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
Chapters
Math is Figure-Out-Able with Pam Harris
Ep: 31: A Cool Multiplication Strategy
Jan 19, 2021 Episode 31
Pam Harris

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 handy multiplication strategy.

Talking points:

  • Are alternative strategies worth learning?
  • What does it mean to do mental math?
  • How to make mental math accessible for everyone.
  • What is a Problem String?
  • The _____ 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 handy multiplication strategy.

Talking points:

  • Are alternative strategies worth learning?
  • What does it mean to do mental math?
  • How to make mental math accessible for everyone.
  • What is a Problem String?
  • The _____ strategy
Pam:

Hey, fellow mathematicians, welcome to the podcast for a 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. 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:

In today's episode, we're going to 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:

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 could pick up one or two of them, but it's not really important. Cause it's just, you know, like this, this huge universe of strategy. That's not actually true. There's really just a few important ones. Today we're going to focus on one of those important multiplication

strategies. Kim:

Yeah. And because we feel so strongly about developing relationships first and then naming it, we're going to actually name this strategy in a bit. So before we get into it,

Pam:

Yep. what do you think people do when they do mental math? When I say the words, mental math, what comes to your minds? So 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 this stuff sort of happening in your head. And that might be true, but we're going to quote Kathy 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. So you do it with you head.

Kim:

Great quote. a great quote from Kathy.

Pam:

Yeah, Do you think that when we say mental math, that you, you see somebody there 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 and someone says ok 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 you think that they are doing the steps. Like they've got the single digit and they carry the one. And then they, 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, that they're able to successfully compute in their heads, they're actually not doing all those steps. They're not mimicking steps of the algorithm at all. 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 crossey outies and carry the one, the magic zero. Like whatever it is they were doing that 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 crossey-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.

Kim:

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. 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

Pam:

Serves you right by the way for not believing me.

Kim:

I know. And we asked our 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 if it was 99 times 37, they 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 onto 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:

Yeah. I mean, so deep down I did, you know, I wondered 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, Oh, well, if I'm not good at it, then 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, see if I can do it. No, I just can't. Like you said, our hearts sort of go out to these kids cause... Could we, could we share something fun? Maybe, 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, 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, cause I can't hang onto 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, those memorized steps that you're mimicking. It's about using relationships. And once we let people know that, that's a thing then they have access to it. That's what mathematicians do. Before in the podcast, we've quoted the research study by Anne dagger, who went to a bunch of mathematicians and said, Hey, mathematicians, will you solve these problems. 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 doing:

repeating steps of the algorithm. Like half of the time? Like 90% of the time? How about 4% of the time. Like 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 and 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:

Yeah. So actually in an earlier episode we talked about using a hundred times anything or a hundred 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:

He said, he said, he, Oh,

Kim:

Yeah, I'm sorry. He said to his daughter who had been interviewed, right. Lexi 99 is so close to a hundred and you could have rounded up one and found a hundred times 37 to get 3,700 and then subtracted that 37. And she lit up and she said, Oh yeah, that would have been so much easier. She just didn't know it was a thing.

Pam:

Isn't that interesting? Yeah. So interesting that a bright young lady, right? I mean, she's precocious she's cute, she's on the video. We're taking the video and the whole thing, but she had never thought about that kind of relationship. Yeah. And then also interesting dad listening in the back is kinda like, Hey, could you kinda think about it this way? Oh! And 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. Yeah. So Kim, how do we build that in kids or, I mean really in, in adults, right? Cause 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 helping students think more and more sophisticated. So y'all, you know, we love to do some math together. So this is going to 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, pen and paper later, if you, if you want to. So don't - no dangerous driving at this point, but we are gonna do a little bit of math. All right. So are you ready, Kim? I'm going to give you the problems. I'm going to ask you to solve them and explain your thinking if, if needed. Okay. All right. Okay. So, first Problem String. Here we go. What is eight times seven?

Kim:

Mm. Okay. That one is a most miss fact, but I know it is 56.

Pam:

All right. So you might use a relationship. You might just have it at your fingertips either way. We want you to be able to retrieve it fairly readily. We call that automaticity, but either way you've sort of got eight times seven is

56. The next problem in the string:

what is 80 times

seven. Kim:

Hmm. Okay. So that's just 10 times more. 80 is 10 times more than eight. So that's going to be the answer as 10 times more. So that's going to be 560.

Pam:

So you're thinking about 10 times and there's this lovely 10 times thing in our number system. So eight times seven is 56. Then you can think about 80 times seven as 560. Cool. Yeah. Next problem. What is, let's see, let me just review that. You just said 80 sevens was 560, right? 80 7s is 560. Next problem is 79 times seven.

Kim:

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:

Sure enough. Cool. Good thinking out loud. All right. So if you've just sort of used some problems to help you think about a kind of a crankier problem, right? Like 79 times seven, you could have done a couple of different ways, but one way that you can do it is to think about 80 sevens to help you think about 79 sevens. Cool. All right. Next problem.

Ready? Kim:

Yep.

Pam:

I can give you another most missed fact. Eight times six. All right.

Kim:

48. So if eight times six is 48, then what's eight times 60?

Pam:

Cause you just sort of scale it by 10,

Kim:

480. right? 8 times six is 48 so 8 times 60 is 480. Cool. Anybody want to guess what the next problem is? If eight times 60, if I've got 60 8s is 48, then what's eight times 59? 8 times 60 was 480. That was 60 8s and I want 59 8s. So that's just one eight less. So 472.

Pam:

Because 60 8s was 480 and one less eight gave you 472, right. 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 eight times seven, 80 times seven to get 79 sevens. Then I gave you eight times six, eight times 60 to get eight times 59, then I might say to students, Hey, so if you were sort of staying in the same vein kind of 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 six times 49? Okay. So six times 49 is really close to six times

Kim:

Hmm. 50 or 50 6s. And I know 50 sixes is 300, so I need 49 sixes, which is just one six less. So 294.

Pam:

And how did you know that 6 times 50 was

300? Kim:

Because I know six times five is 30. So 10 times bigger is 300.

Pam:

10 Times 30 is 300.

Kim:

Right.

Pam:

And then since we only need, we don't need 50 6s, we only two 49, sixes it's just one less six. Nice, nice. It's excellent thinking. 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 six times 49.

Kim:

Right.

Pam:

But we kind of had our brain or your, hopefully we had everybody's brains sort of thinking about, Oh, this kind of a pattern that we were using. And so what's it close to a little bit, a little bit too much. Oh, so I can think about a little too much - six times 50 - to help me think about just a little bit under that six times 49 or 56 is 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. And then, like I said, that's not the only relationship that she could have used for 6 times 49. Kim what was another relationship that you could've used for six times 49.

Kim:

Yeah. I could have split by place value a little bit. I could have said six times 40 and then six times nine and added those two together. So it's not like we want that strategy to go away.

Pam:

Yeah.

Kim:

Right.

Pam:

It's just that we want students to own both strategies. We want them to be able to, to look at both relationships to then choose in the moment, which one do you want to use? And one of their- just sort of a tidbit we'll throw in here - do you know that mathematicians not only use alternative strategies, but they often will play between the two of them. So they'll often think about a problem,

Kim:

Right. like six times 49. I'm sharing this today because I didn't know this. So it's important for us to share this message that Kim might've thought about, Let's see, I could do six times 40 and add six times nine, or I can think about six times 50 because I know six times five, 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, 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've 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. And 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 Problems Strings with everybody, you know, listening so that you really get it down. But we're on podcasts and that wouldn't work so well. So let's go ahead and cement the learning. What we would do after we get most students 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 six times 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 six times 50, that nice number and then subtract the extra,

right? Pam:

Yeah. Okay, cool. So, so this nice number idea, friendly number, landmark number. Like if you're thinking about six times 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 generalize that that's what's going on. Okay. So if that's a set of relationships, what are some things we can do with that?

Kim:

You ready to do some math?

Pam:

Okay.

Kim:

So we were just talking about 49. So what if we said what's 49 times anything like what if I asked you 49 times 24. How would you think about that one?

Pam:

Okay. 49 times 24. Yep. So let's see. I don't think I know 50 times 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 a hundred, so I can think about 100 24s. That's 2,400. And if I know 100 24s is 2,400, then 50 24s is going to be half that and half of 2,400 is not too hard. That's just 1200. And so I'm just keeping track of what we're doing. And I got to tell you, I just got an adjustable desk and I didn't hit the lock on my desk. And so as I wrote down 1200, my desk went up. Sorry, random podcast humor there. Okay. I'm going to lock my desk and it's going to stop doing that. All right. What were we doing? We're doing 49, 24s. And I found 50, 24s by finding half where I finally get 100 24s and then finding half of that. So that's 1200. So 50 24s is 1,200, but we only want 49 24s. So now I have to think about 1200 minus 24. If you've listened to the podcast you've played. I have you need with us before. So if I have 24, what's the partner to a hundred. Is that 76? So 1200 minus 24 is 1,176.

Kim:

Yeah. 1,176. Yeah.

Right? Kim:

Yep.

Pam:

How'd I do?

Kim:

Nicely done. Nicely done.

Pam:

Totally cool. So let me just, let me just put one, I, 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. Cause you just sort of hold things in your head a little bit easier than I do. And I wrote down 49 times 24 and then under that I wrote 50 times 24. And then under that I wrote a hundred times 24 and then wrote equals 2,400. Yeah. Then I found the 50 times 24. So I went back up to where I'd written 50 times 24 and wrote 1200. And then I went back up to where I wrote 49 times 24 and I wrote 1200 minus 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 it's, 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 at it, but that's actually not my goal. My goal wasn't to be able to hold things more in 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:

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 girl not a pen, but I often have a pencil in my hand because when we're in the midst of stuff, I want to, I want to keep track as well. So it's not about, you know, that I never write anything down. And you tinker,

Pam:

Yeah. like if we were looking at your paper, you just sort of write down what you need.

Kim:

Right.

Pam:

It's not like a formal argument. It's not, not something that you could turn in or that somebody can necessarily follow your thinking because in the moment when you're just tinkering and playing, that's not necessarily, you're just tinkering playing. That's mathematizing. Mathematicians tinker and play. And how much fun is that? All right. So I did some math. Kam back at you, are you ready?

Kim:

Yep. How would you solve a problem like 18 times13?

Pam:

Okay.

Kim:

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

Yeah. Pam:

And I had to think about that. Nice. Yeah. So you had too many, 2 too many thirteens. And so you got rid of those thirteens and two of those 13s is 26. And that's how you solved the 18 times 13.

Nice. Kim:

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

Pam:

Sure. Go for it.

Kim:

Here we go.

How would you find - Pam:

I have my pen, not pencil.

Kim:

Okay. Oh, Do you normally write with a pen?

okay. Pam:

Haha.

Pam:

I do. No judgment there. Oh, ok.

Kim:

I need my trusty ticonderoga. But go, if I'm going to write with a pencil it's a ticonderoga.

Pam:

I know, go ,go.

Kim:

Three and nine tenths or 3.9 times two and two tenths.

Pam:

Okay. 3.9 times 2.2. Okay.

Kim:

Yep. So I'm going to think about not three and nine tenths, 3.9 times 2.2. I'm going to think about four times 2.2. Okay. And that's not too bad because four times two is eight. And so four times 0.2 is 0.8. And so four times 2.2 is 8.8 or eight and eight tenths. So that's four two point twos, but I only need 3.9 2.2s. So I need a 10th less two point twos. So a 10th of two and two tenths or a 10th of 2.2, which is like dividing by 10 that's 0.22, or 22 cents. I'm glad you said money.

Pam:

Yeah. Yeah. So, so far I've kind of got four times $2 and 20 cents is $8 and 80 cents minus a 10th, which was 22 cents. So I've got $8 and 80 cents minus 22 cents. And so I'm going to think about that as I'm actually going to think using constant difference. So I'm gonna think about that as 8 cents more so that's like $8 and 88 cents minus 30 cents. that great, Cause I had $8 80 cents minus 22 cents and I've right? Like 3.9 times 2.2 and 18 times 13 do not look so friendly, got $8 and 88 cents minus 30 cents. but it could be really close to something nice, And that is $8 and 58 cents. something landmark. Woo decimal! And the more comfortable that we are with those landmarks,

Kim:

Nice. Decimal, multiplication thinking and reasoning using relationships. So why the over strategy? the better we can be at doing stuff. What like makes it bam, such a cool Strategy? That's like a little bit over a little bit under, Well, sometimes you're given a problem that doesn't really look all right? So strategies are not one size fits all. 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 if the over strategy might be a good one to use. When you find one tweet it out with the #mathisfigureoutable. and remember to join us on MathStratChat on Facebook, Hey, Twitter, or Instagram on Wednesday evenings, where we explore problems with the world, many problems for which the overall strategy is a great one to try.

Pam:

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 that will 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, then don't miss the Math is Figure-Out-Able Podcast because math is Figure-Out-Able!