How sophisticated is your multiplicative reasoning? In this episode Pam and Kim build Flexible Factoring and show how powerful understanding can be!
We love hearing from listeners!
A Problem String
Develping distributive, associative and commutative properties
Where does Flexible Factoring fit into Multiplicative Reasoning?
Why understanding and owning multiplication facts is more powerful than memorization
For more information about the Development of Mathematical Reasoning, see Episodes 5 and 6.
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Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
And I'm Kim.
And you found a place where we love talking about math that 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 know we can mentor students to think and reason like mathematicians. Even those we weren't sure, but they actually can. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.
Hey, everyone. I'm going to say this. And I've said it before, but I'm going to say it again. We love hearing from you. It's super fun to get emails and read the reviews on podcast platforms. Kind of makes my day when I open it up, and I see something fresh and new, so do more.
Yeah, do more of that.
Tell us what you're learning. Or what you like or don't like. We're tough skin.
How you're thinking has changed because of listening. It's super fun. So, here's one that we got recently from Leanne. Hey Leanne.
So, Leanne said, "I came into this course feeling like I was..." quote "...okay with my multiplicative reasoning as I had listened to a few podcasts and taken the additive reasoning and Development of Mathematical Reasoning workshops. But holy moly," says, Leanne. She says, "I gained so much more insight about not only the different strategies (flexible factoring blew my mind) but also, the appropriate use of models to use with them. Area models, ratio tables, and maybe early on, even an open number line. There were so many practical suggestions and takeaways." Alright, Leanne! (unclear).
Yeah, thanks, Leanne. Super to have you in Building Powerful Multiplication. It was awesome to interact in the message board. Leanne and I are actually chatting a little bit more about some other things. She has some really nice perspectives that is helping me kind of understand some things, and so really appreciate that. And, yeah, like Kim said, do some of that. Like, let us know what you're thinking about. Ya'll, it helps us get better, you know, when you tell us the things that are really working for you, and the things that we can tweak. And let's just be clear, it makes our day. So, yeah. We love to hear from you. Alright. So, Kim, when we read that, we realized we hadn't done an episode on flexible factoring.
How have we not?
Yeah. She said "blew my mind" and I was like, "Oh, yeah!" So, we've done some MathStratChat's pointed towards it. But yeah, flexible factoring is a mind blower. In fact, maybe I'll tell a quick story. I had a student in one of my university classes, where we were doing a bunch of distributive property stuff. And this guy was flexibly factoring. And I remember going, "Tell me more." Like, "Do more of that!" Like, it was one of those moments where I kind of knew it was a thing, but I really hadn't appreciated quite. And it was amazing. Maybe I'll say a little bit more about what he was doing at the end because I don't want to spoil the surprise. So, we can't really do it justice in a podcast episode, not like we do in my online workshop Building Powerful Multiplication. But, ya'll, we thought we'd give you at least a taste of the flexible factoring strategies. Alright, so here we go. Kim, I'm going to give you a problem. We're going to do a Problem String. I'm going to give you a problem and ask you how you would solve it. Well, like normal. And then, we'll go from there. So, first problem, ready?
14 times 45.
14 times 45. (unclear) And I'm thinking...
...45 is close to 50. And to get 50, I'm going to go 100. So, I'm going to be 14 times 100, which is 1,400. Half of that, 14 times 50, would be 700. And then, that's five 14's too much. And I know that five 14's is 70.
What?! How do you know that?
I don't know. I actually did Double, Halve in the midst of it. So, is that (unclear)?
funny. Because can I tell you what I was thinking?
So, you probably are writing equations because I know you.
Yeah, I am.
But I was putting your thinking in a ratio table. So, my ratio table right now looks like 1 to 14, 100 to 1,400, 50 to 700.
Oh, yeah to the 5. Yeah.
Mmhmm. When you said 5, I was like, "Oh, divide 50 by 10, divide 700 by 10. 5, 70.
And that's really cool about a model, right? It's a tool for thinking. So, that had I been in a ratio table, I probably would have gone with 5 from the 50.
I mean, it's okay. You didn't need it.
It's great to have both. So, then 14 times 50 was 700. But I needed to remove the 70, so I got 630.
To get the 45, 14
Yep. 14 times 45.
So, I gave you 14 times 45, and you found 45 times 14, which is totally legal.
It's alright. Hey, I'm wondering...
But do you want to find 14?
I mean, let's get one more strategy out there because with this particular string, I would try to get one more strategy for this problem.
It's rich, right? There's a lot we can do, so.
Yeah, what else do you see off the bat?
my first... I mean, there's a couple of things, several things that I could do. I could do the same strategy of going Over and Five is Half of Ten with the 14. So, I could do ten 45's, which is 450. And then, five 45's would be 225.
Because half of that, mmhmm.
Yeah. And then, that's fifteen 45's would be 675. But then, that's one too many 45's. So, I back a 45, which is 630.
Nice. 675 minus 45 gives us 630.
Cool. So, interesting. You just did two strategies that used Five is Half a Ten and Over just in different ways. That's very nice, very nice. Okay, cool. Next problem.
How about if I asked you 30 times 21? What are you thinking?
30 times 21. So, I know 3 times 21 is 63. But you asked me for 30 of them, so 10 times as many. So, then, it's going to be 630.
But 630 was what we got for the first problem. Are you sure it's the same for the second?
Oh, interesting. What does that mean to you? Like, that's just random. It's totally random.
It's just haphazard that I chose two problems that the... And if I was doing this with real kids, because you're not a real kid, then I would probably just note that they both got 630, and I would actually move on. I wouldn't spend a lot of time trying to get somebody to dig something out of that. I would just like, "Huh. So, fourteen 45's. Thirty 21's. That's kind of interesting." Okay, cool. Next problem. 7 times
That's 630. 7 times (unclear).
Huh. 630. Okay.
I will pause here. Do you see any relationships between those (unclear).
I wrote the problems all together. And the 7, and 21, and the 14 are jumping out at me. So, they all three have a bunch of 7's, so I'm wanting to move that 7 around a bunch.
Ah. Can I? While, Kim's thinking about that strategy or that relationship. I might actually in class have you pause a little bit. I might just say, "Kim, keep thinking about that. There's something about a 7 in there. Does anybody see relationship between the first problem, 14 times 45, and the third problem 7 times 90? Just those two?" And that may have come out first, like when I just said "14 times 45". Because I would think that we would have worked on Doubling and Halving, some student might have shared, "Ooh, I can just Halve the 14 to 7. I double the 45 to 90, And then, have an equivalent problem." If that would have come out, I may have tried to squash it, so that I could do this problem later. Or I actually might have just let it come out, and it could have been the second problem in the string.
Then, I still would have thrown out 30 times 21 and gotten that same equivalent. And then, I would do more of what you're doing, Kim. So, I may have wanted to have acknowledged the Double, Halve connection because there really isn't a Double, Halve connection then with the 30 times 21. But is there a connection with the 7? So, can you say more about what your... Is there anything else that comes to mind with the 7?
Yeah. So, while you were talking, I'll admit that I wasn't really listening to you because I was staring at my paper. I have no idea what you just said.
Sorry. I'm sure it was good.
It's all good. It's all good. This is our relationship, listeners.
Pam talks, and Kim knows when to listen.
Both Pam and Kim 9:09
That's hilarious. I don't miss the good stuff. So, yeah. So, I just actually wrote down the prime factorization of all three of the problems, and because I got 630 for everything, they're clearly the same.
Or equivalent, mmhmm.
Yeah, the same prime factors.
Oh, same prime factors.
Like, do that for me.
In a different order. Yeah.
So, what did you... Okay.
14 times 45. I wrote 7 times 2 times 5 times 3 times 3.
And for 30 times 21, I wrote 3 times 2 times 5.
3 times 2 times 5?
Mmhmm, that's 30.
Times 3 times 7. Which are the same factors as the previous problem. And then, for 7 times 90, I wrote 7 times 3 times 3 times 2 times 5.
And ya'll are listening to this, so I don't know if you wrote those numbers down. But I'm noticing something, and I think this is a Kim thing. That if you look at the way.... So, she said they were in different orders. But as she did that, you might... I bet there's a fifth, sixth grade teacher out there somewhere that's like, "But she didn't write those in ascending order. Why did she not?" Like even in the number itself she didn't write them in the ascending order.
Like, on the 30 times 21. For 30, she wrote 3 times 2 times 5.
And I would have expected 2 times 3 times 5 if you were concerned about sort of writing them in kind of a, I don't know, traditional ascending order of those. But...
If you can see my paper...
Oh, do you want tell us why you did it? Because I think I know.
yeah, I kept the 10 together.
So, yeah, if you could see, listeners, what I wrote on my paper. For the first problem, there's a bunch of factors, but the 2 times 5 are next to each other. So, even though 14 times 45 is 7 times 2 times 5 times 3 times 3, it's 7 times 2 times 5 times 3 times 3. Because Kim's like, "Oh, I'm going to find that 10 in there. Like, once you have a factor of a 2 and a factor of a 5 somewhere in the multiplication problem, might as well pull them together using. So, you're reassociating the numbers, and then maybe commuting if you need to get them where you want them. Similarly, on the 30 times 21, she's got 3 times 2 times... Or sorry. 3 times 2 times 5 times 3 times 7. And again, there's that sort of 2 times 5 popping out. And then on the last one, 7 times 3 times 3 times 2 times 5. So, that same prime factorization's kind of playing around in a different order, but we had the same prime factorization. It's almost like to get each of those problems, we could write the prime factorization and just group, just associate different factors.
Like on the first one, we associate the 7 and the 2 and the 5 times 3 times 3, and we get 14 and 45. So, I wonder if that could be helpful thinking about other problems. Like, I'll throw out. Here's a new problem. What about 44 times 15? Now, I know where Kim's going to go. So, what would happen in an actual classroom? Kim, can you go somewhere not... Don't flexible factor yet, Can you share a strategy?
Yeah. Well, actually I think a lot of kids might say 44 times 10 is 440. And then, 44 times 5 is half of that, which is 220. Oh, so 660. Do I say that sorry?
No. So, fifteen 44's would be...
Like, I already shared my thinking, I guess you want the answer. Okay.
So, 15 times 44 would be 660. That would be a way to do that. And I might share that. I might share a kid thinking about four 15's to get fourty 15's. And then, adding the fourty and the four 15's together. That might be another strategy that I might share. And then, I might give this problem. 88 times 30.
Mmhmm. Which, depending on the age, and if you're doing this particular string, you're probably working with a little bit older kids. Might say, "Oh, that's going to be..." What? Tell me the problem again? You said 88 times 30?
88 times 30, mmhmm.
You might look back at the problem before it and see if there's...
So, the 44 is doubled to 88. And the 15 is doubled to 30.
Oh, man. Did I write down the wrong problem?
I don't know.
I think I meant 22 times 30.
22 times 30.
Now that I hear you say that out loud. There. Does it Double, Halve better
Yeah, yeah, yeah. Okay. So, Double, Halve. So, it's going to be the same 660.
Yeah. Okay, sorry.
Oh, yeah. Halve, Double.
Alright. Typo. Is that a podcast-o? Speak-o? Alright, so the second problem should have been 22 times 30, so, that kids say say, "Oh, yeah, if we Halve the first factor, Doubled the other one, then their equivalent.
And then, the next problem is 10 times 66.
Yeah. Which is 660. But I honestly don't care about that problem so much. What I'm thinking about is the first two, and how there's a 10 in there as well. And it's really nice to rearrange the factors to get that 66 times 10.
Both Pam and Kim 14:41
Say more. You were going there.
You gave me 22 times 30, so I wrote 2 times 11 for the 22. And then, for the 30, I wrote 3 times 2 times 5. But it actually then grouped the 2 times 11 times 3 to get 66. And that was times the last 2 times 5, which is 10. So, it's just 66 times 10.
And so, then could we do something similar with the 44 times 15?
And I'm going to wonder if in class if I might actually suggest at this point. Not necessarily prime factoring. We can. But I wonder if we just are looking for the 10.
Yep. Once you know to look for it.
Yeah. So, does that influence how you might factor 44 times 15?
Yeah, I'm writing it down. So, I would write 2 times 22 times 3 times 5. And then, I'd grab that 2 and the 5 as the first and last factor.
Yeah. And I might write down 22 times 2 times 5 times 3.
Yeah, that would have been nice.
So, I would just... Yeah, just changing the order a little bit. So, then the 2 and the 5 are in the middle. That's 10. So, now you end up with the 66 times 10 again.
I'm really glad that you just said that. Because I know that if you were in the classroom, and I said, "2 times 22 times 3 times 5," you would say, "Oh, okay, could I write that like this to make it more visible for more students?" And I because I came up with factors, you'd be like, "Yeah, you're saying the same factors."
Right. And I'm also building intuition for the properties. Like, at this point, I could say, "Oh, let's let's commute those factors in a different order, and then reassociate them." So, I'm putting words on things that kids are doing. I'm tagging as they're doing them. That's vocabulary "just in time". So, we're building properties. Strategy development is not just so kids can get answers. It's not a bunch of new things for kids to memorize, so they can get answers. It's literally about learning the math. Like, we're developing strategy, and we're developing relationships, and we're developing properties all at the same time. Yeah, absolutely. Hey, one other thing that I might push on depending on the kids is, at this point, I might say, "I wonder if there's something that we can do with this factoring we've been doing to find that Double, Halve." Like, when we had the 44 times 50, then, you said... I think we ended up with 22 times 2 times 5 times 3.
"I wonder if..." And I might let that sit for a second. "...If we can some way, we could then have the Double, Halve show up."
Does that make sense?
What are you thinking right now?
...(unclear) from. I think first problem was... What was the first problem? 44 times 15.
And then, you moved to... My papers got jumbled. Then you moved to...
Well, it would be on the board, so you'd be able to refer to it. So, I'll share with you what's on the board. So, on the board would be 22 times... I'm sorry. You want me to go?
I think I got it. So, 44 times 15, then it would be 88 times 30. 22 times 30. Sorry. So, that extra factor of 2 that's in the 44 is going to be reassociated, taken away from the 44, which makes it 22. And it's reassociated with the 15 to make it be 30.
So, there's a shift of that factor of 2.
(unclear) ten we've been talking about.
Yep. And so, the way I would record your thinking is, I've got the 22 times 2 times 5 times 3. There's the 44 times 15. And now equals. Next to that, I've written those same factors. 22 times 2 times 5 times 3. But I've now put parentheses around the 2 times 5 times 3. So, it used to be 22 times 2 was associated. And now the 22 is kind of off by itself times 2 times the 15.
Yeah. And so, that reassociating of that 2 is what's happening when we Double and Halve.
When we have one of the numbers, we're grabbing that factor of 2 from that number. That's why it's Halved.
And we're reassociating with the other factor, which is why it's Doubled.
Ba dum tss.
So screaming cool. Okay, so then maybe a last problem that I might give you is 55 times 18. And then, I would wonder, could you do something with that problem to be ultra cool?
So, I'm going to write down purposefully in order 11 times 5 for the 55, times 2 times 9 for the 18. And then, I've grouped the 5 times 2 in the middle there. And so, I'm going to go 99 times 11, the outer factors. I mean, 9. 9 times 11 is 99.
You're doing it all at once. Yeah, you're doing it all at once. Yep.
And then, times 10. So, 990. Yeah.
So, you had that nice factor of 10 in there. If you can grab the 5 and you could grab the 2 and stick them together, reassociate them to get the factor of 10, you were left with 11 times 9. Yeah, super, super cool, right? And maybe notice, everybody, that what we've just done is pretty multiplicative.
When we use the distributive property, there's some addition or subtraction in there. The Over strategy, you know, we do a bit too much, and we subtract off that chunk. Or if we're doing smart partial products, or Five is Half of Ten, we sort of grab these happy... You know, we got 10 of them. We got 5 of them. We add them together. We kind of have this additive thing that's happening within the multiplication. It's totally good. That's multiplicative reasoning. But this flexible factoring is even more multiplicative, and if you think about the Development of Mathematical Reasoning. And, ya'll, if you haven't heard those episodes yet, go listen to the Development of Mathematical Reasoning. But think about that graphic where we've got counting strategies, built on to get additive thinking, built on to get multiplicative thinking. In that multiplicative thinking oval, it's not a one dimensional, "Now, you own multiplicative thinking. Bam!" It's a span. There's less multiplicative multiplicative thinking, and there's sort of like middle multiplicative multiplicative thinking, and then there's extreme multiplicative thinking. And in fact, even then more extreme multiplicative thinking would be proportional reasoning because proportional reasoning really is kind of an extreme sophisticated way of thinking multiplicatively about non-unit rates. But my point is that if we're using the distributive property to chunk, sort of area to think about multiplication problems, that's multiplicative. But flexible factoring, like we just did in this string, is even more multiplicative. It's further on that continuum. So, because of that, it's a little further down the line. It's a later strategy that we would develop with students. We wouldn't do it right away. But, Kim, I got to tell you, when I was in that university class, and I had this student who said to me at the beginning, "I'm not a math person. You know, I really want to teach elementary school, but I know I'm not good at math. But I'm willing to work hard." And then, in the midst of Problem Strings where I was trying to develop the distributive property, this guy was popping out flexible factoring. And I was like, "Don't you tell me you're not a math guy." He's like, "Oh. Well, I just couldn't think. I couldn't figure out what they were doing, so I just came up with my own, you know like, trick ways." And I was like, "Your trick ways are real math. Like, you're..." Oh, I just watched this guy just sit up tall. I watched the whole class just like look at him differently because, you know, it was so... This is part of why I do what I do. It's so amazing to position all people as sense makers. Everyone can use what they know to think and reason, and then get bigger and badder mathematics to come out of it. Yeah, it's so awesome. So, just to pull back on. Just recently, we did a podcast. We were talking about facts. I think we were talking specifically about subtraction facts. But I want to just mention here. This is an example. This flexible factoring strategy is an example where you might be thinking, "Whoa, kids have got to know their multiplication facts in order for this to happen." Yes, but it's not about rote memory. If they have rote memorized those facts, that's not going to help them with factoring things like 44 or factoring things like 55 and 18. Like, it's not going to. Those things are not going to ping for them in the way they will if we have built them using relationships. And this is what we mean by if kids own those facts, they can use them with bigger, badder stuff. Not just repeating them in a procedure.
Okay, I got a little excited there. Woah!
I love flexible factoring. It's super fun. To me, flexible factoring is very kind of like puzzley, and playing, and manipulating.
Oh, yeah. Yeah, yeah.
Yeah, you get to look for how you want to rearrange those factors in a way that could come up with a slickest sort of. Yeah.
Yeah, it's a way of really looking at that and connecting everything to something that's sort of so much more useful longer term than just memorize the single digit facts. Nice.
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 mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!
Transcribed by https://otter.ai