Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned math-er.

Kim  0:09  
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  0:17  
Because we know that algorithms are amazing historic achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:33  
I was going to ask if you did it in one breath. 

Pam  0:36  
I did. 

Kim  0:36  
I think you did.

Pam  0:38  
Could you hear me take a deep breath?

Kim  0:40  
Yes, I did. It was...

Pam  0:41  
I was like, "Do I have the singer's breath?" You know, because when you take a singer's breath, you take a huge breath, and then you use your core to like keep it going. And I was like, "I got core. Keep going. Core, core, core." I made it. I made it till the end.

Kim  0:55  
Hey, in this podcast... Oh man. We teach you about breathing and math-ing, building mathematical relationships with your students, and grappling with mathematical relationships.

Pam  1:07  
We are so glad that you join us to make math more figure-out-able.

Kim  1:11  
Oh, man. I feel like today is going to be a mess. 

Pam  1:14  
Let's have this be a fun one. It will be a fun episode.

Kim  1:17  
Okay. Alright. We have a review from Audible, and it just says, "Amazon Customer." I'm kind of sad about that. I can't say thank you to anyone, Amazon Customer. 

Pam  1:28  
Okay, but we like the review. I mean you know.

Kim  1:30  
I mean, yes, fantastic, random person. It says, "Love, love, love it. It makes math meaningful. Math makes sense and is much easier to teach because it focuses on the right things."

Pam  1:43  
Aww, nice. 

Kim  1:45  
Yeah, I think sometimes we get...

Pam  1:49  
Thanks for loving it.

Kim  1:50  
...focused on the wrong things in education. And I love that you're thinking about what makes sense.

Pam  1:56  
And we're focusing on the right things. That's cool. 

Kim  1:58  
Yeah.

Pam  1:58  
That's cool. Thank you for the reviews, y'all. It helps people find the podcast. And, you know, aren't you glad you found the podcast? So, to help other people, if you get a chance, just a sec, just give us a rating, review. Yeah, we appreciate it.

Kim  2:12  
Yep. Alright.

Pam  2:13  
Cool. 

Kim  2:13  
Last week, we kicked off a short series. I don't know that we said it was going to be a series, but we did kick off a short series about a routine that we love called Factor Puzzles. They are very short on time, but big on impact. We're going to keep talking today. If you did not catch last week's episode or see the blog post about that episode, you can find that at mathisfigureoutable.com/blog to learn more.

Pam  2:42  
Yeah. So, y'all, Factor Puzzles are one of those routines that you can do a little bit of often because Kim says they don't take very long. But do them often. Like, the strength is going to come for kids having a lot of exposure to it. And you get a lot out of it, including heading towards higher math, which is kind of cool. So, here, in brief, is what it's going to look like today. You're going to have a number on the top, two numbers in the middle, and a number on the bottom. So, number on the top, two numbers in the middle, number on the bottom. We usually draw boxes. So, it's a box in the top, two numbers that are sort of straddle that in the middle, and then a box on the bottom that, again, straddles. Okay. So, Kim. Today,

Kim  3:21  
Yes.

Pam  3:21  
We're going to focus on some integer work, but let's just get started with kind of a normal factor puzzle. So, I'm going to tell you that the top number is 12, and the bottom number is 7. Can you remind everybody? Well, just tell us what you're thinking, and then we'll kind of go over the rules.

Kim  3:38  
Okay, I am thinking about the fact that I know that the two middle numbers multiply to the top number. So, I'm thinking about two numbers that multiply to 12.

Pam  3:47  
Okay.

Kim  3:48  
That also have to sum to 7. 

Pam  3:51  
Mmm, mmhm.

Kim  3:52  
The bottom box is the sum. And I know that 4 times 3 makes 12 and adds to 7. So, it must be 4 and 3 in the two middle boxes. 

Pam  4:03  
Nice. And it could be 3 and 4. That order doesn't matter because...

Kim  4:06  
Yep.

Pam  4:06  
...4 times 3 and 3 times 4 is both 12 and 4 plus 3, or 3 plus 4 is both 7. Alright, so those are the rules of the game. Like, Kim just said, the two middle numbers multiply to the sum on the top and they add... Sorry, multiply to the product on the top, and they add to the sum on the bottom. Cool. Next one. The top number is again 12. But the bottom number is -7.

Kim  4:34  
Okay, so I need the sum to be -7.

Pam  4:36  
Mmhm.

Kim  4:37  
And I need the product to be 12. So, I think one of the middle numbers... Ooh, they both probably need to be negative because multiplied, they're positive 12.

Pam  4:55  
Because it's positive 12, mmhm.

Kim  4:57  
Mmhm. So, I need two negative numbers that make 12.

Pam  5:01  
Okay.

Kim  5:02  
Have a sum of -7. So, I'm going to go with -4 and -3.

Pam  5:12  
Final answer?

Kim  5:14  
Final answer.

Pam  5:15  
Because -4 times -3 is 12. And -4 plus -3 is -7. Nice.

Kim  5:21  
Yep.

Pam  5:22  
Nice. So, you have to do a little bit of play in there, right? Like, what are all the factors of 12? Ooh, but then you had to think about some negatives and two negatives multiply, a negative times a negative is the positive 12. Nice. Okay, next one. Top number is -12. And the bottom number is 1. 

Kim  5:43  
Okay. 

Pam  5:44  
And I love that you were telling us how you're thinking, so if you don't mind, keep thinking out loud. That was brilliant.

Kim  5:49  
Okay. So, I'm thinking that if the product is -12, then one of my numbers is negative and the other one is positive.

Pam  6:00  
Because?

Kim  6:00  
But I want my... Because a negative times a positive is a negative.

Pam  6:05  
Ah. Okay.

Kim  6:08  
But I want my sum to be positive, so that means that my two numbers that are being multiplied, I want the positive number to be bigger. So, I'm going to go with 4 and -3.

Pam  6:25  
So, let me check, 4 times -3 is indeed -12. And 4 plus -3 is indeed 1. 

Kim  6:31  
Yeah. 

Pam  6:31  
Nice. And I'm just going to repeat some of the reasoning that you used. So, you had to have a positive and a negative number because the product was negative. And when you were summing them together, you're like, "I'm going to end up with a positive 1. Therefore, of the two numbers, the larger one has to be positive." Yeah, that's really nice integer thinking. And I like the differentiation between when you're multiplying and when you're adding. And it's not about memorizing rules. It's about experience and pulling out what Kim just said and making sure that's all clear. And then keep going. Kids get more experience. Okay, next one. The top number is, again, -12. And the bottom number is -1.

Kim  7:19  
So, again, I want a negative times a positive. But this time, I want my... I almost said my bigger number. I want it to be -4 and 3 because I need more negative than I have positive.

Pam  7:36  
Because of the -1?

Kim  7:37  
Yeah, -1 at the bottom. Mmhm.

Pam  7:39  
Nice. So, -4 times 3 is indeed -12 and -4 plus 3 is indeed -1. Nice. Yeah, nice thinking.

Kim  7:48  
Do you ever have kids say... Because I wanted to say "my bigger number," I meant my bigger digit. But do they ever say like the bigger number? Mmhm.

Pam  7:57  
Absolute value. Or absolutely yes. That's funny. 

Kim  8:01  
Mmhm.

Pam  8:01  
Yeah, yeah, absolutely. In fact, I used to talk about negative numbers as you go to the left on a number line. I would say, "So, you know, as the numbers are getting bigger negative."

Kim  8:12  
Mmhm, yeah.

Pam  8:13  
I wrote that once. Like, I kind of got away with saying it, but I wrote it once, and some folks at the Dana center were like, "That's not right." I was like, "You know, it's like, bigger negative." And they're like, "No." 

Kim  8:24  
Yeah, yeah.

Pam  8:26  
"That is not precise mathematical vocabulary." I was like, "Fine." So, yeah, (unclear).

Kim  8:31  
That's when you go, "You know what I mean." 

Pam  8:33  
Fine. 

Kim  8:35  
Okay.

Pam  8:36  
Okay, cool

Kim  8:36  
Alright. I love that sequence in a Problem String.

Pam  8:40  
Okay.

Kim  8:40  
I'm assuming. Because it gives kids the opportunity to say, "What happens when they make the shift?" Like, the idea of 4 and -3, but then now -4 and 3 in a row.

Pam  8:51  
Mmhm.

Kim  8:53  
That's a nice order to a string. 

Pam  8:54  
Yeah, and so let me talk a little bit about the purposefulness of it. So, if you were to look back at those four problems. If you guys are listening to the podcast, I hope you wrote those down. Maybe restart it and write the problems down again. To see the 4 kind of... It would be great to have these on a board right now just to look at the patterns, and then have students generalize. You know like.

Kim  9:12  
Yeah.

Pam  9:12  
What are you thinking about when the product's positive? What are you thinking about when the product's negative? What are you thinking about when the sum is positive? What are you thinking about when the sum is negative? And then one other very purposeful thing in this particular series of problems is the fact that I kept the 12 constant. 

Kim  9:28  
Yeah.

Pam  9:29  
So, you might say, "Pam, my kids don't know their multiplication facts." Or, "They can't look at a number and see what the factor pairs are." And okay. Then give them experience where that doesn't hinder them. And notice how much experience they just got with 3 times 4. So, you could choose a product and really give kids a lot of experience with that product, but with different signs and different thinking about the products versus the sums. So,

Kim  9:59  
Yeah.

Pam  10:00  
that is purposeful. Yeah.

Kim  10:01  
It seemed... Sorry.

Pam  10:02  
No, go ahead.

Kim  10:02  
It seems to me that by keeping the... I mean, we're even saying the product the same. It's not really because they're positives and negatives. But yeah. But keeping the 4, 3, and 12 the same, you're really drawing out the integer part of it. And like kind of keeping the multiplication part of it consistent...

Pam  10:23  
Mmhm.

Kim  10:24  
So that it highlights. You know, one of the things that we've talked about, I think a bunch, is that when you make changes, you change one thing at a time. So, here to really stay focused on what happens when integers are involved, I think it's nice that you kept the multiplication the same. And summing the same. 

Pam  10:44  
And It doesn't mean you're going to always do it. But in a particular. Like, this is a teaching sequence. And so, in that teaching sequence... And maybe I already said it, but I'm going to say it again. This is a way to give kids access to grade level content. The fact that you're talking about multiplying positive and negative numbers, adding positive and negative numbers. Even if they're not great at multiplicative reasoning yet, with like having all the factor pairs at their fingertips, it's a way of giving access to grade level content because I'm holding the factor pairs, kind of holding them constant. Yeah, cool. I like how you just brought in the "only changing one thing at a time." Let's do a couple more problems. So, this time, the number in the top box is -36, and the number in the bottom box is -5. And if you can talk out loud, love to hear your thinking.

Kim  11:36  
Okay. 

Pam  11:36  
Especially to determine if I have a typo.

Kim  11:40  
Okay, so the first thing I'm thinking about is I first went to 6 and 6, and then I heard you say 5. Like, when you said 36, I was like, "I wonder if she's going to do 6 and 6."  So, then I was like, "No, this has to be 9 and 4." But then I have to think about which one of those need to be negative. Because I know that it's negative 36, so one of these two factors is going to be negative. 

Pam  12:03  
Okay.

Kim  12:05  
So, I actually wrote negative on one of the boxes, but I didn't fill any numbers in yet.

Pam  12:09  
Oh.

Kim  12:09  
Just to remind myself, one's going to be negative, one's going to be positive. And then I want the larger digit to be the negative one because overall, I want it to be negative. so then I wrote -9 and 4.

Pam  12:23  
Nice. So, you're thinking about that absolute value had to be... Yeah.

Kim  12:27  
Yeah. 

Pam  12:27  
You're saying "digit". I'm trying to decide if that works. I get why you're saying it. Okay, cool. So, just to reiterate. -9 times 4 is -36. And -9 plus 4 is -5. One other thing that students are kind of getting experience here... How do I say this? If we teach the way I would teach. If we teach the way I was taught in middle school, I often did a page of problems that were all multiplying negative numbers. And I had previously done a page of problems where I added negative and positive numbers. And so, I could have gotten good that one day, that, oh yeah, when you add then it's sort of take the sign larger. Well, like, whatever, however you're thinking about adding those numbers. And when I was multiplying, I would be like, "Okay, if one of them is negative, then the product's negative. If they're both negative, then the product's positive." I could kind of get good at those on separate days. 

Kim  13:25  
Yeah.

Pam  13:25  
But here I'm pulling them together, and I have to kind of think about if the product is negative, what does that mean about the factors? And if the sum is negative, what does that mean about the addends? And I'm kind of having to juxtapose those at the same time. And I think that's a good thing. We want kids to have not just these separate experiences. Cool. Alright.

Kim  13:41  
Yeah.

Pam  13:42  
Last problem of this string, Kim. What if I had 36 in the top box and in the bottom box we had -13?

Kim  13:52  
Okay, so I know that because my sum is negative but my product is positive, then I'm going to have two negative numbers.

Pam  14:04  
Okay.

Kim  14:04  
As the middle numbers. And I'm thinking again that it's 9 and 4 because 9 and 4 make 13. So, -9 and -4 make -13.

Pam  14:18  
Yeah. Nice. 

Kim  14:19  
-9, -4.

Pam  14:20  
Yeah, so -9, -4 multiply to 36 and add to -13. Well done. Alright, so there is a sequence, a Problem String of factor puzzles to get kids thinking and reasoning about sums and products of positive and negative numbers. I didn't throw any decimals in that one, but you totally could because kids that are doing negative and positive numbers are also dealing with decimals, and fractions, and whatever. So, teachers, you might consider. Do a string like this to sort of start kids thinking about factor puzzles, and then just drop these in independent work, in homework. Like, have it as a warm up, have it as an exit ticket. Like, you could just have one or two of these all year long, as often as you can see to throw them out. I like it when there's a couple together when you get to play with the factor pair. Like, the two that I gave you at the end, both had to do with 9 and 4 and multiplying to 36. So, I like it when there's a couple of them that once the kid has found the factor pair, they get to use it in the next one. I think that's a nice sort of combo pair to give kind of... Maybe I've said it before, but to give them the experience with that particular factor pair, but also with then dealing with how the negatives play out in the multiplication and then the summing. Yeah.

Kim  15:36  
I like what you just said about spending, you know, some time building the idea of factor puzzles, maybe do a string or two with them, but then drop one on occasion. You know, middle school teachers, I hear them talk about time in their classrooms. And I get that it's very real. You know, they have 38, 42, 47 minutes, and so the idea of like any amount of spiraling back to things that they've already talked about is absurd at times. But this is a very short thing to keep thinking about integers and operations of integers. 

Pam  16:07  
Yeah. And so, we mentioned in the last episode, but I'll mention it again, that where this leads is to factoring quadratic trinomials. And you might be like, "Pam, how much time do you need to spend on factoring quadratic trinomials?" I don't think we need to spend as much as some teachers spend, but I do think it's reasonable to do a little bit of. And if we have done this work before we get to factoring quadratic trinomials, where I'm really thinking about ax^2 plus bx plus c, and I'm trying to figure out what those two linear binomials are that multiply to that, and so to do that, I'm really thinking about what multiplies to c that adds to b. If we get kids thinking about what multiplies to a number that adds to this other number, if we get them thinking about that before we ever introduce the x's, and x^2's, and parentheses, and all the things, then kids are dealing with those relationships, and that is desirable outcome. And now, when we hit the factoring quadratics, it's not this arduous, let me give you this, you know, 15 step series of... 15 step sequence of steps... Oh my gosh. This sequence of 15 steps .

Kim  17:18  
Lots of steps.

Pam  17:19  
All the steps. Instead of giving kids a series of steps to memorize to be able to factor those, they'll already be dealing with these relationships. Every high school teacher in the country is saying, "Please do this before they get to us. Please do these before they get to us." And it's not just for them, but it is also giving your middle school students all the experience that we already talked about. One other thing that I just want to mention before we're done. Kim, every time that you talked about your thinking, it had to do with you considering factor pairs that then summed to that sum, that added to that sum.

Kim  17:53  
Yeah.

Pam  17:54  
So, again, we would want to... And I say "again" because we brought it out in last week's episode. But if you haven't listened to last week's, you really ought to go listen to it. Because a thing that you want to help students consider is as they look at one of these factor pairs, like the last one we did. What are the numbers that multiply to 36 that also add to -13? Versus, what are all the numbers I can think of that add to -13? And then let me go see if they multiply to 36. It's much more efficient to think about the factor pairs of 36, and then see if they sum to -13 than it ever is to think about all of the numbers that sum to -13, an infinite amount of numbers that sum to -13, and then check them all to see if they multiply to 36. Go ahead. 

Kim  18:42  
And that's a really real thing that kids will do...

Pam  18:47  
I have seen it.

Kim  18:48  
..in a Problem String. Yeah. And they have to go through that experience. Like, let them learn the experience of trying that, and then hearing from a peer who is like, "Oh, this is why I did that." You know? Or having them try it both ways, and then realizing, you know, through having their experience that there is one more efficient way over the other.

Pam  19:11  
Yeah, nice. We...

Kim  19:12  
We... Go ahead. 

Pam  19:14  
We have some great video of high school students where I threw out, "So, wait a minute. Do you want to think about all of the different addends first? Or do you want to think about the factor pairs first?" And they... The looks on their faces were like, "I don't know. Let me think about that." And they turned, and they had this conversation. Within a very short period of time, they were all like, "Oh, my gosh! there's infinite numbers that add to that sum, but there's only a few factor pairs. We should start with that." And it was like this "aha". And they were like, "Oh, we're so glad we thought of that." It was great.

Kim  19:47  
Yeah, yeah. Well, we love factor puzzles for so many reasons. We'd love to encourage you to check out the blog at mathisfigureoutable.com/blog to read more about factor puzzles.

Pam  20:00  
And play with your kids in the car. Just throw out a number on the top and a number on the bottom and just say, "Hey, what two numbers multiply to that number but also add to this number?" That could be a car game, parents. Car game. There you go. Alright, y'all. Thanks for tuning in. Oh, we might have a third episode next week about going even farther with factor puzzles, so check that out.

Kim  20:20  
Ooh, okay.

Pam  20:20  
To find out more about the Math is Figure-Out-Able movement, visit Math is Figure-Out-Able.com. And thanks for spreading the word that Math is Figure-Out-Able!