Math is Figure-Out-Able with Pam Harris

Ep: 31: A Cool Multiplication Strategy

January 19, 2021 Pam Harris Episode 31
Math is Figure-Out-Able with Pam Harris
Ep: 31: A Cool Multiplication Strategy
Show Notes Transcript

In this episode Pam and Kim want to help you develop a cool multiplication strategy. Which multiplication strategy? We aren't going to tell you until the end of the episode : -). We believe in developing strategies as natural outcomes, and then giving it a name. How do we do it? With Problem Strings! Listen in as Pam and Kim go through a Problem String to develop this super efficient multiplication strategy.
Talking points:

  • What does it mean to do mental math?
  • We can purposefully help people reason and mathematize mentally.
  • How mathematicians solve computational problems, Ann Dowker's research
  • What is a Problem String?
  • A multiplication Problem String example
  • Another multiplication Problem String example with decimals!
  • The big reveal: The _____ strategy

Pam Harris  00:02

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

 

Kim Montague  00:09

And I'm Kim.

 

Pam Harris  00:10

And we're here to suggest that mathematizing is not about mimicking or rote memorizing. It's about thinking and reasoning: about creating and using mental relationships. That math class can be less like it was for so many of us and more like mathematicians working together, we answer the question: If not algorithms, then what?

 

Kim Montague  00:36

In today's episode, we're gonna dive into the world of multiplication, and talk about a particularly useful strategy. And one that I probably use the most often until I cemented a variety of others.

 

Pam Harris  00:50

So let's make one thing clear. Sometimes people hear us talk about strategies like today, we're gonna talk about a multiplication strategy. And they think, yeah, it's like this vast, unknowable list of strategies. There's like, so many of them out there that, you know, like, I'm sure I can pick up one or two of them. But it's not really important, because it's just, you know, like this huge universe of strategy. That's not actually true. There's really just a few important ones. And today we're gonna focus on one of those important multiplication strategies.

 

Kim Montague  01:19

Yeah. And because we feel so strongly about developing relationships first and then naming it, we're gonna actually name this strategy in a bit.

 

Pam Harris  01:27

Yep. So before we get into it, what do you think people do when they do mental math? When I say the words 'mental math', what comes to your mind? now Kim's chuckling a little bit, because we've had this conversation before. Mental math. Lots of people hear that phrase, 'mental math', and they think to themselves, "That's when you do it all in your head. That's when you've got all the stuff sort of happening in your head." And that might be true, but we're gonna quote Cathy Fosnot again, who said, "Mental math does not mean that you do it all in your head. Mental math means that you do it with your head." 

 

Kim Montague  02:06

So good.

 

Pam Harris  02:06

So do you think - yeah, great quote from Cathy - do you think that when we say mental math that you see somebody they're doing the like, you can kind of see their eyes kind of go up? And they're thinking or whatever? Do you think they're doing the traditional algorithm in their head to solve that problem? So like, somebody said, "Hey, here's a multiplication problem." Somebody says, "I'm going to do mental math," whether they write something down, or they do it in their head, you see them sort of like thinking and reasoning, and you think that they are doing the steps. Like they've got the single digit and they carry the one, and then they have the magic zero, or like, whatever the steps are in the algorithm. That's what they're doing in their head. And guess what? They're not. For the most part, if anybody's doing mental math, they're able to successfully compute in their heads, they're actually not doing all those steps. They're not repeating, mimicking steps of the algorithm at all. And what's kind of embarrassing is, so I could maybe you never thought that. That's fine. But even if you didn't, if I may suggest to you, many of us did. So I assumed that when people were doing mental math, they were just doing the algorithm in their head. I knew that the definition of addition was that traditional algorithm. Or the definition, the thing that you do when you multiply was that traditional algorithm. And so what else would people be doing in their heads? Like, they've got to be doing all those steps and crossy-outies, and carry the 1, and the magic zero, like, whatever it is, they were doing that. And I couldn't do that. I could not keep track of all those steps in my head. I had to write down those steps in order to keep track of all of them. And so I just, you know, I looked at people that did mental math. And I was like, "Well, I don't know, I don't know why you can keep all those steps and crossy-outies and zeros and stuff, but whatever. That's not important. Because I can do it on paper, and I'm totally okay with that." Yeah, I have heard you say that before. And I gotta be honest, I didn't really believe that that's what you actually thought. Um, but we had this experience lately, right? And I got to see it firsthand. So we were taking video of kids for a project and asked our - Served you right, by the way, for not believing me.

 

Kim Montague  04:08

(laughs) And we asked her camera person to ask a variety of kids some questions. And one of the questions was 99 times something. And I was watching the video later. And we talked right? I called you because I was so surprised that a few of the kids were actually doing what you thought mental math was. They literally heard the problem, put their finger in the air, air wrote the problem, like it was 99 times 37, drew the line, and then started doing the steps that they had been taught to do. That actually happened on video. And some of the kids got further into their steps than others. But every single one of them at some point, looked at the camera and said, "I just can't." And I, in that moment watching the video, really felt for them. To think that they thought that some people were just holding on to that. I feel like I would feel inferior if I thought that was the thing to do. And that some could do it. And I couldn't.

 

Pam Harris  05:07

Yeah, I mean, so deep down, I did, you know. I wonder why people could kind of hang on all that stuff. And I sort of dismissed it. I kind of did this with a lot of things I wasn't good at like, "Well, if I'm not good at it, that it must not be worth doing." You know, that was like sort of my immature way of kind of handling not being able to do stuff. And so it was interesting for you to see that there were other kids in the world who thought like I did that that's sort of what was happening in kids heads. And so they tried. You know, the camera person said, "Hey, would you do this problem?" And they're like, "Oh, I guess I'm supposed to, and you're not letting me write stuff down. So let me see what I can. You know, let me try and see if I can do it. No, I just can't." Like you said our hearts just  sort of go out to these kids. Could we share something fun? Maybe you've played with numbers. Maybe you've used relationships. Listeners, you're hearing me say this. And you're like, "Pam, that's really lame that you thought that." So maybe you didn't think that way. Maybe you're the one that could kind of do some stuff in your heads. What's interesting is, we think that we can purposely help people build those relationships that lend to strategies. That stuff that you were sort of naturally doing in your heads for whatever reason, we can purposely help people do that. We don't have to leave me and these poor kids out in the rain thinking, "Oh, I guess you know, like, because I can't hang all that stuff, I can't do mental math." No, we can actually give people the idea, "Hey, this is a thing. It's a thing to use relationships, not not all those memorize steps that you're mimicking that it's about using relationships." And once we let people know that that's a thing, then I have access to it. That's what mathematicians do. Before in the podcast, we've quoted the research study by Ann Dowker, who went to a bunch of mathematicians and said, "Hey, mathematicians, will solve these problems?" and gave them a bunch of arithmetic problems. "So if you'll please solve those problems. I'll just sort of watch you if you don't mind. I'll just kind of interview you about what you do to solve those problems." And out of all of the times that those mathematicians solved those problems, how much of the time do you think those mathematicians were doing what I thought that they were repeating steps of the algorithm? Like half the time? Like 90% of the time? How about 4% of the time? Almost none. Like almost never were they repeating those steps to solve problems. So what in the world were they doing the rest of the time? Well, they were doing what Kim does naturally. And now what I can do. What we can all do, if we know it's a thing. We can use the relationships among the numbers to solve those problems. So, Kim, today's episode, let's get down to talking about one of those major relationships for multiplication.

 

Kim Montague  07:40

Yeah. So actually, in an earlier episode, we talked about using 100 times anything, or 100 times something to think about 99 times anything. And in the video that I talked about earlier, I just wanted to share with you that we caught a dad in the background, who piped in and was talking to his daughter when she got done being interviewed and she said, "Hey, Lexi 99 - "

 

Pam Harris  08:01

Wait, wait. 'He said' right? You said 'she said'.

 

Kim Montague  08:02

Right. Oh, I'm sorry. He said to his daughter who had been interviewed, "Lexi. 99 is so close to 100. And you could have rounded up one and found 100 times 37 to get 3700 and then subtracted that 37." And she lit up and she said, "Oh, yeah, that would have been so much easier." 

 

Pam Harris  08:25

Isn't that interesting?

 

Kim Montague  08:26

She just didn't know it was a thing. Yeah. 

 

Pam Harris  08:27

Oh, that's so interesting that we hear, bright young lady, right? 

 

Kim Montague  08:30

Yeah. 

 

Pam Harris  08:31

I mean she's precocious. She's cute. She's on the video. We're taking the video and the whole thing, but had never thought about that kind of relationship. And then also interesting dad listening in the back is kinda like, "Hey, can you kind of think about it this way?" Oh, you know, like, just letting her know that that's a thing. Instantly, she was already sort of able to hang on to the idea that that could be a thing. 

 

Kim Montague  08:53

Yeah. 

 

Pam Harris  08:53

So Kim, how can we build that in kids or I mean, really in adults, right? Because I came to it later in life. I was definitely an adult when I learned how to do it. We like to use the instructional routine called Problem Strings. We don't think it's enough to just do a problem here or there to talk about our strategies. Although we do do that sometimes on the podcast just to kind of give you a glimpse of what's happening. We really like to use strings of problems that are purposefully written in a certain order in order to help students develop relationships. It's all about development, and progressively help students think more and more sophisticatedly. So y'all, you know we love to do some math together. So this is gonna be one of those times where if you're driving, go ahead and hang with us as much as you can in the air. You might want to re-listen to this part with a pen and paper later. If you want to. So don't, no dangerous driving at this point. But we are gonna do a little bit of math. Alright, so get ready, Kim, I'm gonna give you the problems. I'm gonna ask you to solve them and explain your thinking if needed. Alright, so first problem of the string. Here we go. What is 8 x 7?

 

Kim Montague  10:00

Okay, that one is a most missed fact, but I know it. It is 56.

 

Pam Harris  10:05

Alright, so you might use a relationship, you might just have it at your fingertips. Either way, we want you to be able to, you know, retrieve it fairly readily. We call that automaticity. But either way, you've sort of got 8 x 7 is 56. The next problem in the string, what is 80 x 7?

 

Kim Montague  10:23

Hmm? Okay, so that's just 10 times more. Eighty is 10 times more than eight. So that's going to be, the answers 10 times more, so that's going to be 560.

 

Pam Harris  10:32

So you're thinking about 10 times, and there's this lovely 10 times thing in our number system. So if 8 x 7 is 56, then you can think about 80 x 7 as 560. Cool. Yeah. Next problem. What is? Let's see, let me just review that. You just said 80 sevens was 560. 

 

Kim Montague  10:49

Right. 

 

Pam Harris  10:50

Eighty sevens is 560. Next problem is 79 x 7. 

 

Kim Montague  10:54

Oh, that was really helpful that you said 80 sevens because now I'm thinking 79 sevens. That's just going to be one seven less than my total, my product. So that's going to be 560 minus seven is 553.

 

Pam Harris  11:11

Sure enough, cool. Good thinking out loud. Alright, so if we could, you just sort of use some problems to help you think about kind of a crankier problem, right? Like 79 x 7, you could have done a couple different ways. But one way that you can do it is to think about 80 sevens to help you think about 79 sevens. Cool. Alright, next problem ready? 

 

Kim Montague  11:29

Yep. 

 

Pam Harris  11:30

Give you another most missed fact. 

 

Kim Montague  11:32

Okay.

 

Pam Harris  11:32

Eight times 6.

 

Kim Montague  11:34

Umm, 48.

 

Pam Harris  11:35

Alright. So if 8 x 6 is 48, then what's 8 x 60? 

 

Kim Montague  11:41

Umm, 480.

 

Pam Harris  11:43

Because you just sort of scale it by 10. Right? Eight times 6 is 48, so 8 x 60 is 480. Cool. Anybody want to guess what the next problem is? If 8 x 60, if I've got 60 eights is 480. Then what's 8 x 59.

 

Kim Montague  11:57

Eight times 60 was 480. That was 60 eights, and I want 59 eights. So that's just one eight less. So 472.

 

Pam Harris  12:08

Because 60 eights was 480. And one less eight gave you 472. 

 

Kim Montague  12:13

Right. 

 

Pam Harris  12:13

Nicely done, very cool. And then at this point in the problem string, so we've just given you sort of some helper problems to help you with the clunker. I gave you 8 x 7, 80 x 7 to get 79 sevens. Then I gave you 8 x 6, 8 x 60 to get 8 x 59. Then I might say to students, "Hey, so if you were sort of staying in the same vein, kind of the same pattern that we were just doing, what if I gave you a problem without the helpers? Could you come up with the helpers that would be kind of in the same pattern that we just used to help you find the answer to a problem like 6 x 49?"

 

Kim Montague  12:48

Hmm, okay, so 6 x 49 is really close to 6 x 50, or 50 sixes. And I know 50 sixes is 300. So I need 49 sixes, which is just one six less, so 294.

 

Pam Harris  13:06

And how did you know that 6 x 50 was 300.

 

Kim Montague  13:10

Because I know 6 x 5 is 30. So 10 times bigger is 300.

 

Pam Harris  13:15

Ten times 30 is 300. And then since we only need, we don't need 50 sixes. You only 49 sixes is just one less six. Nice, nice. It's excellent thinking. Thank you. What do you think, listeners? Like this is a way of helping you and helping students kind of think about using a particular relationship. Now that's not the only way that Kim could have multiplied 6 x 49. 

 

Kim Montague  13:38

Right. 

 

Pam Harris  13:39

But we kind of had our brain, or your, hopefully we had everybody's brain sort of thinking about, ooh, this kind of pattern that we were using. And so what's it close to? A little bit too much. So I could think about a little too much, 6 x 50 to help me think about just a little bit under that, 6 x 49, or 50 sixes to help me think about 49 sixes. And that's a Problem String a string or series of problems in order to help learners develop relationships. Now, like I said, that's not the only relationship that she could have used for 6 x 49. Kim, what's another relationship that you could have used for 6 x 49? 

 

Kim Montague  14:14

Yeah, I could have split my place value a little bit. I could have said 6 x 40 and then 6 x 9 and added those two together.

 

Pam Harris  14:22

So it's not like we want that strategy to go away. 

 

Kim Montague  14:25

Right.

 

Pam Harris  14:25

It's just that we want students to own both strategies. We want them to be able to look at both relationships, and then choose in the moment. Which one do you want to use? And one other just sort of tidbit we'll throw in here: Do you know that the mathematicians not only use alternative strategies, but they often will play between the two of them, Tight.  So they'll often think about a problem like 6 x 49. This is a, I'm sharing this day because I didn't know this. Right? So it's important for us to share this message that Kim might have thought about, "Let's see, I could do 6 x 40 and add 6 x 9 or I could think about 6 x 50 because I know 6 x 5, and then just take away a six." Like she has both of those strategies in her repertoire. And so now she has the power to choose. That's empowerment. Empowerment is when you own both of them, and you can choose. If you just only ever learn one strategy, and that's the only thing you can do, that's not power. That's just being stuck, pigeonholed into having only one and only one strategy. So generally, after we've done some Problem Strings with students, we might have done one problem string to sort of introduce the idea. We might do another one to kind of up the ante and get more students thinking about it. But by the time we kind of have most students in the class sort of playing with the strategy, then we kind of cement the learning. So let's do that here. Ideally, we would have done two or three more Problem Strings with everybody, you know, listening so that you'd really get it down. We're on a podcast, that might not work so well. So let's go ahead and cement the learning. What we would do after we get most students sort of playing around with that strategy, with those relationships. So Kim, what's sort of going on here? How would you kind of generalize what's happening as you thought about 6 x 49? So I think that the simplest way that I would describe that is that we want to make one of the numbers nice. It's kind of close to a nice number. And then we can multiply like 6 x 50, that nice number, and then subtract the extra. Right? Yeah. Okay, cool. So this nice number idea, friendly number, landmark number, like, if you're thinking about 6 x 49, you might go, "Ooh, not a nice number." Oh, but actually, it's nice, because it's so close to an even nicer number, right? It's so close to 50. So close to a landmark or friendly number. And if you can find that multiplication, then like you said, subtract the extra. Cool. So we would want to have that conversation, kind of generalized that's what's going on. Okay. So if that's a set of relationships, what are some things we could do with that?

 

Kim Montague  16:55

Are you ready to do some math?

 

Pam Harris  16:57

Yeah. I'm ready.

 

Kim Montague  16:58

Okay. So we're just talking about 49. So what if we said, "What's 49 times anything?" Like, what if I asked you 49 x 24? (chuckles) How would you think about that one? Okay. 49 x 24.  Yep. 

 

Pam Harris  17:12

So let's see. I don't think I know 50 x 24. But I think I could figure it out easier than I can figure out 49 times 24. Because I know that 50 is half of 100. So I can think about 100 twenty-fours, that's 2400. And if I know 100 twenty-fours is 2400, then 50 twenty-fours is going to be half that. And half a 2400 is not too hard. That's just 1200. And so, (laughs) I'm just keeping track of what we're doing. And I gotta tell you, I just got an adjustable desk. And I didn't click the lock on my adjustable desk. And so as I wrote down 1200 my desk went up. Sorry, random podcast humor there. Okay, I'm gonna lock my desk so it's gonna stop doing that. Alright, what were we doing? We're doing 49 twenty-fours, and I found 50 twenty-fours by finding 100 twenty-fours and then finding half of that. So that's 1200. So 50 twenty-fours is 1200. But we only want 49 twenty-fours. So now I have to think about 1200 minus 24. If you listen to the podcast, you've played I Have, You Need with us before. So if I have 24, what's the partner to 100? Is that 76? So 1200 minus 24 is 1176, 1176. Yeah. Right?

 

Kim Montague  18:33

Yep. 

 

Pam Harris  18:34

How'd I do? 

 

Kim Montague  18:34

Nicely done, nicely done.

 

Pam Harris  18:37

Totally cool. So let me just put one, I kind of told on the whole adjustable desk raising thing, because I wanted to make sure that you know that I'm pretty sure Kim, when I gave you that problem string, I think you did that in your head. Because you just sort of hold things in your head a little bit easier than I do. And I wrote down '49 x 24'. And then under that I wrote '50 x 24'. And then under that I wrote '100 x 24' and then wrote '= 2400'. 

 

Kim Montague  19:04

Yeah.

 

Pam Harris  19:05

And I found the 50 x 24. So I went back up to where I'd written '50 x 24' and wrote '1200'. And then went back up to where I wrote '49 x 24' and I wrote '1200 - 24'. Thought about I Have, You Need and wrote '1176'. So it's okay, I want to just give the audience permission, to keep track of your mental thinking. That can be a thing. I am not less than because I don't hold all that in my head. Maybe that's something someday that I'll get better at. I think I have got a little bit better on it. But that's actually not my goal. My goal isn't to be able to hold things more my head. My goal is to be able to use relationships to solve problems, to mathematize. And we're here to suggest that that doesn't necessarily mean that you can just do it all in your head. If you need to keep track, that's totally okay. Because I do.

 

Kim Montague  19:52

Yeah, well, and it's funny because when you call or when we're talking, I often have a pencil in my hand. I'm a pencil not a pen girl, but I often have a pencil in my hand. Because when we're in the midst of stuff, I want to keep track as well. So it's not about you know that I never write anything down either.

 

Pam Harris  20:10

Yeah, and you can tinker. Like, if we were to look at your paper, you just sort of write down what you need, right? It's not like a formal argument. It's not something that you can turn in, or that somebody could actually follow your thinking. Because in the moment, when you're just tinkering and playing, that's not necessary. You're just tinkering and playing. 

 

Kim Montague  20:25

Yeah. 

 

Pam Harris  20:26

Y'all, that's mathematizing. Mathematicians tinker and play and how much fun is that? Alright, so I did some math. Kim, back at you. Are you ready?

 

Kim Montague  20:34

Yup. 

 

Pam Harris  20:34

Okay, how would you solve a problem, like 18 x 13?

 

Kim Montague  20:39

Eighteen times 13. So I like 18, because it's really close to 20. So instead of 18 thirteens, I'm gonna go with 20 thirteens. And I know 20 thirteens because I know 2 thirteens. Two thirteens is 26. So 20 thirteens is 10 times bigger, that's 260. So I'm actually gonna write that down, '20 x 13 is 260'. And then I have too many thirteens. In fact, I have 2 too many thirteens. So I'm going to subtract 2 x 13, which is 26. And again, because I'm familiar with partners pretty well, I'm going to say 260 minus 26 is 234.

 

Pam Harris  21:26

Yeah. And I had to think about that. Nice. 

 

Kim Montague  21:29

Yeah. 

 

Pam Harris  21:30

So you had too many thirteens. And so you're rid of those thirteens. And two of those thirteens is 26. And that's how you solve the 18 x 13. Nice.

 

Kim Montague  21:40

Yeah. Okay, you want, can I give you one with decimals? 

 

Pam Harris  21:44

Sure. 

 

Kim Montague  21:45

Okay. 

 

Pam Harris  21:45

Go for it.

 

Kim Montague  21:46

Here we go. 

 

Pam Harris  21:47

Okay.

 

Kim Montague  21:47

How would you find -

 

Pam Harris  21:48

I have my pen, not pencil ready, go.

 

Kim Montague  21:51

Do you normally write with a pen? 

 

Pam Harris  21:53

I do. 

 

Kim Montague  21:53

Oh okay. How would you solve (laughs) -

 

Pam Harris  21:57

Okay, no judgment there. 

 

Kim Montague  22:00

I need my trusty Ticonderoga.

 

Pam Harris  22:03

I mean, if I go write with a pencil it's a Ticonderoga, for the most part. 

 

Kim Montague  22:06

Okay. 

 

Pam Harris  22:06

Go, go go. 

 

Kim Montague  22:07

Alright, three and nine tenths, or 3.9 times two and two tenths.

 

Pam Harris  22:14

Okay, 3.9 x 2.2. Yeah? 

 

Kim Montague  22:16

Yep. 

 

Pam Harris  22:16

Okay, so I'm going to think about, not three and nine tenths. 3.9 x 2.2. I'm going to think about 4 x 2.2. 

 

Kim Montague  22:24

Okay. 

 

Pam Harris  22:25

And that's not too bad, because 4 x 2 is eight. And so 4 x 0.2 is 0.8. And so 4 x 2.2 is 8.8, or eight and eight tenths. So that's 4 two point twos, but I only need 3.9 two point twos, so I need a tenth less two point twos. So a tenth of two and two tenths or a tenth of 2.2, which is like dividing by 10. That's 0.22 or 22 cents.

 

Kim Montague  22:54

I'm glad you said money. Yeah.

 

Pam Harris  22:56

Yeah. So so far, I've kind of got 4 x $2.20 is $8.80 minus a tenth, which was 22 cents. So I've got $8.80 minus 22 cents. And so I'm going to think about that as, I'm actually gonna think using constant difference. So I'm gonna think about that as eight cents more. So that's like $8.88 minus 30 cents, because I had $8.80 minus 22 cents, and I've got $8.88 minus 30 cents. And that is $8.58.

 

Kim Montague  23:34

Nice. Yeah.

 

Pam Harris  23:36

Whoo. decimal, decimal multiplication, thinking and reasoning using relationships. So why the Over strategy? What like, makes it, bam, such a cool strategy?

 

Kim Montague  23:47

Well, sometimes you're given a problem that doesn't really look all that great, right? Like 3.9 times 2.2, and 18 times 13 do not look so friendly. But it could be really close to something nice. Something landmark. 

 

Pam Harris  24:01

Yep. 

 

Kim Montague  24:01

And the more comfortable that we are with those landmarks, the better we can be at doing stuff that's like a little bit over a little bit under. Right?

 

Pam Harris  24:10

So strategies are not one size fits all. 

 

Kim Montague  24:13

Right. 

 

Pam Harris  24:13

We use them when the numbers make sense to use that relationship. And the Over strategy is just so cool. So to wrap up, next time you find a multiplication problem, look to see the Over strategy might be a good one to use. When you find one tweet it out with the hashtag mathisFigure-Out-Able.

 

Kim Montague  24:32

Hey, and remember to join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday evenings where we explore problems with the world, many problems for which the over strategy is a great one to try.

 

Pam Harris  24:43

Absolutely. And we've created MathStratChat Central where you can find all the problems ever created and have lots at your fingertips. Click on one of the problems and it'll totally throw you into the Twitter feed where you can see all the strategies shared 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.