Pam Harris  00:00

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

Kim Montague  00:07

And I'm Kim. 

Pam Harris  00:08

And we're here to suggest that mathematizing is not about mimicking, or rote memorizing. It's about reasoning; about creating and using mental relationships. We're all about empowering teachers, students, parents, and everybody, to empower them to think and reason mathematically. We answer the question, if not algorithms, then what? Alright, in today's episode, we're going to highlight the commutative property, and how it can make some mental computation really cool. 

Kim Montague  00:40

Mhmm. 

Pam Harris  00:41

Okay, so when kids start to solve problems using counting strategies, because that's where they first do, for solving problems, they use counting strategies. They get really literal. And they always start with the first number. They sort of act out the problem. And so that means they're going to act it out in order, because that's what makes sense to them. 

Kim Montague  00:57

Yeah. So for example, if you had a problem, like: Javan had four pencils. She found eight pencils on the floor and picked them up. How many pencils does she have now? Kids are going to often solve that by starting with those four pencils in the problem. And then we're really, really happy when students realize that they can count on from the larger number, starting with the eight, because then they have less to count on. 

Pam Harris  01:21

Yeah, sure enough, so instead of a student saying, "Okay, four pencils. And now I'm going to pick these pencils up: 5, 6, 7." And sort of keeping track of those extra eight, a student can just start with the 8: 8, 9, 10, 11, 12. They have to count on less, they have to count on four, instead of counting on all of the eight. So counting on less makes sense. We kind of want to bring out that with students. Kim, what are some ways that teachers can encourage young students to construct this idea, help them construct the idea of counting on from the larger? 

Kim Montague  01:52

Well, first of all, they just need to know that it's a thing, right? So that's one reason why we share strategies. So I feel like we'd solve a problem, and I'd have one kid share who did all that counting on starting with the smaller number. And then I'm going to invite someone to share, who saw that they could just start with the eight and just count on four more. And I would celebrate hard how sophisticated their thinking was, and ask the students to consider which one they'd like to do next time, right? So there's also some times where you may need to be even more explicit. I was thinking about a few years ago, when I worked with a student who was struggling to move from counting on to using relationships. And we were doing some individual work. And I found it really interesting, when I was kind of poking around a little bit to see where some of those gaps were, one of the things that I realized was that he wasn't ever counting on from the bigger number. And the reason that he wasn't doing that was because he didn't yet trust that problems, like four plus eight and eight plus four, would give him the same result, he didn't see that they were equivalent. For him, and maybe for some of your students, it's helpful to see that equivalence of problems like eight plus four and four plus eight. So I would absolutely have my rekenrek or my number rack out and ask, what does a four plus eight look like. And then for an eight plus four, it's just that same number rack flipped over, right? So if I have four on top and eight on the bottom to represent four plus eight, then if I flip the rack over, that's going to represent eight plus four, eight on top and four on the bottom. And those are equivalent, they can actually see no beads have been added or removed. They're just arranged differently on top and bottom. 

Pam Harris  03:36

That's a really nice sort of visual concrete way for students to go.  Okay, I'm totally clear that I've represented four beads on the top and eight beads on the bottom. And if I just flip that rack upside down, now I have that other problem that I don't yet trust is equivalent, but I can now sort of experience the fact that it's equivalent. Nice idea, Kim, well done. So that's the community property for addition, that a number A plus a number B is equivalent to the number B plus the number A. A plus B is equivalent to B plus A. And since A plus B is equivalent to B plus A, you can solve either problem because they're going to have the same sum. And so you might want to choose which one that you're going to solve. So a problem with bigger numbers that where this comes in - let's talk about more complicated problems - is anything plus 99. So like if I have anything plus 99, I can think of that problem as 99 plus that anything. So for example, if I say 24 plus 99, then I might want to use the commutative property instead of thinking about 24 plus 99. To think about 99 plus 24. Why, you're asking, why would I want to think about 99 plus 24? Because 99...100 then I could think about 99 to 100 and then just add what's leftover. I was supposed to add 99 plus 24. But since 99... 100, I've already added one, then I just have 23 left and what is 100 and 23? It's just 123. So the problem 24 plus 99, becomes 99 plus 24, which is just simply 100 plus 23, or 123. Maybe times where we don't even really acknowledge we're using the community property that can come in handy when we're adding numbers. 

Kim Montague  05:27

Yeah, as long as we're open to the idea that I would rather solve that 24 plus 99 as 24 plus 100. And then backup one because I really do like the Over strategy. 

Pam Harris  05:38

I will never forget, when you told me that one day, I was like, so excited about being able to Give and Take and this idea of 99, and you're like, "Well, actually, I like to over those", which is totally reasonable. Right? Totally cool. You can completely do that. And it's a great relationship. But yeah, it's not my go to. But yeah, we will allow that. I had a feeling you would say that. Okay. So, Kim, what's another place where using a community property might come in handy. 

Kim Montague  06:05

So if our listeners were available last week, they learned about the Swapping strategy for addition, and that's based on the commutative property of addition as well. 

Pam Harris  06:14

Right. So in Episode 22, we talked about the Swapping strategy, and in case people didn't catch episode 22, although you really ought to go listen to it, it's pretty great. Kim, will you remind us about the Swapping strategy a little bit?

Kim Montague  06:27

Yeah, sure. So that's like when you're given a problem, like 59 and 92. And you want to just rearrange the place values to create an equivalent problem. So like 59, and 92, could be rearranged to create a problem of 99 and 52. 

Pam Harris  06:45

And then you can sort of do either one of the two strategies that you and I just talked about, right? Because I have 99 plus anything and so 99 plus 52 is delightful to solve. Cool. All right, what's another community property gem?

Kim Montague  06:57

Oh, probably one of my favorites. So 10 times anything can be thought about as that many 10s. So like, go ahead. 

Pam Harris  07:06

So sure enough, so instead of thinking about ten 23s, to find 10 times 23, I could think about twenty-three 10s. That's like 23 in the 10s slot, and no 1s leftover, right? So I'm thinking about twenty-three 10s, then I sort of put 23 in the 10 slot. I have zero ones left over. And that's where the zero shows up. So 23 times 10, is 230. I could also think about that, like if I was thinking about 10 times 23, especially a beginning student might think about, "Okay, I need 10 23s 23, 46... I don't even know what comes next." Like they're thinking about literally ten 23s. But if I'm thinking about twenty-three 10s, then I can think about 10, 20, 30, 40, 50. And I could count by ten 23 times. Well, then I could kind of do that in big chunks, big jumps. Like if I need twenty-three 10s, I could think about ten 10s is 100. So twenty 10s is 200. So twenty-three 10s is 230. Like I can make sense of this times 10 thing in our number system, much better than saying "add a zero." Slash that; don't say that. It's much more about thinking about how many 10s we have. So that's a brilliant time that we can use the community property to sort of think about the number of 10s instead of like twenty-three 10s, instead of thinking about ten 23s. 

Kim Montague  08:24

So Pam, what if I asked you to find 23 times 99? 

Pam Harris  08:28

So find twenty-three 99s? Yuck. Okay, cool. So this is a little complicated. Hold on your hat, because I'm actually going to use the commutative property twice here. So the first thing when I think about finding twenty-three 99s, is that I don't really want to find twenty-three 99s, I actually want to find ninety-nine 23, because I think that's going to be easier for me because 99 is so close to 100. If I can find one hundred 23s then ninety-nine 23s will just be one less 23. Okay, so I'm going to focus on finding one hundred 23s  in order to find ninety-nine 23s. So that's the first time that I use the commutative property. But now just like we just said, in order to find one hundred 23s, I don't actually want to think about one hundred 23s, I want to think about twenty-three 100s, and twenty-three 100s is just the number 2300. Right? It's like 23 times 100 is twenty-three hundred or two thousand, three hundred. So I've sort of used it twice. So now that I have one hundred 23s. I can think about that as twenty-three 100s. Wait, Nope, sorry, I can think about twenty-three 100s as one hundred 23s . And so then I can find 99 of them by just getting rid of one 23. That might be a little confusing, but I sort of use the community property twice in there in order to find ninety-nine 23s rather than twenty-three 99s. 

Kim Montague  09:49

That was really well said and it might give our listeners a little bit of insight because you're so able to verbalize that, a little bit of insight into why the commutative property is kind of complex and not just, hey, do this thing kids. 

Pam Harris  10:05

Yeah, especially if we just decided, alright, I'm going to tell you what the commutative property is. And I'm going to write up an example on the board. And I want you to memorize the name of it. So we are not suggesting that. That doesn't work. It doesn't give kids the flexibility that you just heard us using and all these examples. What we really need to do is  engage kids in the problem solving process, and share those strategies and use problem strings to build the commutative property. And use concrete examples like flipping the number rack. Or with multiplication, we really want to be able to represent multiplication on rectangles, because then we can rotate those rectangles and we can notice that if I have a 23 by 99, that's going to have the same area as a 99 by 23. So some of those visual models are gonna be really important in helping kind of concretely build that commutative property in students so that we can use it in examples like we just used. Alright, so I've got a finale for us today, Kim. 

Kim Montague  10:59

Okay. 

Pam Harris  11:00

Alright. So in episodes 20 and 21, we talked all about percentages. We played with percentages, and we threw out lots of percent problems. We actually had an ulterior motive in there. An ulterior motive where we kind of had a goal because you know, there's always a BAM in there somewhere. There's always something that we're kind of leading towards, something that we want to be able to refer back to, or something that we want to be able to kind of keep listeners engaged, keep them on the edge of your seats. Try to figure out what our next BAM is going to be. 

Kim Montague  11:34

So yeah, let me list a couple of those problems and answers and see if maybe you notice a pattern, maybe one that has something to do with the commutative property. So we asked 12% of 50; 50% of 12. And those were both six. And we also asked 45% of 10 and 10% of 45. And those were both 4.5. 

Pam Harris  11:58

So we have some equivalent answers. What's going on? Alright, so in percents, there's a multiplicative relationship between the whole and the percent. So is 10% of 45 really equivalent to 45% of 10? Yes, because 10 times 45 is equivalent to 45 times 10. Wow, that is so cool. 

Kim Montague  12:24

It is so cool. So here's a really cranky problem. Would this be one that you would want to use the commutative property? And that would make it really nice. What about 67% of 20?

Pam Harris  12:36

Okay, so since I know it's a thing, since I know, I can choose to do 67, I can find 67% of 20. Or I can choose to do 20% of 67. Since I know that's a thing, since I have that choice, because I understand the commutative property of multiplication, I'm going to choose to do 20% of 67 every time! Because I can find 10% of 67. That's just six and seven tenths or 6.7. Then I can double that because that's 10%. So double that to get 20%. So double 6.7, that's just 13.4, double 6.7 is 13.4. And so 20% of 67 is 13 and four tenths or 13.4. In other words, I can now look at a percent problem, find some percent of a number and I can ask myself, do I want to find that? The thing that's given to me like A% of B? Or do I want to use the community property to find B% of A? 

Kim Montague  13:34

So cool.

Pam Harris  13:35

So let's try another one. 

Kim Montague  13:36

Okay. 

Pam Harris  13:37

Alright. So I'm going to try to find a problem that it won't really help to use the commutative property, and we'll see how that goes. So Kim, how about 82% of 19?

Kim Montague  13:47

Oh, sounds awful. 82% of 19. So I know that I can do 82% of 19 or 19% of 82. Those are my choices. Ah, and at first glance, they don't seem really nice. But you know what, I would rather swap here, because I would rather solve 19% of 82. Because I can find 20% of 82 and then get rid of the 1%. 

Pam Harris  14:24

Oh, nice. So instead of finding 82% of 19, you're going to find 19% of 82 by finding 20% and getting rid of 1%. Okay, cool. So I tried to come up with a problem where you wouldn't want to use the community property, but you still did. Let me see if I can make it even harder. So we started with 82% of 19. What about 82% of 17?

Kim Montague  14:46

So 82% of 17 or 17% of 82. You know what? 82% is not bad because I can just do 2%, 4%, 8%, 80%.  Like double, double, double times 10. So 2% of 17, 4% of 17. And I'm not going to actually throw out the numbers right here, but the strategy that I would use is double, double, double, times ten to get the 80%. And then just tack on the 2%. To find 2% of 17. Did you follow that?

Pam Harris  15:21

Nice. Yes, yes. Totally cool. Well, okay, let me give you another one. What about 83% of 17?

Kim Montague  15:28

Hum, that one's not quite as nice... 83% of 17 or 17% of 83. I think you've got one. I think that's a nice problem. But we can just chunk some pieces just like we've done before, right? 

Pam Harris  15:42

Sure. It's just that maybe neither of them are pulling at you to use the commutative property either way? 

Kim Montague  15:46

Nah, either way.

Pam Harris  15:48

Cool. Okay, cool. 

Kim Montague  15:49

So we've talked today about the commutative property and where kids can use it at different ages. It's not that they have to name the property, right. You said that earlier. We're not about naming the property, but that they can use the property and figure it with some relationships. 

Pam Harris  16:04

Cool. The commutative property of addition and multiplication! Who knew? So helpful! We hope you enjoyed our journey today. We love it when you like the podcast and review it, that helps more people see it. Check us out on the website, mathisFigureOutAble.com. We'd love for you to join us on #MathStratChat on Wednesday evenings for the global number talk. So if you're interested to learn more math, and 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!