Pam Harris  00:01

Hey, fellow mathematicians, you're listening to the podcast where math is Figure-Out-Able. I'm Pam Harris.

Kim Montague  00:09

And I'm Kim Montague.

Pam Harris  00:10

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  00:42

So a few weeks ago, we talked about place value, right? Some noticing- 

Pam Harris  00:46

Some 100 place value. 

Kim Montague  00:48

Oh, yeah.

Pam Harris  00:49

Yeah. 

Kim Montague  00:49

And um, so those were about when multiplying by 10, just noticing patterns that emerge. Today, we're going to talk about a subtle nuance in solving multiplication problems about becoming more sophisticated, a more sophisticated relationship that we want to nudge students towards.

Pam Harris  01:08

And I'm going to say, this is something I think we talk about, we do. We help students progress, we help them develop this relationship. I'm not sure I've seen this anywhere else. So this is a little subtle, but so cool, and so powerful. And let's get right at an example. So if I were to ask a relative beginning, I shouldn't say beginning multiplier, but someone who's learning maybe double digit multiplication, bigger number, so not like single digit, like, what does multiplication mean? But we're kind of getting to maybe problems like 40 times 17. So 40 times 17? If I were to, so maybe pause the podcast for a second. You figure out how would you figure 40 times 17? Get a handle on the relationships a little bit. So come on back. So if we were to hand that to a student who's just thinking about these sort of bigger numbers, they're kind of making sense of 40 times 17, often we would find that a first go to strategy for students might be something like, or would be something like, "Okay, I need 40 seventeens. So I'm going to think about Ooh, 10 I got, (that's why we just brought up this idea of multiplying by 10) 10 seventeens." So a student might say, "I know 10 seventeens is 170. But I need 40 of them." So 20 seventeens, they would double the 170 to get 340, double 170, you'd get 340. So now they have 20 seventeens. And then they might think about doubling the 20 seventeens. Double that 340 to get 680. And now they have 40 seventeens. So they might think about that 40 anything's really, there's 10 of them, double that to get 20 of them, and double that to get 40 of them, whatever that is. And that's a typical, it's fantastic. We want students to be thinking about, "Okay, if I'm thinking about 40 of these guys, you know, whoa, like, how can I get up there?" We might have students thinking more additively. But when they think more multiplicatively, which is what we're heading towards, that's brilliant. We want kids kind of thinking about, like, I'm repeating myself, but 40 things, 10 of them, then it could kind of double up to get 40 of them. Or maybe even once I had 20, I could add the 20 and 20 to get 40 in some way. They're kind of using that 10 in a helpful way. 

Kim Montague  03:31

Right. 

Pam Harris  03:31

So I remember the day Kim, I remember the day when I realized my brain was starting to make a shift that for maybe a hot minute, I had been shifting, and it felt like a sophistication shift. It felt like I was doing something more sophisticated. That when I began to look at a problem like 40 times 17, I began to say to myself, "I'm going to think about 4 seventeens. And then I'm going to scale that times 10." I began to see 40 less as 10 doubled to 20, doubled to 40. And more and more multiplicative as four times 10. And if I could think about 40 is four times 10. Then I literally and maybe it was because I was dealing with ratio tables so often, but if I were to do 40 times 17. Today, I would write down 1 seventeen. One to 17, a ratio table with one to 17. And then I would think about 2 seventeens is 34. And then I would think about four seventeens as 68. And then I would scale four to 40 times 10. And then 68 times 10 to get 680.

Kim Montague  04:51

Yeah, that's pretty interesting that you can remember exactly when that happened.

Pam Harris  04:56

Oh my gosh, I remember the day. I remember thinking, "Ah, that's more sophisticated." 

Kim Montague  05:00

Yeah. 

Pam Harris  05:00

Like I'm seeing so many kids use and I was I had, like really like kind of finding the 10. And then multiplying up from there instead of multiplying up and then scaling by 10. 

Kim Montague  05:14

Yeah. 

Pam Harris  05:15

And I found that kind of interesting. 

Kim Montague  05:17

Yeah.

Pam Harris  05:17

So if we were to use the associative property to kind of look at that, we might think about that 40 times 17, well, we might think about it in more of a distributive property way. Where it's sort of that 10 times 17, and then maybe add that to another 10. Or maybe then, yeah, maybe if I was to stay with the associative property, 10 times 17 is kind of is in parentheses, and then times four versus four times 17, in parentheses, then times 10. Is that a way to write that? Does that, do you agree with that, Kim?

Kim Montague  05:56

I didn't write it down. I'm sorry. 

Pam Harris  05:58

It's okay. 

Kim Montague  05:58

Say it again. 

Pam Harris  05:59

If I'm thinking about 40 times 17, the first way, the less sophisticated way, I wonder if I could write that as 10 times 17 times four. It's like I'm thinking about the 10 times 17. I'm finding that 170. And then I'm messing with that to get four of those groups. I might do that by doubling to get 20 of them and then doubling again. But I'm really focused on the 10 times 17, and then focused on getting four of those groups somehow. 

Kim Montague  06:26

Yeah. 

Pam Harris  06:26

Versus what we're kind of want to nudge kids towards is thinking about 40 times 17 as four times 17, that's in parentheses, then find that, then scale that times 10.

Kim Montague  06:41

Can I actually tell you about something that happened? 

Pam Harris  06:43

Yeah. 

Kim Montague  06:43

So Cooper, my youngest was in third grade, he started towards the very end of the year, he was starting to dabble in a little bit of I don't know if it was two digit multiplication of two digit by two digit, but certainly some larger multiplication problems, multi-digit. And I remember he had some problem that was probably something like, I don't know, 70 times 12, right. And I was like, oh, okay, so this is going to be you know, he was comfortable with the ratio table. And I thought, okay, cool. Let's like, you're going to chunk your way up there for a while, it's gonna take you a minute. 

Pam Harris  07:21

To get 70 of those. 

Kim Montague  07:22

To get 70. Right? That's a lot of twelves. 

Pam Harris  07:25

Yeah.

Kim Montague  07:25

So I was prepared to sit next to him and kind of like, you know, talk to him about what pieces do you know, and like, work his way there. And I remember being so fascinated-

Pam Harris  07:37

Before you do, before you go into what he did, you were prepared to have him do something like 10 times 12.

Kim Montague  07:42

Oh like 10 times 12, 20 times 12, 40 times 12, like, add some groups of tens together, exactly like you mentioned. Just groups of tens, building up to 70 twelves. And he didn't. And I was not prepared for that. What he said was, "Oh, it's like, it's like 7 times 12 times 10." And I thought, "How in the world do you know that?" And we have, he was so confident about his understanding about 70 was 7 tens or 7 times 10. He was thinking multiplicatively about 70 to begin with, that then when you said times 12, he had an understanding about saying that's the same as or equivalent to 7 times 12. Scale that by times 10. And I wasn't prepared for that. I gotta be honest with you. I was like, "Are you sure you know that?"

Pam Harris  08:44

So we had similar moments. It was my brain I was recognizing about my brain was doing and you were recognizing what Cooper's brain was doing. You're like, man, like, that's- And in that moment, both of us, I think had this sort of, like, ah-ha, that's more multiplicative. 

Kim Montague  08:58

Sure. 

Pam Harris  08:59

And that is part of our goal. We want not only to move students from Counting Strategies to Additive Reasoning, to Multiplicative Reasoning, but in Multiplicative Reasoning, we want to also get more multiplicative in our reasoning. 

Kim Montague  09:11

Yeah. 

Pam Harris  09:12

And that that could be a goal. So for example, you might think about problems, like I was talking about 40 times 17 earlier, so because we have those numbers floating around, I'm gonna kind of go on that theme. What if we were to think about 42 times 17? So I might expect a beginning student to do 10 seventeens, 20 seventeens. 40 seventeens. Then a completely new problem, think about, "Okay, now I need if it's 42 seventeens, completely new problem. I gotta find 2 seventeens." Not recognizing that they already had 20 seventeens, they could use that to help them scale down like divided by 10 to find the 2 seventeens. If 20 seventeens is 340. Then 2 seventeens would have to be 34, not even thinking about that. It would be a whole new problem. And then they would add, once they found that 2 seventeens, they would add that back to get the 42. Now I might be nudging students who really own that, and we want to start there, we want that. That's going to be their first sort of thing to do. We want to start there. But then I want to nudge students, using Problem Strings to help them think about, "Ooh, could I, so if I was to solve 42 times 17, thinking about 42. So like, I might think about 17 forty-twos, but if I was thinking about 42 seventeens, now I might go, "Okay, 1 to 17, 2 seventeens is 34, 4 seventeens is 68. Therefore 40 seventeens, now what's funny, Kim, is in my ratio table, where I had the 4 to 68, I'm just tacking on a zero to make it 40 to 680. 

Kim Montague  10:50

Yeah. 

Pam Harris  10:51

And I'm gonna give my daughter credit for that one. I was watching my daughter solve problems. And she would literally, she knew she was headed for 40, she would find two, four, and then turn that four into a 40. And whatever that was the 4 to 68 became 40 to 680. And then in, so now, my ratio table on my paper literally is 1 to 17, 2 to 34, I can't read my writing, 2 to 34, 40 to 680. And then I just add the 40 and the 2 together to get the 42. And then the corresponding, I hadn't done that yet. Is that 714? I can add live there to get the 42 seventeens. So then when I figured that out, then I got excited. And then I was like, "Whoa, like so 44 seventeens, once you found the 4, you've got the 40 just add it back to the four and then you've got 44. Then I got excited to do things like 88 seventeens, because I could find the 8 seventeens. Then I would have the 80 seventeens, and I would just have to add it to the 8 seventeens. Like lots of ways of finding the small number first, find the 8 seventeens, and bam, just scale that up. 

Kim Montague  12:06

Yeah.

Pam Harris  12:06

To 80 seventeens became a way that was more multiplicative. And a way that I could sort of mess with smaller numbers first before I had to mess with the bigger numbers. Does that make sense when I say that, Kim, that I was kind of messing? Like, if you're finding 10 seven teens, doubling that to get 20 seventeens, you're doubling 170. 

Kim Montague  12:29

Right.

Pam Harris  12:30

Right? And then now I've got 20 seventeens is 340. To get the 40 now you're doubling 340. You're messing with these big numbers. 

Kim Montague  12:39

It almost-

Pam Harris  12:40

Verses- Go ahead, what? 

Kim Montague  12:41

It almost feels like what you're about to say the sticking with the single digits, it almost feels like a little bit more of a pre planning. Like you know you're going to 40 but you want to stay in 1, 2, 4, 40.

Pam Harris  12:54

It's the anticipatory thinking.

Kim Montague  12:56

It's like you're considering where you're heading.

Pam Harris  13:00

And so that is more sophisticated, right? If you're not just kind of, "Okay, I need 40. Well, let's find some chunks I know and start working up there." Instead, you're kind of anticipating, "Well, if I need 40, bam, if I have four I can scale times tend to get the 40." It's really nice. Thanks for popping that in there.

Kim Montague  13:16

I think that what you said is true. Like, you know, we're gonna dogging the 10 groups, 20 groups, 40 groups.

Pam Harris  13:22

No, that's necessary. We have to start there, right? Yeah. 

Kim Montague  13:25

But then we can start considering and kind of lobbing out, "What if you had 40? Could you have 1, 2 , 4, 40?" Would that, just throwing out the suggestion that that might be a thing to consider, and see who takes that up.

Pam Harris  13:40

And how do we throw that out? I would suggest it would be a Problem String.

Kim Montague  13:43

Problem String.

Pam Harris  13:43

Problem String would literally be what we just did: find 2 of them, find 4 of them, find 40 of them. And then let kids realize, "Oh, I can from the four I can scale to get the 40. Ah, that's a thing. Hmm." So this idea of when you do kind of the less sophisticated, less anticipatory thinking strategy of kind of just working up and so you end up working with these bigger numbers, also plays out in the Five is Half a 10 strategy. Because there's this cool extension of how we can use a tweak on the Five is Half a 10 strategy that lets us deal with smaller numbers first, and then scale. So for example, if we were dealing with a problem like 50, five zero, 50 times 15. That might not have been my smartest number choices. Now I'm dealing all these 15 sounding things. So 50 times 15. If I was doing that problem, we might expect a less sophisticated learner to do exactly what we just said, where they might find 10 fifteens, and then double that to get 20 fifteens, and double that to get 40 fifteens, and then add back the 10 fifteens. So 40 and 10 to get the 50 fifteens. So we might expect a beginning sort of bigger number multiplier to mess with that. But then we want to deal with the Five is Half of 10 strategy which is also has an analog of 50 is half of 100. And so to find 50 fifteens, we might nudge students to find 100 fifteens. Now that I know that that's 1500, then I can think about 50 fifteen is half of that. And so half of 1500 to 750. And bam, I've got, I can use Five as Half a 10, to help me think about 50 fifteens. But the analog here of those are big numbers, right? If they came about 100 fifteens, and then I'm cutting that in half to get 50 fifteens. The analog of using the smaller numbers and then scaling would be: could I think about 10 fifteens, is 150, to get 5 fifteens. So if 10 fifteens is 150, then 5 fifteens is half of 150 is 75. Now scale that up to get 50 fifteens, of scaling up to 75 times 10, to get 750. But then there's even a third analog, where we can even deal with smaller numbers. And this is really the middle school move. This is where I want to move sort of fifth and sixth grade, where to think again, remember, our original problem was 50 times 15. So to find 50 fifteens could I find half of 15, or point five, half of 15? What? Half of 15? Well, yeah, because point five is going to eventually, again, this anticipatory thinking, if I need 50 of them, I could find 10 of them to get five. But can I just find point five of them, half of 15 and then scale that point five up to 50? So half of 15 is that 7.5, which therefore 5 fifteens would be 75, which therefore 50 fifteens would be 750. So that's an analogue if I could to kind of deal with the smaller numbers first and then scale up. And that is more both multiplicative thinking, and more anticipatory thinking, where we kind of see the end from the beginning and kind of have a roadmap of how to get there. Which is more sophisticated. Cool, right? 

Kim Montague  17:01

That's really cool. Yeah, it's so cool. 

Pam Harris  17:03

So we wanted to do an episode today to kind of help you in on this subtle, sophisticated move that we want to help students make, and what are the best ways to do that? Problem Strings that showcase these relationships, and that help students realize, "Oh, that's a thing." And then we can bring that into our work. Super cool. 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 mathisFigureOutAble.com. Let's keep spreading the word that math is Figure-Out-Able.