Ep 32: Another Cool Multiplication Strategy

Pam Harris  00:02

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

And we're here to suggest that mathematizing is not about mimicking, or rote memorizing. But it's about thinking and reasoning about creating and using mental mathematical relationships. That math class can be less like it has been for so many of us and more like mathematicians working together, we answer the question: if you're not teaching algorithms, then what? Alright, y'all in the last episode, Episode 31, we talked about the Over multiplication strategy. And we also talked about strategies in general. And why we emphasize helping students develop strategies, instead of spending all that time getting down the steps of the algorithm, because let's be clear, it takes a while to get those steps sort of ingrained in kids heads and getting them successful. Most students take quite a while. We practice over and over and over to get them to be able to do that. Instead of taking all that time, what if we put that time and actually helping build the relationships in student's head so there they can think and reason more and more sophisticatedly? So check that episode out, Episode...? What'd I say 31? 

Kim Montague  01:20

31, yeah. 

Pam Harris  01:21

If you want to know more about background, and you want to know more about the Over strategy.

Kim Montague  01:26

So in today's episode, I'm super excited, because we want to highlight one of my personal very favorite multiplication strategies. It's so slick, and maybe one that not everyone makes use of, but we want to work with it today so that you can all have access to it.

Pam Harris  01:44

Yeah, so to start off, let's be clear that this is a mathy episode. So you might want to grab a pencil if you're Kim or a pen if you're me. And maybe a paper and listen to it now. Or like if you're driving or whatever, go ahead and hang on to as much as you can, but drive carefully. But maybe keep track of some of the relationships. Drive careful, everyone. Okay. 

Kim Montague  02:07

Yeah. So I'm going to give Pam some problems to develop this strategy this time. Are you ready, Pam?

Pam Harris  02:11

I'm ready. 

Kim Montague  02:12

Okay, so we're gonna do a quick Problem String, right, a series of problems to develop this strategy today. Here's your first problem: it is 16 times 3.

Pam Harris  02:23

Sixteen times three, okay. I don't just know 16 times three. I actually know 16 times 4. And I know 16 times 2, but 16 times 3 is kind of in the middle, so it's not so much. So I'm gonna think about 10 times 3 is 30. And 6 times 3 is 18. And 30 and 18 is 48. Sixteen times 3 is 48. 

Kim Montague  02:40

Cool. So your next problem is 8 times 6.

Pam Harris  02:45

Eight times 6. And that's just a most missed fact and I deal with it a lot. So I just know that's 48. Hey, so hang on a second, you gave me 16 times 3 is 48. I'm gonna be very verbal during this Problem String. I'm gonna try to say everything that's going on in my head. So you gave me 16 times 3, and I figured that out is 48. And then you gave me a problem with the same product. Eight times 6 is 48. So I'm thinking something's up.

Kim Montague  03:09

Alright, you ready?

Pam Harris  03:11

Yeah, I'm looking. I'm thinking. 

Kim Montague  03:13

Are you writing these down? 

Pam Harris  03:14

I am. Yeah.

Kim Montague  03:15

That would be useful. Yeah. So the next problem is 24 times 30. 

Pam Harris  03:20

Twenty-four times 30. I don't know that that's related to the other ones, it's gonna be much bigger. Okay. So I'll just go ahead and figure it out. Let's see, I'm going to think about 24 times 30 by thinking about 24 times 3 and then scaling it up by 10. 

Kim Montague  03:35

Okay.

Pam Harris  03:35

So 24 times 3 is like three 20s is 60 plus 12 is 72. So I think 24 times 3 is 72. Scale that up times 10. That's 720. 

Kim Montague  03:48

Very nice. 

Pam Harris  03:49

Okay. 

Kim Montague  03:49

Twenty-four times 3, and then scale up by 10. Okay, you ready? 

Pam Harris  03:52

Uh-hum.

Kim Montague  03:53

Your next problem is 12 times 60.

Pam Harris  03:58

Twelve times 60. I'm going to think about 12 times 6, which I'm pretty sure 72. And let me tell you what I'm thinking in my head. I know that like multiples of 12 - I never worked on multiples of 12 - and so later in life, I've dealt with a lot of 12s. And so I know that 5 times 12 is 60. That one's just ingrained. And so when I was thinking about 12 times 6, I just really quick check myself to think about five 12s is 60. So six 12s sure enough is 72. So six 12s is 72, then sixty 12s is 720. Scale it up by 10. Hey, okay, so those of you that are driving and aren't writing this down, Kim had given me 24 times 30. That was 720. And now I just did 12 times 60 that 720 and I am seeing a relationship. So let me backup. Kim gave me 6 times 3. Sixteen times 3. Yeah. Sixteen times 3 and 8 times 6. So I was thinking about that 8 times 6. So 16 times 3 and 8  times 6 had the same product. Twenty-four times 30, and 12 times 60 had the same product. So when I look at the relationships between those pairs of problems, there's definitely some doubles and halves happening here. So like 16 times 3 then became 8 times 6. Sixteen, half of that is 8. Three, double that is 6. So that's interesting. So if I'm thinking about three 16s, could I think about six, 8s instead? Because like, three 16s, I need 3 of those 16s. But 6, I think about double, I only need half as many. So I can think about six 8s. Because I don't need 3, I don't need six 16s, I only three 16s. So if I find 6, half as big of those things, then I can then they should have the same product. I wonder if that helps. Let me think about the second one. So 24 times 30. That's like me needing twenty-four 30s. And then the next time you gave me was just half that 12. Well, if I need twenty-four 30s. So if I'm only going to work with 12, then I need things that are twice as big. If I don't, if I only find half as many of them, I need to find the twice as big. Did I say it, Kim? I don't know. 

Kim Montague  06:23

I need half as many that are twice as big.

Pam Harris  06:26

There you go. Okay, yeah, I need half as many things that are twice as big in order to have the same product. 

Kim Montague  06:33

Okay, I have a last problem for you. You're noticing some sort of pattern here. And I wonder if that would be useful for you, when I give you this problem? It is 5.5, 5 1/2 times 18.

Pam Harris  06:45

Okay, so I need five and a half 18s or eighteen 5 1/2s. I'm going to sort of think about finding - instead of eighteen 5 1/2s, I'm going to find 9 of things that are twice as big, Because 5 1/2s times 2 is just 11. So instead of finding eighteen 5 1/2s, I'm gonna find nine 11s. Right. 

Kim Montague  07:10

Yeah. 

Pam Harris  07:10

Nine times 11 is 99. Bam! Woo!

Kim Montague  07:13

Nice. 

Pam Harris  07:14

Nicely done.  Good string, Kim. I like it. 

Kim Montague  07:17

Yeah.  Thank you. 

Pam Harris  07:19

So this is an interesting strategy, right? This idea that if I need to find, like we just said eighteen 5 1/2s so that I can find half as many things that are twice as big. I bet we could look at this as an area as well.  In fact on my paper right now I'm drawing a 16 by 3 rectangle. So if I need the area of a 16 by 3 rectangle, I think I could cut that rectangle in half. So now I have an 8 by 3 and an 8 by 3. And I'm just gonna think about that 8 by 3 tucked up next to that other 8 by 3. And so now I have, not a 16 by 3, but I have two 8 by 3s next to each other. That's like an 8 by 6. And I didn't lose any area. So we could think about kind of ripping a rectangle in half and moving the area around. Actually using the associative property, we associate that area. And voila, we have a new rectangle but with the same area. So that would be a way to kind of have a proof without words that we can sort of like shift area around to kind of think about why we can do this double one factor and half the other factor and the product stays the same. Because we are doubling one dimension as we halve the other dimension and the area stays the same. Very cool. Alright. So we call this Doubling and Halving. That's the Doubling and Halving strategy. Very cool. Let's do a couple problems to sort of demonstrate how this could be useful. So Kim, ready to go? 

Kim Montague  08:52

Sure. Yep. 

Pam Harris  08:52

Thirty-five times 18. Woo do it.

Kim Montague  08:56

Hmm. 35 times 18. Well, so since we're talking about Doubling and Halving, that's kind of on the forefront of my mind. And I'm going to think about this, like, I don't like 35 times 18. So I'm going to double 35. And since I doubled that dimension, I'm going to halve the other dimension. So instead of 35 times 18, I've created a new problem that is 70 times 9.

Pam Harris  09:23

So double 35 - sorry, I'm slowing you down - double 35 to 70. And half of 18 is 9. So it's 70 times 9. Okay. 

Kim Montague  09:29

Yep. And so since I'm halving the problem, halving the problem, since I'm solving the problem, 70 times 9, I know 7 times 9 is 63. And I'm making it 10 times bigger, or is 10 times bigger. So 7 times 7 is 63, 70 times 9 is 630. 

Pam Harris  09:47

Six hundred thirty. Scale it up by 10. Nicely done. That's a good application of Doubling and Halving. Alright, give me one. 

Kim Montague  09:55

Alright, ready?

Pam Harris  09:56

Yep.

Kim Montague  09:56

Let's go with 25 times 64. 

Pam Harris  10:01

Twenty-five times 64. Okay, so I'm going to double the 25 to get 50 and halve the 64, half of 64 is 32. And you know what? I'm not satisfied, I think I can continue to go. So I'm going to double the 50 to 100. Whoa, and halve the 32 to 16. And now I have the equivalent problem. 100 times 16, which is just 1600. And I just solved 25 times 64. 

Kim Montague  10:27

Nice. 

Pam Harris  10:27

In a couple of doubling and halving steps. Yes. 

Kim Montague  10:29

Yeah. And actually you double, doubled. Right? And we know -

Pam Harris  10:33

And halve, halved, right? 

Kim Montague  10:34

And halve, halved.

Pam Harris  10:35

Uh huh.

Kim Montague  10:35

So we know that doubling and halving twice is the same as quadrupling and quartering.

Pam Harris  10:43

You're gonna go there, huh? 

Kim Montague  10:44

I am.

Pam Harris  10:47

So they're saying that we could also quadruple the one length of a rectangle and  quarter the other one and we wouldn't lose any area.

Kim Montague  10:55

Right. Which is part of why we don't love this name: Double/Half, right?

Pam Harris  10:59

Yeah, totally. Because we could double/half. We can triple/third. We can quadruple/quarter.

Kim Montague  11:03

Yeah. 

Pam Harris  11:03

Like as long as we maintain the same factor that we multiply one factor by a number, that we divide the other factor by that same number, by that same scalar, then we've kept the product the same. So yeah, we don't love the name 'Doubling and Halving'. But it's all we had to go with right now. So we're just gonna go with it. Okay, cool.

Kim Montague  11:27

Alright, give me one. 

Pam Harris  11:29

Okay, let's see. How about a problem like - hey, in our last episode, you might have noticed that Kim gave me a decimal multiplication problem -so, back at you. How about 14 times 2.5. 

Kim Montague  11:43

Fourteen times 2.5? That one's not so bad. Thanks for that. So I know that 2.5 times 4 is just 10.

Pam Harris  11:50

Whoa, that's kinda fast.

Kim Montague  11:52

I love 10. So 2.4 times, sorry, 2.5 times 4 is 10. And since I quadrupled one of the dimensions, then I'm going to quarter or divide by 4, the other dimension. And actually, I happen to know this one, but if I didn't know, I could just halve, halve.

Pam Harris  12:13

If you didn't know 14 divided by 4, right?

Kim Montague  12:15

Yeah. So 14 divided by 2, one time is 7. And then divided by 2, again, is 3 1/2. So 14 times 2 1/2 is equivalent to 3.5 times 10. And it's interesting because I still have decimals in my problem, but I would so much rather do 3.5 times 10, because that's just 35.

Pam Harris  12:39

Yeah, just scale it up by 10. Nice. 

Kim Montague  12:41

Yeah.

Pam Harris  12:41

Nice. So now the original problem 14 times 2.5 is just 35. With a little quadrupling and quartering. 

Kim Montague  12:48

Yeah. 

Pam Harris  12:48

Or like you said you could have double/halve and then double/halve again. Either way, yeah. Cool. So one of the nice things is to recognize - it's funny, because when I gave you 14 times 2.5, you were like, "Oh, that's not too bad." But I think that's because you've doubled a lot. And so you know, this relationship between 2.5 and 10. 

Kim Montague  13:06

Yeah. 

Pam Harris  13:07

Whereas I for sure, when I started my journey in numeracy I didn't. I had no idea that 2 1/2 doubled was 5, and 5 doubled was 10. So there's this beautiful relationship between 2 1/2 and 10. So much so that now I play around all the time with 1 1/4. Because 1 1/4 doubled this 2 1/2. And now we're at that 2 1/2 landmark place again. So 1 point 25 or 1.25, 1 1/4 is also brilliant to think about this doubling and halving strategy, because I know it can scale up to 10 so easily. So that's kind of a cool relationship. Okay, cool. So we could generalize the strategy to not just doubling and halving but tripling and 'thirding'. That's so not a word, multiply by three and divided by three, quadrupling and quartering. And then after that, I think we have to sort of generalize, multiply by five divided by five, multiply by six divided by six. Like whatever it is, as long as we maintain that sort of same scale factor. Once it's multiplied by the factor, and then divided the other is divided by the factor, then we maintain this equivalence. And that's really a cool part of this strategy. Kim, can I tell you where I first sort of ran into the strategy?

Kim Montague  14:12

Sure. 

Pam Harris  14:12

So I first, well, I did say that very well. I first ran into it reading Cathy Fosnot's work and thinking about alternative strategies. But then I was doing some numeracy work in a friend of mine, in her high school classroom. So Abby Sanchez, teaches High School, she used to teach just down the road for me, and I was doing some numeracy worked with her Algebra I students. And one day as I was doing a totally different strategy, this kid raised his hand. He was kind of this snarky kid. He was a kind of kid like everybody kind of joked around with and they think liked him. You could tell because he kind of had this really like kind of almost 'Kim' attitude towards the work.  And he looked at me and goes, "Hey, is that why that works?" And I was like, "Is that why what works?" Because we were doing something totally different. He goes, "Why you can double one number and divide the other one by two and you get the answer?" And he kind of said it kinda like, you know, like, kind of like a freshman, snarky kid would say it. And I was like, "Oh my gosh, there are kids out there that have developed Doubling and Halving on their own?" He's like, "Well, yeah, you know, works. But is that why it works?" Like he was really clear it worked. He just wanted to get kind of behind the scenes about why it worked. So right then and there, we kind of dove in and, like, explored a little bit, the Doubling and Halving strategy. It was a lot of fun. 

Kim Montague  15:21

That's cool. That's really 

Pam Harris  15:22

So Kim, when you run into a problem, and you decide you're going to Doubling and Halving, how do you decide which number to double and which to halve?

Kim Montague  15:29

Oh, people ask us that, right? And they asked about how do you teach your kids - I'm air quoting - how do I teach my kids to know which ones which. And there are some generalizations that we kind of want kids to develop. But that's the word, we want them to develop that idea, right? We have got to let kids mess around to come to their own conclusions about would you want to halve evens or halve the odds. the idea that you often might want to try to get out of fractional parts or decimals. And if I may with a story real quick, I was so happy when Doubling and Halving became part of my youngest son, Cooper's, repertoire. He now Double/Halves so often, but I've never had a direct instruction with him. In fact, I don't know that I've ever done a Problem String with him. And I'm not sure that he has at school yet. But he's heard me and my oldest son talk a little bit about messing with numbers. And this is why we would advocate so much for just talk about what you think about.

Pam Harris  16:32

Yeah, just be verbal, right? Just talk. It's one of the reasons why I kind of tried to think out loud as I was solving problems today is to kind of model what it means to say, what's kind of going on in your head. Yeah, sorry to interrupt.

Kim Montague  16:43

Yeah. That's okay. And so it has become part of one of Cooper's strategies. And it is so fun to see him mess with numbers. And now pretty often when he's doing, you know, like a math assignment, or especially when we're doing MathStratChat, I will do a particular strategy. And he's, he's kind of snarky, like me, and he'll say, "Why didn't you Double/ Half?" And I'll just, it'll not be what I'm thinking about, or I'll miss it sometimes. So we want to encourage you guys to be on the lookout. If it's something, if doubling and halving or tripling and 'thirding' is something that you haven't ever thought about before as a multiplication strategy. Or it's something that you find that you don't use very often, maybe make a note or try to keep it on the forefront of your mind, because there are so many great opportunities to use this strategy.

Pam Harris  17:32

Absolutely. Like tell your math friends, or even tell your students like, "Hey, guys, with this great strategy I'm trying to use it more often. Helped me recognize when it would be a good time, when I'm just sort of missing that it's an opportunity to bring those relationships to bear." So one other small note, Kim mentioned earlier that we don't want to direct teach this strategy. We're not advocating that you stand up - i'm just going to tell a quick story. No names, no names. Just so you know, Kim I will tell you what we think. But if it's negative, we'll never give a name. Like your name is safe with us. So real quick story with no names attached. Quite a while ago, we were doing some filming of Problem Strings. We had some teachers that volunteered. They're like, "Yeah, you come in our classroom, and video me giving a Problem String." And so we had, I don't know, five or six teachers. Kim was one of them and another teacher in the district. And so I walked in the door. And these were real low tech. It was me holding a camera. And I walked in the door and I said, "Okay, you ready? Alright, I'll just be in the back." And I held the camera and she stood at the front of the classroom. She said, "Alright, today, guys, today, we're going to learn the Doubling and Halving strategy. Here's what you do." Step one, step two, and she'd literally broken down the strategy into steps. It's like she had the Z perspective, that math is all about procedures and mimicking someone steps and she just couldn't, like, get out of that. And so she's, "Alright, here are the steps." She said, "So for the first step is you double the first factor, double the first factor. And then the second, and so ready to do that. Here we go." She had a sample problem, everybody did that. Then she said, "Then you halve the second factor and see, and that now see i'ts an easier problem. And so you solve that one. See how that worked. That was really cool." So she did one and they did one together. And then she kind of assigned everybody to do another one. Kim, the hilarious part of the story for me, well, first of all, after a couple of minutes, I was like, "Is she doing the non-example?" Like I was in the back of the room trying, like, what are you doing? Because it was so clear that that's not what we're advocating. And somehow we just hadn't worked with her long enough or whatever. She couldn't quite get out of that perspective. But the funniest part of everything is remember, she said the first step was to double the first number and halve the second number. Whereas you just a minute ago, we're saying, we need to let kids fuss with that. We need to let kids try you know which way. How do you decide which number to double and which one to halve? Because you could do either one. So the problem that she had chosen to demonstrate was 12 times 24 or at least one like it. I don't exactly remember what it was. And listeners, you're like, "What's wrong with that, Pam?" Well, guys, let's follow her strategy real quick. So if her strategy is you have to double the first number, so double 12 to get 24 and halve the second number to get 12. Wait, wasn't that the same problem we had to start? So if the problem is 12 times 24, and you double and halve in her lockstep fashion, you get 12 times 24, turning into 24 times 12. And then she kept going, then she got 48 times 6 and etc. So interesting like, right? Like, she was so not catching the vision, that it's not about, I'm gonna force the strategy on the numbers no matter what. It's you're still thinking. It's not about memorizing strategies, it's about thinking and letting the relationships influence the strategy that you're going to use. Alright, cool. So a little bit of story there. So we don't love the name. We said that before. If you can think of a better name for this strategy, tweet or post about it. We would love to hear your suggestions for a better name for this strategy. We didn't come up with Doubling and Halving. It's been out there for a little while. It totally represents well, describes well the relationships you use, if all you do is double and halve. But what about if you halve and double? Or what are all the other factors? Like we said, quadrupling and quartering and all the rest of them. So we wanted to sound good, not too technical, and not too jargony. Also, not too informal. Like it can be 'Mary's strategy'? O what's, the 'lasso' strategy. Like can't be goofy. I don't know. I'm looking around my office to think of what, a 'light switch' strategy, like it has to be mathematical enough that it would work great. When a student, if a teacher says, "What are you doing?" And the student said, "I'm using this strategy." The teacher could sort of mathematically hang on to what's happening. Because one other, I'm making it really, like you have to find a good name that meets all these criteria. Another criteria is that when students are first learning it, it has to make sense. Because I've had some people suggest some strategy names that make sense when later I want to generalize it to all the things. But it also has to sort of make sense in the beginning when we're calling it Doubling and Halving. Or we're in the beginning when all we're doing is Doubling and Halving. But later, it feels general enough so that when students want to scale by the scale factor, and then divide by that same scale factor, or in middle school, instead of dividing by that factor, you multiply by one divided by that scale factor either way. So we want it to be recognizable to the beginning student, but also scalable so that it works kind of when we get more general. So if you can think of one we would love for you to tweet about it and post it on Facebook or Instagram with the hashtag math is Figure-Out-Able. And we will check those out. 

Kim Montague  22:38

Absolutely. Hey, there's problems all over the world. But if you are looking for an opportunity to try out the Doubling and Halving strategy, remember to join us on MathStratChat on Facebook, Twitter, and Instagram on Wednesday evenings, where we explore problems with the world. And many of those problems, you would be able to use Double/Halve.

Pam Harris  22:57

Yeah. 

Kim Montague  22:58

Or some version.

Pam Harris  22:59

Or some version. Nice. And remember, we've created MathStratChat Central where you can find all the problems. If you can't find that just Google MathStratChat Central and maybe Pam Harris, but I don't think you even need that. It'll come right up where we list all the problems we've ever asked on MathStratChat. Click on it and you can see strategies from around the world. So if you're interested to learn more math and you want to help yourself and students develop as mathematicians, then don't miss the Math is Figure-Out-Able podcast because Math is Figure-Out-Able.