Ep 22: The Swapping Strategy

Pam Harris  00:01

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. It's about thinking and reasoning; about creating and using mental relationships. We are all about empowering teachers and students. We answer the question, if not algorithms, then what? Alright in today's episode, we are going to spotlight a really cool addition strategy that some people may use naturally, based on what they know and understand about place value. But for some of you, you may not even know it's a thing. Like I didn't! Like I had no idea that this was a thing to do, or thing to even think about for quite a while. I'm so excited that I do now because it's really empowering. Alright, so Kim, we're gonna highlight something that you did naturally growing up that I had never even thought about. To do that, will you tell us about the experience you told me about with your son Luke the other day? 

Kim Montague  01:02

Yeah, sure. So I was talking with my son, Luke. And I asked him this problem, and it was 94 plus 39. And I said, "Hey, tell me what you're thinking." And he wrote down 93 plus 40. And I said, "Wait, tell me about that." And he said, "Well, the problems 94 plus 39. So I'm just going to take one from the 94, and give it to the 39." And the problem that he wrote down was 93, plus 40. So he would just Give and Take a little bit. 

Pam Harris  01:32

So those are equivalent, right? You could use that to solve, cool. 

Kim Montague  01:35

And I said, "Cool, what else you got?" And he said - 

Pam Harris  01:38

Because that's what you do with your kids. Right? What else you got?

Kim Montague  01:41

You know, I'm just realizing we didn't even ever solve the problem.

Pam Harris  01:46

You just wanted to know what his plan of attack was. So you could talk about that, ok.

Kim Montague  01:50

Okay. I said, "What else you got?" And he said, "Oh, I could do this." And he wrote down 100 plus 33. And I said, "How did you get that?" And he basically said that he did a little Give and Take the other way. He took six from the 39. And he gave it to the 94 to create 133.

Pam Harris  02:09

 Cool, cool. 

Kim Montague  02:11

And I said, "Oh my gosh, that's so funny. I didn't do either of these ways." And he said, "Well, what did you do?" And this is what I did. I wrote down 99 plus 34. And Luke asked why. And I said, "Well, aren't they equivalent problems?" 

Pam Harris  02:24

Okay, so let me just hear that correctly. The problem, the original problem was 94 plus 39. But you wrote down 99 plus 34?  I did.  Wait, what? How?

Kim Montague  02:38

So, if the problem was 94 plus 39, right? It's kind of like having a 90 and a 4, and a 30 and a 9. And I just rearranged the ones that I wanted to put together really? 

Pam Harris  02:53

Huh.

Kim Montague  02:54

Can  you picture that? 

Pam Harris  02:55

So I picture a Splitting model: 94 is made up of 90 and 4, 39 is made up of 30 and 9. And you brought the 90 from the 94 and the 9 from the 39 together, and the 4 from the 94 and the 30 from the 39 together to make 99 and 34. Why would you do that? Why would you make 99 and 34? 

Kim Montague  03:16

So 99, and 34 is really nice, because then I can just grab one from the 34 and give it to the 99. Can you picture base 10 blocks, like if you had base 10 blocks, and you were building those numbers? It's kinda like you're rearranging the ones from one number? 

Pam Harris  03:36

Mm hmm, sure enough, yeah. So if I can picture the base 10 blocks, and you just literally like would move the blocks around a little bit. 

Kim Montague  03:43

Sure. 

Pam Harris  03:43

And then since you didn't lose or add any blocks, you have the same numbers. Cool. 

Kim Montague  03:48

Yeah. 

Pam Harris  03:48

Very cool. And then you have this brilliant problem. 99... 100 plus what's leftover, and you can just solve it that way. That's really nice. Alright, Kim. So you're sort of natural at this. Give us another problem that we can all work on that would work for this Swapping strategy. What would be a really sweet problem that we would all want to swap a little bit? 

Kim Montague  04:07

Well, it's funny you asked because I actually gave Luke another one. Here's the problem I gave him. 

Pam Harris  04:10

Of course you did.

Kim Montague  04:12

I said, "What about 159 plus 92?"

Pam Harris  04:17

Okay, let me think, 159 plus 92. What blocks would I want to rearrange? I think I might want to take the 90 from the 92 and swap it out with the 50 from the 159. So the 159 becomes 199 and then I'm left over with the 50 and the two. So 199 plus 52, 199 plus 52 is 200 plus a leftover 51, 251. Bam! That is really cool. 

Kim Montague  04:53

Nice, right?

Pam Harris  04:53

 It's really cool. 

Kim Montague  04:54

Okay, let me give you another one. 

Pam Harris  04:56

Okay. 

Kim Montague  04:57

The next one I asked him was 929 and 191. 

Pam Harris  05:03

929 and what was the second number?

Kim Montague  05:06

191.

Pam Harris  05:07

929... 191. I'm noticing lots of nines in here. Okay. So if I have that 929, I'm going to swap out the 20 in 929, with the 90 in the 191. So that now I have 999. And then what would be leftover would be 121. Right, 121? So now I have 999 plus 121. So that's 999...1000. Plus 121. Is... is - wait. Hang on. 999...1000 plus a leftover 120? Bam, that's just  1120. Woo!

Kim Montague  05:48

Yeah, nice. Right? 

Pam Harris  05:49

That is really interesting. 

Kim Montague  05:50

Yeah. So after I did those problems with Luke, I wanted to see if he had kind of a like a handle on what I was doing. And I asked him to come up with some problems where - I kind of think about that, like a swapping. So I wanted him to give me some problems, where that would be a nice kind of slick idea. And you know what he did? I said, "Hey, give me a problem. Maybe like a bigger problem where that might be a strategy that would be useful." You know what he did? All he did was add a nine in the thousands place, and increase the other number by 10. So he just created 9939 and 191, such as stinker.

Pam Harris  06:29

Oh, wow. That's awesome. So then you could just make 9939 into 9999 plus 131. That's really nice. Because 9,999 is so close to 10,000. 

Kim Montague  06:47

Right. 

Pam Harris  06:47

Yeah, that's so so his way of making a whole new problem was to just tack on a couple numbers.  Okay, cool. That actually shows really good place value, right? Often the laziest thing that somebody can do is kind of cool, because you're showing that you really understand what's happening. So then when he did that, what did you do to push him a little bit? You did, right? 

Kim Montague  07:11

Yeah, I kind of eye-rolled a little bit. And I said, "No, for real. Give me something that's a little bit like more fifth-sixth grade." And so he randomly stuck a decimal point in the middle of those big numbers. And he just made 993.9 plus 19.1.

Pam Harris  07:29

That's awesome. 

Kim Montague  07:30

Sure. 

Pam Harris  07:31

Okay, so I actually want to think about that one. So 993.9 plus 19.1. I'm going to swap out the three from the 993.9. And the nine from the 19.1 to get 999.9 plus the leftover 13.1. Okay, so if I just give that .1 to the  999.9. Now I have 1000. 

Kim Montague  08:02

Yep.

Pam Harris  08:03

Is that right? 

Kim Montague  08:04

Yeah. 

Pam Harris  08:05

Yes. So then I have 1000 plus the leftover 13. 

Kim Montague  08:10

Yep. 

Pam Harris  08:11

And what is 1000 and 13? It's 1013. Okay, cool. Wow, that is, that seems really interesting and a great understanding of place value. Because, Kim, you and I've talked about before that place value understanding is different than a lot of what units out there are called place value. Like I see a lot of third grade, fourth grade, sixth grade teachers, fifth grade teachers now because the standards are calling for a little bit younger to do decimal understanding, that a lot of place value units, there are these units out there that are called 'place value', but in reality, they're much more about 'place labeling'.

Kim Montague  08:48

Yes. 

Pam Harris  08:49

Right? Ya'll, if you can think about the units that you have in traditional textbooks or anything, and it says 'place value', really think about what kids are doing in those place value units. They're really labeling a lot of places. Now, that's necessary, it is necessary that kids know the names of the places in our place value system. That's necessary that they can label those places. But it's not enough. It's necessary, but it's not sufficient. There's a good math phrase for you: necessary but not sufficient. We need a lot more place value work. And I'm going to suggest that the kind of stuff that Kim and I were just playing around with where we're kind of thinking about these numbers in their place values. And so we could just swap out the place values and two numbers, that's place value, that's really understanding that the value of the digit depends on the place that it's in. That's what place value means and to understand place value, to be really like conversant with place value, really own the relationships, then we need students to have experience messing with place value, not just labeling the places. So I'm going to suggest that we work with students with the Swapping strategy, that there are some Problem Strings we could do to help students with the Swapping strategy to develop it. Not because students need the Swapping strategy, but because we're trying to develop mathematicians. And mathematicians play with relationships like place value. And because they understand place value deeply, that exploring the Swapping strategy is one way to get at that deep place value understanding. You might notice that while we're playing with place value, we're also having to name the places a lot, right? 

Kim Montague  10:25

Right.

Pam Harris  10:27

As we're playing with place value, doing the work that we just did, we had to name the places. We had to sort of label them. So we get a lot of practice with the labeling in when we do real place value work, because it comes out naturally. Both things come out naturally, when we play with place value. If we're only playing with place labeling, that's all that kind of happens. So instead, we're recommending do something like mess with the swapping strategy. And students will understand both better: both place value and place labeling.

Kim Montague  10:56

You're so right. In today's episode, we played around with place value and the addition Swapping strategy. So listeners we'd like to invite you to join in on the fun by creating and sharing with us a problem that the Swapping strategy would be so cool for. You can ping us on Twitter @pwharris, and tell us about that really gnarly problem that was made so much more doable by just swapping a little bit. We'd love to hear what you come up with and maybe make it a little bit different than Luke where he just added another nine or put on a decimal point. 

Pam Harris  11:29

Like maybe come up with your own cool problems or you can be a little lazy but you can't use the ones that we used on the show today, right? So @pwharris, is where to ping me on Twitter. We'd love to hear your suggestions about what you think would be really cool, slick problem that if you just do a little swapping turns into a problem that's much easier to solve. Alright everybody, we really appreciate when you like the podcast and when you give it a review that helps other people find it. You can check out more about Math is Figure-Out-Able at our website mathisFigureOutAble.com. We'd love for you to join us at #MathStratChat on Wednesday, on your favorite social media platform. So if you're interested to learn more math, if you want to help students become mathematicians, where they mathematize their world, then the Math is Figure-Out-Able Podcast is for you. Because math is Figure-Out-Able!