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

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  
Y'all, we know that algorithms are super cool human achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.  One breath.

Kim  0:34  
It is a long sentence. In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships. 

Pam  0:43  
We invite you to join us to make math more figure-out-able. In one breath. Sorry, I'm just always proud when I can get it in one breath because I don't always. Alright, Kim. Hey, how's it going? 

Kim  0:54  
Hi.

Pam  0:55  
I'm a little allergy today. So. 

Kim  0:57  
Sorry. 

Pam  0:58  
Clear my throat a little bit (unclear).

Kim  0:59  
It's the worst. Worst allergies in Texas.

Pam  1:02  
Central Texas. Hey, cedar. love it. 

Kim  1:05  
Yay. Okay, this is a plea to our very sweet podcast listeners. My heart gets so happy when I get to find new reviews. 

Pam  1:20  
Oh, yeah.

Kim  1:21  
I would like to find some more please, dear sweet podcast listeners.

Pam  1:25  
Yeah, y'all, if you have time, please give us a review and rate. It helps people find it. And it makes Kim happy, so it's a win-win.

Kim  1:34  
Okay, I do have this very sweet one from EWylen, and it says, "From asynchronous professional development, to books, to this podcast, Pam and Kim make all listeners think and question, 'Is there another way?' I started listening and couldn't stop. This podcast is my yard work companion and my dishwashing learning. I truly can't get enough. Keep up the amazing work. I'm so excited to share all I learned with my colleagues and to try some of the younger standards with my own kids."

Pam  2:02  
Oh, that's fun. 

Pam and Kim  2:03  
I love that. 

Kim  2:04  
Yeah, that part, like, "I want to try some things. And the yard work." We have a couple of Journey members, the coaching program, who say they go outside and they do the yard work at the same time as they listen.

Pam  2:17  
I love it. "Dishwasher... 

Kim  2:18  
I love that.

Pam  2:18  
...learning." That's brilliant. 

Kim  2:20  
That's so funny.

Pam  2:21  
And I'm noticing it's a five star rating and the title is Hooked! I love it. Hooked. 

Kim  2:27  
Thank you so much. Very sweet. 

Pam  2:29  
Thanks, for being hooked, podcast listeners, and especially EWylen for giving us a review that makes Kim happy. And, seriously, it helps other people find the podcast, so yeah. We really appreciate it when you can take the time to do that. Thanks. 

Kim  2:41  
Okay, so today's episode, we are going to dive into a routine that we absolutely love, and this is your heads up, if you're in the car, if you're on a run, if you are bike riding, if you're in the garden or washing dishes, you should dry your hands off, grab a piece of paper. This episode is best for you when you can listen when you have time to sketch out. That's probably true for many of the problems we do in a Problem String, but this one is going to be particularly helpful if you can sketch.

Pam  3:15  
Write some stuff down.

Pam and Kim  3:16  
Yeah. 

Kim  3:16  
Yep, yep. Write some stuff down. Pause it now and come on back.

Pam  3:20  
Okay, alright, you're back with your pencil. 

Kim  3:23  
Yeah. 

Pam  3:23  
Or, wait. Pen.

Kim  3:24  
Pencil.

Pam  3:25  
I'm on my iPad today. Okay, so we call these factor puzzles. And I don't know if there's an official name. I think the first time I saw these was in CPM, maybe. I've seen them other places. Yeah, so let's do some factor puzzles. Fantastic routine for anybody grades three and up. Today, we'll probably focus on ones that you would use in grades three, four, and five. How do I say this? You could use them in any grade.

Kim  3:56  
Yeah.

Pam  3:56  
Yeah, but we're going to do some podcast episodes in the future where we'll do ones for higher grades only. Like, you wouldn't want to do in three, four, and five. But today, you could do them three and up. Alright. So, Kim.

Kim  4:07  
Yeah?

Pam  4:07  
This is what I want on your paper. 

Kim  4:09  
Yep.

Pam  4:10  
You need to have four numbers, and they are arranged, so that two numbers are on the top and bottom, and two numbers are in the middle, side by side. So, it's kind of like there's one number on the top, then two numbers underneath that, side by side. Like, centered underneath the top number. And then centered underneath those two numbers is a fourth number. 

Kim  4:31  
Okay.

Pam  4:32  
So, it's... I don't know. What, is like a... I can say this. It's almost like a rhombus. Like, if you think about a rhombus, it's kind of like one vertex, and then two vertices, and then one vertex.

Kim  4:44  
Yeah, and typically we draw them like in like a box. Like, so it's...

Pam  4:48  
Typically, I put a box, and then two boxes next to each other underneath that top box.

Kim  4:53  
Yeah.

Pam  4:53  
One box centered underneath that one. 

Kim  4:56  
Yeah.

Pam  4:56  
Is there a better way to describe those boxes? I don't even... They're squares. There's four squares. One, and then two underneath it, and then one underneath that. 

Kim  5:05  
Yeah. 

Pam  5:06  
Four. Four total. 

Kim  5:07  
Four total.

Pam  5:08  
Okay.

Kim  5:08  
Alright.

Pam  5:08  
So, Kim, in those boxes. In the top center box is 30.

Kim  5:13  
Okay.

Pam  5:13  
In the two side-by-side boxes, 5 and then 6. 

Kim  5:17  
Mmhm.

Pam  5:18  
And in the bottom box is 11. And the question I would ask students, and I'll ask you, is do you see any relationship between these numbers? So, again, 30, 5, 6, 11.

Kim  5:29  
Yeah.

Pam  5:29  
What relationships do you see?

Kim  5:31  
The two middle numbers, 5 and 6 multiply to make 30, which is the top number. And they add together to make 11, which is the bottom number. 

Pam  5:41  
Okay, so just to do one more. So, like can we see if those relationships you just said, do they hold? So, top box is 60. The middle ones are 15 and 4. And the bottom one is 19. Did the relationships hold?

Kim  5:56  
Yes.

Pam  5:57  
Because?

Kim  5:58  
15 times 4 is 60, which is on top. And they add together to make 19, which is on bottom. 

Pam  6:04  
Okay, cool. So, we're kind of multiplying the two middle numbers together to put them on top, and we're adding the two numbers together to put them on the bottom. Alright. So, now I'm going to leave some numbers out. Okay, so the top number is blank, the bottom number is blank, but the middle two numbers are 24 and 2.

Kim  6:22  
Okay.

Pam  6:23  
Okay, what's on the top number?

Kim  6:24  
That's 48.

Pam  6:25  
Because? 

Kim  6:27  
25... 24 times 2 is 48.

Pam  6:32  
Okay. 

Kim  6:33  
And the bottom number is going to be 24 plus 2, which is 26.

Pam  6:37  
Cool. So, you might be bored right now. You're like, "Really?" I mean, it's a little bit of your practicing kind of multiplying. You're practicing adding. It's not really exciting. But here's kind of one that's a little different. The middle two numbers are 24 and 0.5.

Kim  6:52  
Okay.

Pam  6:53  
Or 24, and a 1/2. Alright, so what goes on top? 

Kim  6:56  
So, half of 24 is 12. And that's going to go on top. 24 times 0.5. And then I'm going to add them together and get 24 and 5/10 on the bottom. 

Pam  7:08  
Cool. Next one. I'm going to leave out two numbers again. So, I'm going to give you two numbers. This time, I'm going to give you one of the middle numbers. 

Kim  7:17  
Okay.

Pam  7:17  
So, on the left I put 7.

Kim  7:19  
Okay.

Pam  7:20  
So, you don't have the top number. You don't have the middle right. But you do have the bottom number, which is 10. Now, listeners, like pause for a second. If you have 7 in the middle and 10 on the bottom, what goes on the top and what goes with the other middle number? Go ahead, Kim.

Kim  7:36  
Okay, I'm going to say that 7 plus something is 10. 

Pam  7:40  
Okay.

Kim  7:40  
So, 7 plus 3 is 10. So, the right side box is 3. 

Pam  7:45  
Nice. 

Kim  7:46  
And then I'm going to multiply them now to get the top. 7 times 3 is 21.

Pam  7:50  
So, it was almost a little bit of a puzzle. That's kind of cool. 

Kim  7:53  
Yeah.

Pam  7:53  
It's kind of like you said. I would have written... When you said 7 plus something is 10, I would have written 7 plus blank equals 10. And I also might say, then, therefore, 10 minus that blank is 7.

Kim  8:06  
Yeah.

Pam  8:07  
Just kind of like fact family kind of thing.

Kim  8:09  
Would you also write 7... Or, sorry. 10 minus 7 is blank.

Pam  8:15  
Yeah. 

Kim  8:15  
Until you have all three of those. Mmhm.

Pam  8:17  
Yep, mmhm. Okay, next problem. This time, I'm going to give you a middle number, but it's the right middle number.

Kim  8:24  
Okay.

Pam  8:25  
4. And the bottom number is 6. 

Kim  8:29  
Okay.

Pam  8:30  
Go.

Kim  8:30  
So, the left side is going to be 2 because 2 plus 4 is 6.

Pam  8:35  
Okay.

Kim  8:35  
And the top number is 8 because 2 times 4 is 8. 

Pam  8:39  
Cool. So, so far, it's not real sexy, but we're getting some multiplication and some addition practice. And we're also kind of doing a little bit of subtraction to maybe find that missing addend. Or, you know, kind of missing addend thinking. (unclear).

Kim  8:50  
Yeah, yeah. Can I interrupt to say this is meaningful addition, subtraction relationships practice. Like, not like let's just copy down fact families. Yeah.

Pam  9:03  
Ooh, nice. Yes, cool. Okay, so let's get a little general. If I have these spaces, these boxes set up on a page, and I put in the middle... I leave the top one's blank, the bottom one's blank, but the two middle numbers are the letters C and the letter D. I said letters. C on the left. D on the right.

Kim  9:22  
Yeah. 

Pam  9:22  
What goes on the top?

Kim  9:25  
CD.

Pam  9:27  
C times D, mmhm. And then what goes on the bottom? 

Kim  9:29  
C plus D.

Pam  9:30  
Nice. Next one. This time, I'm going to put C in the same place. The left middle. But I'm going to put on the bottom D.

Kim  9:42  
Okay.

Pam  9:42  
So, C is in the middle left and D is on the bottom. Okay, what else can you get? 

Kim  9:46  
So, I'm going to put D minus C on the right.

Pam  9:50  
That seems interesting. Why?

Kim  9:55  
Because that's what goes there. Because I didn't want to put C plus something is D. 

Pam  10:04  
Ah, okay, okay.

Kim  10:05  
Yeah, because if I subtract C from D, that's what I'll have on the right side. 

Pam  10:10  
Nice. And so, similarly, when we had written 10 minus 7.

Kim  10:13  
Yeah.

Pam  10:14  
On the other one. You're just saying, "Well, it's that number minus that number is going to give me that missing addend. Cool. 

Kim  10:19  
Yeah. 

Pam  10:19  
Okay. So, now I'm going to... Oh, go ahead. 

Kim  10:21  
So, the middle, I have D minus C, and so on the top, I'm going to have C times D minus C.

Pam  10:31  
Which is a little complicated looking. But, yeah. Nice. So, this is an example of algebraic reasoning where we've kind of done a bunch. And I might have done more with students to make sure they were kind of clear on the pattern. But we've done a bunch of them, and then I'm going to say, "Let's get a little general." That's algebraic reasoning. We're going to say, "If I have this number here, C, then I can add it to D, and I'm going to put that sum down here." But, you know, like all the relationships that we're doing. One more. 

Kim  10:55  
Yeah. Okay.

Pam  10:56  
One more general. So, what if I put in the top number? What usually goes in the top box? 

Kim  11:01  
Two things multiplied, the left and the right multiplied.

Pam  10:31  
The product. Yeah. Okay, so this times the top is D and the middle left is C.

Kim  11:12  
Mmhm.

Pam  11:13  
So, we're missing the middle right and the bottom.

Kim  11:15  
So, I'm going to put on the right side D divided by C. 

Pam  11:21  
Nice. 

Kim  11:22  
And then the bottom, I'm going to put C plus D divided by C. 

Pam  11:28  
Cool. You might have to write that down, everybody just to like... Again, ideally, this is better. I am reminding myself that I wish when I had done the sort of generalizing earlier that we had done one where we generalize the division that you just did. So, a better teacher earlier would have said, "Ooh, how did you fill that in?" And then with numbers, we would have talked about it was the product divided by the given factor to get the other factor.

Kim  11:54  
Mmhm.

Pam  11:54  
Yeah. Then, it would generalize better the way you just did D divided by C. Nicely done without having a number example before. Okay, next one. What if I gave you the top and the bottom numbers? We haven't done that yet. Let's have some fun. What if the top number is 21 and the bottom number is 10, but we don't have the two middle numbers? What do you got?

Kim  12:17  
So, I'm thinking of two things, two numbers that multiply to 21. So, I'm thinking about factors of 21. And I know there are 3 times 7 is 21 and those come together to make 10.

Pam  12:32  
Ah. So, you think the two numbers that go in the middle are 3 and 7. Could they be 7 and 3? 

Kim  12:38  
They could.

Pam  12:39  
So, you could switch them around. And we could have brought that up earlier times where we had sort of one number in the middle. And, you know, could they have swapped places or whatever. Nice. So, you have 21 on the top. And you said that was because you were looking for numbers that multiply together that also added to 10. 

Kim  12:55  
Yeah.

Pam  12:56  
And for 21... Well, so... Shoot, I didn't say that the way I meant to. Let me try that again. And, of course, the only answers are 3 and 7 because those are the only numbers that multiply to 21. At least whole numbers. Whole number factors.

Kim  13:11  
Well, I mean 1 and 21 but those wouldn't add to 10. 

Pam  13:15  
But they don't add to 10. Nice.

Kim  13:16  
Yeah. 

Pam  13:17  
So, let's make it a little bit more exciting or fun.

Kim  13:20  
I'm running out of space on my paper, Pamela.

Pam  13:24  
What if I tell you this is the last one? 

Kim  13:25  
Okay, I'll draw some boxes.

Pam  13:27  
Maybe it's the last one.

Kim  13:29  
Maybe.

Pam  13:30  
Okay, the top number this time is 36. Dun, dun, dun.

Kim  13:34  
Fun. Okay.

Pam  13:35  
What do you think? What just happened in your head when I said the top number is 36?

Kim  13:37  
I was like, "Ooh!" Instantly, I was like, "What are all the options that could be?" Like, I was thinking about what is your sum going to be? Because then I can kind of like filter through all the factor pairs?

Pam  13:51  
And am I right that 36 has a whole lot more factor pairs than 21 did? 

Kim  13:54  
Yeah, yeah. Yeah. 

Pam  13:55  
So, this one, or you got some options here to...

Kim  13:57  
Mmhm.

Pam  13:58  
Okay, so this time, I'm going to put in that bottom number 13. 

Kim  14:02  
Okay.

Pam  14:03  
What do you get?

Kim  14:04  
So, I know that a factor pair for 36 is 4 and 9. And those add to 13. So, I happen to write the 4 on the left and the 9 on the right, but it could be 9 and then 4.

Pam  14:17  
Cool, cool. Nicely done. One more.

Kim  14:22  
I guess I'll draw more boxes.

Pam  14:25  
You can do it. You can do it. What if the top number is 36?

Kim  14:30  
I knew you were going to do that! Because I was like, "I wonder if she's going to give me some of 12 next."

Pam  14:36  
Oh! Okay.

Kim  14:37  
Is it?

Pam  14:38  
I'm not going to give you... Yes. It was 12. Okay, do 12, and then I'm going to give you one more. Do 12.

Kim  14:42  
Okay, so 12 is 6 times 6. 

Pam  14:46  
Love that you guess what I'm going to do.

Kim  14:48  
So, 6 times 6 is 36 and add together is 12.

Pam  14:52  
Okay, last one 

Kim  14:54  
Yep. 

Pam  14:55  
Top number is 36.

Kim  14:56  
Yep. 

Pam  14:57  
Bottom number is 37.

Kim  14:59  
Yep. So, 1 times 36.

Pam  15:02  
Bam.

Kim  15:03  
Yeah. And how fun is that, right? Like, if your student...

Pam  15:07  
Yeah. 

Kim  15:08  
...is thinking about all the factor pairs, and then wonders, what if you give me this sum? What if you give me this sum? What if you give me the sum?

Pam  15:15  
And that's a nice, natural differentiation there that if a kid is not kind of bogged down by trying to find the factor pairs, they can kind of get outside the problem. It's a little bit of an extension, kind of. They can be thinking, "Ooh, I wonder which one's going to come next."

Kim  15:29  
Yeah. 

Pam  15:30  
They can also be thinking kind of like I was indicating a minute ago, that if the number in the top box is 21 or 15, there's not a whole lot of factor pairs. But if I give you 24 or 48. Or, for heaven's sakes, 96. You know like, something that is just really has a lot of factor pairs, then you're like, "Oof, I'm going to have to try a lot of those to see which one sums to the bottom one." Hey, Kim?

Kim  15:54  
Yeah?

Pam  15:55  
I'm a little curious. Why... If you don't mind, go back to the one that we had 21 on top.

Kim  16:00  
Mmhm.

Pam  16:01  
And 10 on the bottom.

Kim  16:02  
Yep. 

Pam  16:02  
How come you didn't say, "I'm thinking of all the numbers that add to 10..."

Kim  16:08  
Yeah.

Pam  16:08  
"...and then deciding which of them multiply to 21." Can you talk about that a little bit?

Kim  16:13  
Sure. When I'm thinking about sums to 10...

Pam  16:19  
Mmhm.

Kim  16:19  
...there are quite a few more. There's, you know, 1 and 9, 2 and 8, etc, etc. And so, I would have to go through, filter through, many more options. But when I'm thinking about 21, there's only 1 times 21 and 3 times 7. So, it's just there's less to work towards, less choices.

Pam  16:47  
What about a number like 36 that has a lot of factor pairs? Is that still true when we had 36 and 13? Is it still true that there's fewer factor pairs than there are things that add to 13?

Kim  16:59  
Yeah. There's fewer. And even though we're talking about only whole numbers... I mean, I'm assuming that we're only talking about whole numbers. You gave me one with the decimal, but.

Pam  17:12  
I did.

Kim  17:12  
You know.

Pam  17:13  
And only positive numbers so far.

Kim  17:15  
Yeah. So, if we're talking about numbers outside of positives, then there's like an unlimited amount that I could get a sum of 13. Soon as we switch to negatives, there's...

Pam  17:27  
1 and 14. (unclear).

Kim  17:28  
So many choices. Yeah, it just keeps going and going. Yeah.

Pam  17:34  
So, I hear you saying that if you see a product and a sum, that it makes sense for you to consider all of the factor pairs for the product to see if they sum to the sum, versus thinking of all of the addends that could sum to the sum, to then decide which of those is the correct factor pair for the product. 

Kim  17:57  
Yeah. 

Pam  17:58  
And teachers you might... Oh, go ahead. Go ahead, Kim.

Kim  18:01  
You have me thinking right now. I was just... When you said, are there more for 13 than there are for 36. 

Pam  18:08  
Yeah. 

Kim  18:08  
I was like, "I'm pretty sure there's just an extra few more." So, then I was like, "I wonder if there are ever times where there's more sums than there are factor pairs. And there probably are. 

Pam  18:21  
More sums? I think there are always more sums than there are factor pairs. (unclear).

Kim  18:28  
What if it's a prime number and it's just like 1 times 37? Like, if 37 is the number on top, then you have one factor pair, but you can sum to... What would that be? You can sum to 38 a whole bunch more times. Oh, I'm thinking the other way. Hello!

Pam  18:51  
Agreed, Kim. Agreed, agreed. No matter what the products are, then there's always going to be more. Yeah. And so, what's interesting... Yay.

Kim  19:03  
We agree! 

Pam  19:05  
Alright, so if you teach third grade, fourth grade, fifth grade, right now, you might be like, "Hey, this is a great way for me to get kids to practice multiplication and addition with..." 

Kim  19:14  
Yeah.

Pam  19:14  
"...kind of subtraction ideas and division ideas floating in there." But, man, if you teach high school, and you teach factoring quadratics, you are recognizing some relationships where, if you're factoring a quadratic trinomial, you are really thinking about what numbers multiply to this product, that add to that middle b term, right? That multiply to c, that add to the b in ax^2 plus bx plus c. And that is a... So, algebra teachers. I guess we should have started the episode by saying, "Hey, listen, even if you teach high school..." We highly recommend that you give some factor puzzles to your students at the beginning of the year, and pepper them in your homework, and give them these experiences where it's just the numbers that multiply to one product and add to one sum before you ever get the x^2, and the x's, and the parentheses, and all the things. Before all of that floats in there, get them thinking about these relationships. And one reason I can suggest that is when I began doing these factor puzzles with high school students and really listening to how they were reasoning, I found that there was a whole group of students who were looking at, "Hmm, let me think of all the numbers that add to this sum, and then see if they multiply to this product." Like, that was their strategy. And so, I started doing Problem Strings with high school kids to pull out the idea of thinking there's a much fewer set of whole number factor pairs, then there is integer sums. Is that the right way to say that? Integer numbers that sum to that sum. And so, kids were like, "Oh, I really should think about factor pairs." Was also really interesting to hear kids that don't, that can't look... Yet. Can't yet look at a number, a product and think of its factor pairs. 

Kim  21:04  
Yeah. 

Pam  21:04  
So, that's work that we would want to do, and this is one of the ways we can do that. We can help kids by saying, "Hey, when you look at this number, what are all its factor pairs?" and give them experience in finding factor pairs of numbers.

Kim  21:16  
Yeah. 

Pam  21:16  
(unclear)

Kim  21:17  
We have a lot of older grade teachers who talk about they have kids, students in their classes, who are additive thinkers. They recognize that they're additive thinkers currently, and they know that they want to move them into multiplicative thinking. This is exactly the kind of thing that you can do because it takes a factor approach. It's one of the reasons why we also love Number Club, Stick N' Split so much because it takes a multiplicative factor approach to break apart numbers rather than just think about, you know, a single fact at a time.

Pam  21:52  
Yeah, nice.

Kim  21:52  
So, great for 3-5 teachers. Great for older grade teachers who want to build multiplicative thinking in their students.

Pam  22:00  
And I'm going to give a shout out. You just shout out Stick N' Split. Love it. I'm going to shout out Alice Keeler on social media saw me put in a bunch of factor puzzles and created a Google slide add on. So, I think if you just Google "Alice Keeler" and "Google slide add on for factor puzzles," you can create your own with her Google slide add on. So, that's kind of nice. Or you can just create your own as well.

Kim  22:24  
Yeah. If you are a grade 3-5 teacher, you can use this series. We call this a string, Problem String of factor puzzles. Can use this string and use it with your students. Also, you can visit the blog, the mathisfigureoutable.com/blog to find a blog about factor puzzles.

Pam  22:45  
Yeah, check it out. You might not even know we have a blog. We have a great blog. We'd love to have you go there. If you go there, you can download a blank sheet that then you don't have to draw the boxes that poor Kim was drawing the whole time. So, if you want to download the sheet and kind of see the problems that we did and a whole lot more, you can check out our blog. Y'all, 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 and do a lot of factor puzzles. Thanks for spreading the word that Math is Figure-Out-Able!