skip all the middle, and I'm gonna go all the way to the bottom, and I'm gonna go well, there's the 98th group of 99. And the very last group would be the 99th group of, I said 89, 79. The last group of 79, I would have the 100th group of 79. And so that very last one will be the 100th group, but I don't need 100. You gave me the problem ninety-nine 79s. So I'm going to cut off that last group of 79 to be just 99 of them. And so now I'm thinking about what is that total area of 7900, subtract 79? That would be 7821.

Kim Montague  13:50

So I'm going to pause you for just a second. How did you know 7821?

Pam Harris  13:55

Yeah, so literally, I wrote down 7900, I already had that from the problem before, minus 79. And then I wrote down 7800, because I knew once I subtracted 79 I had to go behind, below that 900. I was at 7900. I know I'm subtracting something within 100. So I'm gonna be in the 7800s. And then I said to myself, what's the partner of 79 to 100? 21.

Kim Montague  14:25

Yes, it's almost like within a 7900 There's a 7800 and an extra 100. And you just play I Have, You Need with that last 100. 

Pam Harris  14:35

Oh, that's very well said. In fact, sometimes I'll model it that way. I'll draw on the board a number line, and I'll put on the left hand side 7800 And on the right hand side 7900 And I'll say, "Ok what we're trying to do is track 79." So I'll draw a bubble that looks, a jump that looks about you know, 80ish. And I'll say, "So I'm looking for this other bit, right? I'm looking for this bit between 7800 and 7900. That little bit to get me up to that jump of 79. Oh, that's the partner of 79 to 100" Bam! If I can think about that partner, then I'm good.

Kim Montague  15:10

Yeah. And you and I play I Have, You Need often and so you just know that partner. But if you didn't know that partner, the minus 79 could have been really nicely what we talked about last week.

Pam Harris  15:24

Nice. Yeah, yeah, like, subtract 80. So 7900, subtract 80 will be 7820. And I subtracted too much. How much too much? Just one too much. So yeah, pop back up that one, 7841. Nice. So yeah. Good connecting to last week. I like it.

Kim Montague  15:41

You ready for your next problem? 

Pam Harris  15:43

Yeah, go ahead. There's another problem?

Kim Montague  15:44

Okay.

Pam Harris  15:44

Okay.

Kim Montague  15:46

Yeah, just a few more.

Pam Harris  15:47

Well, because let's check before you go on, because that's the problem we started with, right? Ninety-nine times 79? 

Kim Montague  15:52

Oh, yeah. 

Pam Harris  15:53

You stuck me in a place where I was thinking over. And so I was able to sort of think over and kind of solve it that way. Okay.

Kim Montague  15:59

All right, one tenth times 79. Point one times 79. 

Pam Harris  16:05

Oh, that's interesting. Because you said "one tenth" I actually wrote the fraction, one divided by ten.

Kim Montague  16:09

Oh, sorry about that. 

Pam Harris  16:11

I mean, I can think about it the other way, right? A 10th of 79 is 7.9. It's 79 divided by 10. And just like I can think of 70 divided by 10 is seven. Then I can think about nine divided by 10 is nine tenths. So that's seven and nine tenths or 7.9.

Kim Montague  16:30

All right, last problem. Ninety-nine and nine tenths, 99.9 times 79.

Pam Harris  16:37

Oh, now it almost makes me wish I was on a ratio table. Let's see. I know what 99 times 79 is because we had that before that was 7821. So I really just need nine tenths of a 79. And luckily, oh, no, that's not what I want to do. I could find nine tenths of 79. But I don't have that. My bad. "Pam, we're doing and Over string. Why aren't you thinking over?" "Well, I wasn't till now. But I am now." Do you like listening to me think out loud? So (laughs)

Kim Montague  17:08

It's real life.

Pam Harris  17:10

Ninety-nine is almost 100. Since I know one hundred 79s is 7900. I'm just gonna get rid of a 10th of 79, which was 7.9. So now I've got 7900 minus 7.9. I'm going to think about that as 7900 minus 8. So 7900 minus eight. Now, there's some place value happening right here in my head. I got to think about 7900 minus eight. That's 7892. 

Kim Montague  17:38

Yeah. 

Pam Harris  17:39

But I subtracted a bit too much. How much too much? Just a 10th. So I'm gonna tack that 10th back on. So my answer is 7892 and 1/10. Point one.

Kim Montague  17:49

Nice.

Pam Harris  17:50

You think?

Kim Montague  17:50

Nice. Yeah. So much easier, right, than a traditional algorithm. 

Pam Harris  17:56

I mean so much more meaningful. 

Kim Montague  17:58

Oh, my gosh. 

Pam Harris  18:00

And all the place value that I just used, place value, not place labeling, but where I actually had to think about the values involved. And check it out, I would submit that the steps I just did, you could keep track of everything I just did. And maybe it might be the same. Now I still think it's fewer. I think it's more efficient than the traditional algorithm. But more importantly, I'm using multiplicative reasoning. I'm helping my brain think more and more multiplicatively, get better at those relationships. And then my brain just gets more dense. I just have this better, more sophisticated, more grown up, more mature, more dense. A lot more connections become more natural outcome for me that I can use, right? 

Kim Montague  18:48

Yeah. And I love the fact that you were going to solve it one way. And then, no, for real, because then you were like, "Wait a second, I know something that's going to be even slicker." And that's what we want for our kids, right? We want them to go, I'm going to look at these numbers. I'm going to think I'm going to do something and then go, "Wait a second. I know something else." The "I know something else" is the really cool part that we want for kids. Choices. 

Pam Harris  19:15

Choices. I'm so glad you said that. Yes. Yes. If we want to really create kids who are mathematizing. Mathematizing people have choices and they play around with relationships and they choose the one that is slicker for them that day, based on what's pinging for them that day. And the only way we can have that happen is when we have choices, when we own, that's why we're so excited about being dense. Right? That's what dense means. I have multiple connections, multiple relationships that I can rely on, I can go to and use one. Thanks. Thanks for thinking that was an okay move. Hey, and thanks for giving me the permission. I remember the day, I remember where I was standing. When you said, you solved a problem, andI was like, "Whoa, whoa, whoa, whoa, whoa, tell me what you just did." And you were like, "Well, I thought about this, I thought about that. And then thought about this." I was like, "That was the best." And I was like, "You don't just know the best way right off the bat?" And you laughed right at me. You're like, "Of course I don't, I play around with the relationships." That freed me up to know that's what mathy people do. And I can be mathy too. Y'all, we can all be mathy, too. It's not about knowing 'the thing' right off the bat. It's about using what you know, solving the problem seeking for clever, efficient strategies. That's mathematizing. 

Kim Montague  20:33

Yeah. 

Pam Harris  20:34

Love it. So where does Over fit in multiplication? Well, actually, pretty soon. Over is one of the first strategies, once we get kids solving problems using Partial Products, we really quickly go to Smart Partial Products. And one of those Smart Partial Products is the Over strategy. We want kids thinking a little bit too much and then hacking off that little bit of area. It's the place where we tend to use the area model more, and then we move to a ratio table. So let me just say that we use the area model or the open array to help kids build a spatial sense of getting rid of that extra group. It also simultaneously helps kids actually learn what area is. It's not just this procedural, was not the one where I multiply or is that the one where I add the sides together? Like aah! Like area, like we actually have to like own what area is. We can develop that spatial sense. Soon after that we want kids really messing with this idea, or as we're doing that one, because messing with the Over strategy, that is a Partial Product, it's a kind of Smart Partial Product, where they go a bit too much and hack it off. It's also brilliant, because we can model that with the distributive property with equations. So it's super early, we want kids messing with the Over strategy. Now, in this problem string today, we went all the way to multiplying by tenths. We're not suggesting that you're doing that early with students. We just wanted to show you how versatile it is. Y'all, look at that problem, 99.9 times 79. Oh, yeah. And we can get much more complicated numbers. But that was a three digit by two digit multiplication with decimals. But in a few steps, thinking and reasoning multiplicatively. We know - that's where Over sort of fits in with the hierarchy. And so many of our leaders are saying, "Whoa, I'm so glad that you gave us this free ebook that outlines that hierarchy." So many of the leaders in our support system are just like over the top using it with their teachers. So we thought we'd tell you that link again. mathisFigureOutAble.com/big is where you can download our, what's it called, again? 

Kim Montague  22:39

Hierarchy? ebook?

Pam Harris  22:40

Yeah, the Major Strategies, ebook, thank you. So check that out. You might have noticed that I talked about using the area model so I can take off that group. Modeling is super, super important. And also notice the place value that comes out when we have to think about if I have this clunker problem, like 99.9 times 79. What do I know? And what can I use? I'm thinking about numbers that are close to it. And by doing that, that sort of sense of neighborhood and nearness, those are major, big ideas in mathematics. That's getting at place value. So super, super cool. 

Kim Montague  23:17

Yep. 

Pam Harris  23:18

Let me just mention one other thing. If you have kids that are stuck in what we call dumb partial products, one of the things we want to encourage you to do is really think hard about this Over strategy and help your kids, help your students, help yourself move into Smart Partial Products. And oh, the Iver strategy is one of those Smart Partial Products. So if you're doing a four chunker every time, you look at a two by two multiplication, and you draw a square. Y'all, the chances that those two by two numbers actually represent a square and not a rectangle are pretty small. So you should be drawing a rectangle. And then if you're chunking it into four chunks every time as always the place value Partial Products. May we invite you, please consider the Over strategy as a next move that you can make to really be mathematizing, thinking multiplicatively about those multiplication problems, not getting stuck in turning multiplication into a different algorithm. It's not about steps. It's about really reasoning using relationships.

Kim Montague  24:19

Yeah, absolutely. If you've been tuning in for the last couple of weeks, and you've not already snagged the free downloads that we've been mentioning, you are missing out. You'll want to check out this great resource for example problems and visuals for how you can model, represent the Over strategy for yourself and for your students. You can find this one I'm mentioning now in the show notes or at mathisFigureOutAble.com/over.

Pam Harris  24:43

Because it's over. Yeah, great Over strategy and this podcast is over. So thanks for tuning in, and teaching more and more Real Math. To find out more about the Math is Figure-Out-Able movement, visit mathisFigureOutAble.com. Let's keep spreading the word that Math is Figure-Out-Able.