Ep 139: Factor to Add and Subtract!?

Pam: 0:01

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

Kim: 0:07

And I'm Kim.

Pam: 0:08

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim: 0:36

So, this week, we are going to do... Well, actually, Pam, it's Valentine's Day, right? So, we're going to tackle another Problem String that we love to do.

Pam: 0:45

We love Valentine's day.

Kim: 0:46

For a strategy that we love. That we now both love.

Pam: 0:51

[Pam laughs] Now. Yeah. Now.

Kim: 0:54

[Kim laughs] So, we're going to do a Problem String today to tackle a very cool strategy for addition, subtraction that.

Pam: 1:03

And the reason we're laughing is because we'll tell you in a hot minute about the fact that maybe it wasn't a favorite to begin with, and why we like it now. Cool. So, I remember where I was standing. I can picture the room. I know who was in the room. I was doing professional learning here in the state of Texas. They had come and asked me to do eight days of numeracy training for K-12 leaders across the state. And it was fantastic. We did four days right in a row. We were downtown Austin. We were in a hotel, ballroom kind of space, and with way too many leaders in the room, but they were amazing. And we did four days in a row. And then, we'd let some time pass, and then we did another four days. And it was a training of trainers kind of a thing. And phenomenal experience. I had such a good time. We had so many really, really wonderful leaders in the state of Texas at that point. And I remember, near the beginning of the workshop, we were doing some subtraction. It might have been addition. I don't know. It doesn't matter. It was addition or subtraction, and I had a particular strategy in mind, and I gave a problem like... So, I don't remember exactly the problem, but I gave a problem like 63 minus 28, and I was looking... Oh my word. I'm sitting at my moveable desk, and it's going up on me. I reached for my pen, and all of a sudden I hit the button. I'm sorry, I can't. I had. Okay, here we go. Put me on live recording as we go. So, I'll never forget, I was looking for 63 minus 28. In the midst of that Problem String, I don't remember exactly what strategy because with that problem, I could have been trying to nudge a couple different strategies, but let's just say it was going for the Over strategy. So, I might have had 63 minus 30 as a helper problem, and then I gave 63 minus 28. And I'm expecting everybody to think about 63 minus 30, and then how would you adjust to get 63 minus 28. And there was a physics teacher, who was a... She was a leader, but she was like more physics than math. You know, how science people kind of identify sometimes as like, "I'm a physics..." And she was wonderful. I'll never forget. She had this look on her face, and I was like, "Oh, cool. Like, she's really..." You know, "I've taught her something." And I called on her, and I said, "Hey, what are you thinking about?" And she goes, "Oh, do you want to know." I was like, "Yeah, yeah, yeah." Like, "Tell me what strategy you're thinking about." Now, I've learned if I want the Over strategy, in that moment, I should have said, "Did anybody use the problem before?" if that's the one I want to celebrate. But in this moment, I didn't know that move, and so I'm kinda calling on anybody cold. But I was so excited. She had this look. I was like, "Oh, I want to hear what you're thinking about." And she said,"I cannot just see factors." And I said, "I'm sorry what? Are you in the room with us right here?" Like, "You're off somewhere. You're thinking about something else." She goes, "No, like I can't... I see 7s." And I was like, "What? 63 minus 28." And she goes, "Yeah, I see nine 7s, and I see four 7s. And I just can't stop. I can't stop seeing that. I see nine 7s is 63 and four 7s is 28, and it's just like screaming at me that therefore nine 7s minus four 7s would be five 7s. And I know that's 35. And my brain is just there." And I looked at her. I was like, "What?" I had never. I was like, "That's what your brain did. Like, how do you? How did your brain go there?" She goes, "I don't know. It just did. Like, the factors are just screaming at me." Ya'll, I don't know how long I stood there with my mouth open. I was like, "Oh, does anybody else think that way?" Now, honestly, I think when I said that, there were a couple of hands in the maybe 50 or 60 people that we had. It was way too many people for one training. So, not very many people, but most people were intrigued. Most people were like, "That's interesting. How often does that happen? Is that a thing? Do people actually do that?" Well, ya'll, I'll tell you what. The more I throw problems like that out, the more I find that people, if they are thinking multiplicatively, if they have built relationships in their brains to think multiplicatively, factors can pop out at the most opportune times. So, let's do a Problem String, Kim, to see if we can kind of build that strategy in our listeners a little bit. Here we go. Alright, first problem. Kim, what if I were to ask you a problem like 88 plus 72. 88 plus 72. Tell us what you're thinking about.

Kim: 5:32

So, 88 and 72 have 8s in common. So, 88.

Pam: 5:39

How so?

Kim: 5:40

Eighty-eight is eleven 8s. So, I just wrote down 11 parentheses 8. Eleven 8s.

Pam: 5:46

Okay.

Kim: 5:47

Plus 72 is nine 8s.

Pam: 5:49

Mmhmm.

Kim: 5:50

And so, eleven 8s plus nine 8s is twenty 8s. Which is really nice because that is 160.

Pam: 5:57

How do you know twenty 8s is 160?

Kim: 5:59

Two times 8 is 16, times 10 is 160.

Pam: 6:02

Nice. So, you can look at an addition problem like 88 plus 72 and factors popped out for you, and all of a sudden you have this nice. Now, if you would have gotten nineteen 8s.

Kim: 6:16

Yeah.

Pam: 6:16

I wonder if you might have gone back to addition and done like an Over strategy or something. Or even for these numbers, Splitting by Place Value. Sort of that nice 10 sticking in there. So, it's not like this has to be the strategy that you do if factors pop out at you, but it could be one that you consider. Next problem. Maybe I'll pause a little bit longer before I let you answer it, so. Or, listeners, solve the problem before you maybe hear what we're doing. You know, give yourself a chance to see if factors can pop for you. So, Kim, how about 81 minus 36?

Kim: 6:51

81 minus 36. So, I'm looking for what they have in common, and I'm thinking that there's some 9s there. So, 81 is nine 9s and 36 is four 9s. So, nine 9s minus four 9s is five 9s, which is 45.

Pam: 7:13

Nice. And often kids know they're 5s, so that's a pretty five 9s once you get there. I mean, I know you know. But often kids if they're going to know facts, they'll know the five, so that's not too bad to land up with 5 times 9. Cool. Alright, next problem. How about(unclear).

Kim: 7:30

I'm going to tell you real quick.

Pam: 7:31

Yeah, yeah, yeah.

Kim: 7:31

I actually like that strategy more than I like really any of the strategies that I would use for subtraction of that problem.

Pam: 7:41

Oh, that's interesting.

Kim: 7:42

Yeah.

Pam: 7:43

So, I might be tempted to shift everything up 4 and have 85 minus 40, which is also 45. Which is good, because it should be. But you're like, "No, I actually like that..."

Kim: 7:56

Yeah, for that one. I really do like that strategy.

Pam: 7:58

Okay, nice. Nice. Alright, how about 77 plus 44.

Kim: 8:06

There's some elevens there. So, 77 is seven 11s, and 44 is four 11s, so then I have eleven 11s, which I know is 121.

Pam: 8:20

And if a kid didn't know eleven 11s?

Kim: 8:22

Mmhmm.

Pam: 8:24

Then?

Kim: 8:25

They might do ten 11s, plus one more 11. So, 110 plus 11.

Pam: 8:30

Yeah, I like it. I like it. Or they could even back up and just do an addition strategy.

Kim: 8:35

Sure.

Pam: 8:36

So, our goal isn't to pigeonhole anybody into any one strategy.

Kim: 8:41

Right.

Pam: 8:42

Our goal is to own lots of relationships, so strategies become natural inclinations.

Kim: 8:48

Yeah.

Pam: 8:48

I'll never forget the day, Kim, when you said to me, "I look at numbers, and I let things pop, and I go, 'That? Mmm, how about this? Mmm, how about that? Oh, actually, the middle one.'" Like, you consider different things you can do, and then you choose the one that on that day sort of slides the easiest.

Kim: 9:05

Mmhmm.

Pam: 9:05

And I'm working on my definition of fluency, but that's my definition of fluency. My definition of fluency is looking at the numbers, letting what naturally pops, that intuition tells you, and then going with the one that... You know like, considering all the things that pop. Not all, but enough of the things that pop, and then going with the one that slides the best that day. I know that's not a very technical definition, but I'm working on it. What it's not is that students go, "Oh, for this problem, I do this procedure. Step, step, step, step." That's not fluent. Okay, ready? Next, 64 minus 48.

Kim: 9:44

So, they have 8s in common. So, 64 is eight 8s, and 48 is six 8s. So, eight 8s minus six 8s is just two 8s, which is 16.

Pam: 9:59

I wonder, listeners, as you're hearing these problems, if because we've kind of got you in factor mode that maybe you're like... Maybe it's starting to ping a little bit more. Maybe. Maybe it's starting to like because you're... And that is a real neurological thing that once we kind of get you situated, and factors are kind of the thing that we're talking about, then they might pop a little bit more readily. Alright, let's try another one. How about 144 plus 96?

Kim: 10:33

Cooper would love this one because he loves 12. So, 144 is twelve 12s and 96 is eight 12s. So, twelve 12s, plus eight 12s, is just twenty 12s, which is 240.

Pam: 10:48

Because two 12s is 24 times 10.

Kim: 10:51

Mmhmm.

Pam: 10:51

Yeah. Now, I'm going to be really honest. 96, I had to think about. I did not have a lot of experience with 12s. It is still the one that I have to go, "What do I know? Let's see, I know 7..." For some reason, I know 7 times 12. I think I actually work off of 6 times 12. I know 6 times 12 well, and so 7 times 12 is pretty, and so I'm like, "So, 96... Ah, so that's got to be 8 times 12."

Kim: 11:17

That would probably be... Go ahead. Sorry.

Pam: 11:20

Ninety-six does not pop for me as a 12. However, because the 144 is sitting there. 144 is so 12 squared to me. That one is just ingrained. Then, I'm kind of asking myself, "Is there a 12 in 96? Oh! There is a 12 in 96."

Kim: 11:38

Yep.

Pam: 11:38

Yeah, what were you going to say?

Kim: 11:39

I would say that's probably would be true for me until about a year ago that I don't own 12s as well as some of the other ones, but because of Cooper. He so loves 12s that we talk about them probably far more often than we should in this house.

Pam: 11:57

So, since you've been messing with 12.

Kim: 11:59

Yeah (unclear)

Pam: 11:59

Twelves is at the top of your head more and more, and so now it's a relationship that you can draw on. Yeah, nice. Do you know what else is funny about... No, I'll tell that some other time. Okay. So, to end this Problem String, I'm going to ask the question, and we're not actually going to answer it on the Problem String, but I'm going to ask a question that we just put out on MathStratChat. So, we've been doing... Might have been three or four weeks ago. We've been doing a series of Problem Strings to try to build this strategy, this idea that to add or subtract, I can see if there are common factors involved. And if factors pop, then I can use those factors to think about adding or subtracting. And so, one of the last problems that we posted was 25.6 subtract 12.8. And now that I've done that, now that we've done this Problem String today, and we've been talking about this idea of factor to add or subtract, we're going to leave that problem, listeners. And maybe if you go check out MathStratChat, you can search for hashtag MathStratChat and see what you can do with 25.6 minus 12.8, and go see what other people have done for that problem.

Kim: 13:11

Yeah. So, I remember being in the hotel when you... And I don't remember the details exactly, but I feel like what happened was this person shared this strategy, and you were like, "That's amazing! I want to come back and think about it." And if I'm remembering it correctly, you came back and you were like, "I need to write a Problem String for that." Or...

Pam: 13:34

[Pam laughs] Yeah.

Kim: 13:35

(unclear) typically your response, right, when somebody shares something cool.

Pam: 13:37

Oh, yeah. Because I was so fascinated that I wanted my brain to do that, and so a way I do that is I try to write a Problem String to teach other people to do it, and that helps gets into my brain.

Both Pam and Kim: 13:47

Yeah.

Kim: 13:47

Yeah.

Pam: 13:47

Yeah.

Kim: 13:48

So, what I do remember is that.

Pam: 13:53

Is that one of those moments when you were like, "Oh, Pam."

Kim: 13:58

I don't remember. But I will say that I remember that I was there when you were tinkering with it, and then there when you did a Problem String, either that day or some other time. And I remember. We'd had this conversation. But I remember being like, "Yeah, I mean, I get it, but I'm not all that impressed." I remember thinking like, "It's cool, but nobody's going to do that." Like, "Why would you ever go to factors and thinking about multiplicative relationships to solve a really straightforward addition or subtraction problem?" And I'll tell you what, I wasn't really convinced that people would actually do it. And really, you said, "I've run into this" and whatever, and I guess I kind of believed you, but not really.

Pam: 14:51

[Pam laughs].

Kim: 14:51

And it was never one that I really used. But when you started putting out the MathStratChat problems, that had a factor to add or factor to subtract.

Pam: 15:01

Just recently.

Kim: 15:02

Recently, right?

Pam: 15:03

Yep.

Kim: 15:04

You know, when MathStratChat comes out, I hold my phone in front of my children's faces and asked what they would do. And Luke, my older son, looked at the problem, and he said an answer. And I said, "How do you know?" And he used factor to add or factor. Whatever it was. Addition, subtraction.

Pam: 15:23

Whatever it was. He totally did it. Oh! Go, Luke!

Kim: 15:24

And he did it, and I was like, "What?!" Because we've never talked about it before, and he just uses those relationships. And I remember texting you I'm like, "Alright, fine." I'm like, "Okay, people do."

Pam: 15:38

Oh, ye of little faith. Way to go, Luke! Oh, I love that kid. Good. You know, it's interesting, because at the same time period, my particular son was... I don't even remember why. But he said to me, "Wait, what's that strategy?" And I said, "Well, you know like..." And I gave him the problem 81 minus 27. I said, "So, you know, 81 minus 27..." And he goes,"Oh, well, I can clearly see nine 9s, minus three 9s. So, it'd be six 9s." And then he goes, "But I don't really know six 9s, so I wouldn't do that. I would just do minus 30, plus 3." That's like.

Kim: 16:11

Yeah.

Pam: 16:11

But that's an example. And that's my oldest son, by the way. The one that got me into all this. That's an example of someone who owns relationships, who says, "Oh, okay. I could do that thing." So, he had never thought about factor to add or factor to subtract. But once I raised it, he goes, "Oh, yeah, I could totally see the 9s in there. But that would give me 6 times 9, which I don't know, so I probably would have just done..." It's exactly what you said, Kim. People with relationships go, "I could do this. I could do that. I could... Oh, I'm going to go with the one that, today, slides the most easily, that my intuition is just like..."

Kim: 16:48

Yeah. And I think the extra piece of that, Pam, is that you... It's perfectly okay to get started down the strategy, realize that where it's going to leave you is something that you don't want to mess with or doesn't make, you know, your life easier, and so then you back out of it. And I don't know that... I think we've said it before in the podcast, but it is not a magic like, "Oh, what do I want to do?" Like, pick the right thing. Sometimes you try some strategies, and you go,"Nah, I'm going to back out, and I'm going to do something..."

Pam: 17:20

"Yeah, not today."

Kim: 17:21

Yeah, yeah.

Pam: 17:21

Yeah, absolutely. And that's mathematical behavior.

Kim: 17:25

Yeah, absolutely.

Pam: 17:26

And hopefully it can free everybody up. And maybe one other thing that we'll mention about this is, this is a really cool strategy. But you'll notice that it's not in our major strategies ebook.

Kim: 17:37

Yeah.

Pam: 17:38

"Why, Pam, if it's so cool?" It's a fun, interesting, great strategy to use. And I would use it. I would do Problem Strings like this with students who you're trying to get them to think and reason either additively and you've got some kids that are already reasoning multiplicatively. And so, throw in some numbers that have common factors, and that can keep those kids that are already reasoning well additively. It can keep them going. It can like, "Oh, they have something to think about." Or, use it with kids that you want to work on multiplicative reasoning. You want to work on sort of some single digit facts with multiplication, but you want to do it in a little bit of a different context. Oh, well you could do it here, where you're hoping those factors begin to pop a little bit more. Consider, if you (unclear) a Problem String like we just did in this podcast, right before you do some things like adding or subtracting fractions, where you want denominators to pop as with common factors, this could be something that could kind of get that warmed up. Now, those numbers are popping with common factors, and now you're like,"Oh, so if I'm looking at common denominators... Bam! I've got these common factors popping out that could be sort of helpful there." So, it's not a major strategy, but it is a fun relationship that we can play with if our goal is to get kids to think and reason like mathematicians.

Kim: 19:05

Yeah. It reminds me a little bit of how much we love the swapping strategy, which is not something that is on our ebooklet either, but it's a really cool strategy. And when the numbers are there, they're primed for that strategy, why not use it? It's cool and fun to try new things.

Pam: 19:26

Yeah, and maybe I'll just clarify one other thing. So, why is it not a major strategy? The major strategies that I've chosen, I've chosen because those are the ones kids need to own. Those are the ones we need to work on, so that they can solve any problem that's reasonable to solve without a calculator.

Kim: 19:41

Yeah.

Pam: 19:42

So, that's why those are the major ones. We need to work on those. But there are other fun ones that we can also build in kids toward the aim of creating them as mathematicians. That's the difference between if we've labeled something as a major one versus not. And maybe we can get into that more in a different podcast.

Kim: 20:00

Yeah. And maybe if people don't know about that. They're new listeners, and they don't know what we're talking about when we're talking about the major strategies ebooklet.

Pam: 20:07

Oh, yeah, we should probably give them that.

Kim: 20:08

We'll throw that link in the show notes as well.

Pam: 20:10

Absolutely. And it's mathisfigureoutable.com/big. Because it's a big, free ebook that you can get. mathisfigureoutable.com/big. You can download that free ebook. If you haven't already, check it out. Where you can find those major strategies. Hey, ya'll, thanks for tuning in and teaching more and more Real Math! To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!