Math is Figure-Out-Able with Pam Harris

Ep 32: Another Cool Multiplication Strategy

January 26, 2021 Pam Harris Episode 32
Math is Figure-Out-Able with Pam Harris
Ep 32: Another Cool Multiplication Strategy
Chapters
Math is Figure-Out-Able with Pam Harris
Ep 32: Another Cool Multiplication Strategy
Jan 26, 2021 Episode 32
Pam Harris

It's time for another Problem String! In this episode Pam and Kim develop one of their favorite multiplication strategies, it's so slick! In doing so they talk more about the importance of letting students discover and develop relationships, as opposed to teaching strategies step by step.
Talking Points

  • What makes a good Problem String?
  • How do we help kids learn the appropriate uses of alternative strategies?
  • What to avoid when facilitating Problem Strings
  • Tweet at us your favorite name for this strategy! 
Show Notes Transcript

It's time for another Problem String! In this episode Pam and Kim develop one of their favorite multiplication strategies, it's so slick! In doing so they talk more about the importance of letting students discover and develop relationships, as opposed to teaching strategies step by step.
Talking Points

  • What makes a good Problem String?
  • How do we help kids learn the appropriate uses of alternative strategies?
  • What to avoid when facilitating Problem Strings
  • Tweet at us your favorite name for this strategy! 
Pam:

Hey, fellow mathematicians, welcome to the podcast where math is figure out-able. I'm Pam.

Kim:

And I'm Kim.

Pam:

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 and 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?

Pam:

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 into actually helping build the relationships in students' heads. So they can think and reason more and more sophisticatedly. So check that episode out, episode 31, if you want more back background and you want to know more about the over strategy.

Kim:

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 wanted to work with it today so that you can all have access to it.

Pam:

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 like Kim, or a pen if you're me and maybe a piece of 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:

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

Pam? Pam:

I'm ready. So we're going to do a quick Problem String,

Kim:

Okay. right? A series of problems to develop this strategy today. Here's your first problem? It is 16 times 3. 16 times 3.

Pam:

Okay. I don't just know 16 times three. I actually know 16 times 4 and I know 16 times 2, but 60 times 3 is kind of in the middle. So I'm going to think about 10 times 3 is 30 and 6 times 3 is 18 and 30 and 18 is 48, 16 times three is 48.

Kim:

Cool. So your next problem is eight times six.

Pam:

Eight times six. 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 three is 48. I'm going to be very verbal during this Problem String. I'm trying to say everything that's going on in my head. So you gave me 16 times three, and I figured that out as 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:

You ready?

Pam:

I'm looking at I'm thinking.

Okay. Kim:

You writing these down?

Pam:

I am,

yeah. Kim:

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

Pam:

24 times 30. I don't know that that's related to the other ones, it's going to be much bigger. Okay. So I'll just go to 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, probably four times three is like three twenties is 60 plus 12 is 72. So I think 24 times three is 70 to scale it up times 10 that's 720.

Kim:

Very nice. Okay. 24 times three and then scale it by 10. Okay. You ready? Your next problem is 12 times 60.

Pam:

12 times 60. I'm going to think about 12 times six, which I'm pretty sure is 72. And let me tell you what I'm thinking in my head. I know that like multiples of 12 - I never, I never worked on multiples of 12. And so later in life, I've dealt with a lot of twelves. And so I know that five times 12 is 60, that one's just ingrained. And so when I was thinking about 12 times six, I just really quick checked myself to think about five, 12 to 60. So six 12s sure enough is 72. So six 12s to 72. Then 60 12s is 720. Scale that up by tens and huh. 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's 720. And I am seeing a relationship. So let me back up.

Kim gave me six times three - Kim:

16 times three,

yeah. Pam:

Yeah, 16 times three and eight times six. So 16 times three at eight times six had the same product, 24 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 three then became eight times six, 16, half of that is eight, three double, that is six. So that's interesting. So if I'm thinking about three sixteens, could I think about six, eights instead? Cause like three sixteens. I need three of those sixteens, but the six, I think about double, I only need half as many, so I can think about six eights. I don't need six sixteens, only three 16. So if I find six half as big as those things, then they should have the same product. I wonder if that holds, let me think about the second one. So 24 times 30 that's like me needing 24 thirties. And then the next problem you gave me was just half of that 12. Well, if I'm really, if I need 24 30s, so if I'm only going to work with 12, then I need things that are twice as big. If I only find half as many of them, I need to find the twice as big. Did I say that Kim, how should I say that Kim?

Kim:

I need half as many that are twice as big.

Pam:

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

Kim:

Okay. Pam, 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, five and a half times 18.

Pam:

Okay. Nicely done. So I need five and a half eighteens or 18 five

bam. Kim:

Nice. Yeah 9 times 11 is 99, I'm going to sort of think about finding instead of 18 I'm going to find nine eleven's right? I'm going to find nine of things that are twice as five and a halfs, and a halfs. big because five and a half times two is just eleven. So instead of finding 18 five and a halfs, Good string Kim.

Pam:

Yeah. I like it.

Kim:

Thank you.

Pam:

So this is an interesting strategy, right? This idea that if I need to find, like we just said 18 five and a halfs so that I can find half as many things that are twice as big. I bet we could look at this as area as well, in fact on my paper right now, I'm drawing a 16 by three rectangle. So if I need the area of a 16 by three rectangle, I think I could cut that rectangle in half. So now I have an eight by three and an eight by three. And I'm just going to think about that eight by three tucked up next to that other eight by three. And so now I have not a 16 by three, but I have two eight by threes next to each other. That's like an eight by six and I didn't lose any area. So we can think about kind of ripping a rectangle in half and moving the area around actually using the associative property. Reassociate that area. And now 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 could 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 have the other dimension and the area stays the same. Very cool. All right. So we call this doubling and halving. That's the doubling and halving strategy. Very cool. Let's do a couple of problems to sort of demonstrate how this could be useful.

So Kim - Kim:

Yup.

Pam:

35 times 18.

Kim:

Hmm. 35 times 18. So since we're talking about doubling and having, that's kind of on the forefront of my mind and I'm going to think about this. 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 nine.

35 - Pam:

So I'm slowing you down, Double 35 to 70 and halve 18 is nine so 70 times nine.

Okay. Kim:

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

Pam:

Scale by 10 nicely done. That's a good application of doubling and halving. All right. Give me one.

Kim:

All right, Woah.

Pam:

Twenty-five times 64. So 32 and you know what, Okay. So I'm going to double the 25 to get 50 I'm not satisfied. So I'm going to double the 50 to 100. Half of 64. ready? Let's go with 25 times 64. and halve the 64, I think I can continue to go. 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, in a couple of doubling and halving steps.

Kim:

Yeah. And actually you double doubled. Right.

Pam:

And halved halved. And we know that doubling and halving twice is the same as quadrupling and quartering. You're going to go there, huh? So you'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:

Right. Which is sort of why we don't love this name 'Double halve', right?

Pam:

Yeah. Totally. Because we could double half, we can triple third. We can quadruple quarter. We could like, as long as we can 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 scaler, then we've kept the product the same. So yeah, we don't love the name double halving, but it's all we had to go with right now. So we're just going to, we're going to go with it. Okay cool. Let's see.

Kim:

Okay give me one. How about a problem like - hey, in our last episode, you might've noticed that Kim gave me a decimal multiplication problem. So how about 14 times 2.5, 14 times 2.5. Oh, that one's not so bad. Thanks for that. So I know that 2.5 times four is just 10.

Pam:

Whoa. That's kinda fast.

Kim:

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

Pam:

You needed to know 14 divided by 4 right?

Kim:

Yeah, so14 divided by two, one time is seven and then divided by two again is three and a half. So 14 times two and a half 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. That's just 35.

Pam:

Yeah.

Kim:

I just scaled up by 10.

Pam:

Nice, nice. So now the original problem, 14 times two and a half is just 35 with a little quadrupling and quartering. Yeah. Or like you said, you could have double-halved and then double-halved again, either way. Yeah. Cool. So one of the nice things is to recognize - it's funny because what I gave you 14 times two and a half, 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 two and a half and 10. Whereas I,

Kim:

Yeah. for sure, when I started my journey of numeracy, I didn't. I had no idea that two and a half doubled was five and five doubled was 10. So there's this beautiful relationship between two and a half and 10 so much so that now I play around all the time with one and a quarter, because one in a quarter doubled is two and a half. And now we're at that two and a half landmark place again. So 1.25 or one and 25 hundredths, one and a quarter is also brilliant to think about the 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 then - so not a word - multiplying 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 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? So I first, well, I didn't say that very well. I first ran into it while I'm reading Kathy Fosnot's work and thinking about alternative strategies, but then I was doing some numeracy work and a friend of mine in her high school classrooms, so Abby Sanchez teaches high school, teaches just down the road for me. And I was doing some numeracy work with her algebra one students. And one day as I was doing a totally different strategy, this kid raised his hand. He was kind of a snarky kid. He was like, everybody was - kind of kid like everybody kinda joked around with and they, they liked him. You could tell, cause he kind of had this really like kind of almost Kim attitude towards the world. And he looked at me. He goes, Hey, is that why that works? And I was like, is that why what works? Cause we're doing something totally different. He goes, why you can double one number and divide the other one by two and you get the answer. He kind of said it, kind of like, you know, like kind of like a freshmen 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, it 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. That's cool. That's really cool.

Pam:

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

halve? Kim:

Oh, people ask us that, right? And they ask about how do you teach your kids? I'm air quoting. How do I teach my kids to know which one's 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 and 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, we would advocate so much for, just talk about what you think about.

Pam:

Absolutely like tell your math friends, right? One of the reasons why I kind of tried to Just be verbal,

Kim:

And so it has become part of one of Cooper or even tell your students like, strategies and it is so fun to see him mess with think out loud as I was solving the problems today is to kind of model what it means to say what's kind of going on in your head. Hey guys, with this great strategy, I'm trying to use it more often. numbers. And now pretty often when he's doing, Help me recognize when it would be a good time when you know, like a math assignment or especially when we're doing MathStratChat, I'm just sort of missing that. I will do a particular strategy. It's an opportunity. I can bring those relationships to bear. And he's, he's kind of snarky like me. So one other small note, And he'll say, Kim mentioned earlier that don't want to direct teach the strategy. why didn't you double-half? And I'll just, it'll not be what I'm thinking about or I'll miss it We're not advocating that you stand up - So I'm gonna sometimes. So we want to encourage you guys to be on tell a quick story, no names, no names, just so you know Kim and I will tell you what, the lookout. If it's something, if doubling and halving or tripling and thirding is something that we'll tell you what we think, you haven't ever thought about before as this multiplication strategy or but if it's negative, we'll never, we'll never give a name. it's something that you find that you don't use very often, Like your name is safe with us. So real quick, quick story with no names attached quite a while ago, maybe make a note or try to keep it on the we were doing some filming of Problem Strings. forefront of your mind, because there are so many great opportunities to use the strategy. 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, another teacher in the district. So I walked in the door and, this was real low tech, it's me holding a camera and I walked in the door and I said, okay, ready? Alright, I'll just be in the back. And I held the camera and she stood at the front of the class and she said, all right, today, guys, today, we're going to learn the doubling and halving strategy. Here's what you do. Step one, step two. So, and she literally broken down the strategy into steps. It's like she had the Z perspective that math is all about procedures and mimicking someone's steps. And she just couldn't like, get out of that. And so she's like, all right, here are the steps. And she said, so for the first step is you double the first factor. So everybody do that. Here we go. She has a sample problem, everyone do that. And then you halve the second factor and see, and that now is an easier problem. And so you solve that one. See how that worked because it was really cool. So then 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 filming, like, what are you doing? Cause I knew 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 were saying, we need to let kids fuss with that. We need to let kids try. You know, how do you decide which number to double on which one to have? Cause 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, and listeners you're like, what's wrong with that, Pam? Well guys, let's follow her strategy real quick. So her strategy is you have to double the first number. So double the 12 to get 24 and have the second number to get 12 wait, wasn't that the same problem when 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 six and et cetera. So interesting, like right, like, like she was so not catching the vision that it's

not about:

I'm going to force the strategy on the numbers no matter what. You're still thinking, it's not about memorizing strategies. It's about thinking and letting the relationships influence the strategy that you're gonna use. All right, 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 the strategy. We didn't come up with doubling and halving. That's been out there for a little while. It, it totally represents well, it describes well the relationships you use, if all you do is double nd halve, but what about if you halve then double or what about all the other factors? Like we said, quadrupling and quartering and all the rest of them. So we want it to sound good. Not too technical, not too jargony. Also not too informal. Like you can't be Mary's strategy or the lasso strategy. Like can't be goofy. I don't know. I'm looking around my office to think of what I, the 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 onto 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, 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 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, that scale factor either way. So we want it to be recognizable to the beginning student, but also scalable so that it works 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 #mathisfigureoutable and we will check those out.

Kim:

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 half. Or some version.

Pam:

Yeah. Nice. And remember, we've created MathStratChat Central, where you can find all of the problems. If you can't find that just Google MathStratChat Central, and maybe Pam Harris, but I don't thnk you'll need that, and they'll come right up where we list all the problems we've ever asked onMathStratChat. Click on it and 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, don't miss the Math is Figure-Out-Able Podcast, because math is Figure-Out-Able!