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

Kim  0:10  
And I'm Kim Montague, 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  
We know that algorithms are super cool 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  0:31  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:38  
Y'all, thanks for joining us to make math more figure-out-able. Alright, hey.

Kim  0:44  
I never realized that the breathing thing, but now I've heard you three episodes in a row. I did not know that that was... You've held it well. I think you've mentioned it before, but I didn't know it was really that sort of thing. 

Pam  0:49  
Maybe I'm out of shape, you think? 

Kim  0:58  
No, I buy that talking for a long time without breathing is a thing. Okay. Today, part three. Factor Puzzles, part three. We tackled whole numbers. It might have had a decimal in there somewhere. We tackled integers. What is up for today?

Pam  1:17  
Today, we are going to go farther into the future into higher math. We are going to do some stuff with algebraic expressions in factor puzzles.

Kim  1:29  
I actually... You're right. I saw that. Because... Can I say what your note to self is? 

Pam  1:36  
Oh, okay. Yeah, what? 

Kim  1:37  
Listen, listeners. There are some days, some episodes, where we're like, "Hey, we're going to do some math." And Pam's like, "I'll do a string." And, you know, that's kind of it. That's like, the whole note. And there are other times where we want to talk and, you know, we leave a few notes. For sure, we have the podcast intro always in front of us because...

Pam  2:01  
We don't have it memorized. 

Kim  2:03  
If you've ever seen us present together recently, you will know we cannot get it right. It's a mess. We laugh every time because we should know this after 300 episodes. But we don't, and so I just looked on the document I pulled open. It says, "Pam, look in your iPad notes for Problem String expressions." So, that's the entire note for today. And so, I have absolutely no idea what we're doing.

Pam  2:31  
So, Kim's taking a deep breath. 

Kim  2:33  
We'll see what we do. 

Pam  2:34  
Bam. It's going to be great. Alright, y'all, if you haven't listened to the last two episodes, you might want to grab one of them before you listen to today, but you don't have to. But we're working on factor puzzles, which is a great routine that we can start as young as third grade, and then take it all the way into high school. We really wish that younger grade teachers would... We. High school teachers really wish that this routine would happen with younger grade students a lot to get the relationships happening because it will help later. So, the way that we've talked about that in the last two episodes is... I'm not doing this very well.

Kim  3:09  
We need more notes. 

Pam  3:09  
Yeah. Okay, I'm going to talk about what a factor puzzle looks like, briefly, for those of you who haven't looked at the other two, and then we're going to dive into expressions. So, I might give you a factor puzzle that looks something like the top number is 36, and then there's two numbers that are underneath that kind of straddling it. So, 36 is sort of in the middle, and then two numbers underneath it. And then there's a third number on the bottom. Fourth number. Thank you. I can count. Is 12. So, the top number is 36 and the bottom number is 12. Kim, tell us how you're thinking about filling in the middle two numbers, and then we'll kind of come up with some what were you doing?

Kim  3:48  
Okay, so I know in factor puzzles, the rules of the game are that the two middle numbers multiply to give you the top number. So, the top number is the product. And the two numbers add together to get the bottom number. That's the sum. So, when I saw 36 and 12, I knew that it had to be 6 and 6 because 6 times 6 is 36, and 6 plus 6 is 12.

Pam  4:13  
Okay, cool. And I'm going to give you one more just so everybody can practice. 

Kim  4:18  
Okay. 

Pam  4:18  
So, still, this number on the top is 36. 

Kim  4:21  
Yeah.

Pam  4:21  
And this time the number on the bottom is 20. 

Kim  4:23  
Okay. So, I'm thinking. I'm kind of rolling through. 6 times 6, 9 times 4, 2 times 18. So, two and 18 are the two middle numbers.

Pam  4:39  
Because 18 times 2 is 36 and 18 plus 2 is 20. And like we've mentioned in other episodes, notice how Kim thought about the factor pairs and then checked each of them to see if they summed to that last number. She did not think about what are all the numbers that sum to 20, and then check to see if they multiply to 36. So, that's something that you would want to come out with students. Think about the factor pair, see if they sum. Don't go the other direction. It's much too inefficient to go that way. Okay, if those are the relationships. We're going to multiply the two middle numbers to get the top number. We're going to add those two middle numbers to get the bottom number. When I give you a top and bottom, you're kind of asking yourself, "What two numbers multiply to this number, but also sum to this other number?" So, Kim, what if I gave you a top number today? Oh, no. Shoot. Let me do one more thing. A reason to do what we just did with Kim is to get students thinking about those relationships because those are the relationships that we need kids thinking about when they factor a quadratic trinomial. So, if I have a quadratic trinomial in the form of ax^2 plus bx plus c, and I want to factor it into x plus or minus a product, plus or minus a number times the quantity of x plus or minus a number, then those two numbers are going to multiply to be c, and they're going to add together to be b. So, those relationships are alive and well when we're factoring those quadratic trinomials. I hope I just said all that right. All the words. Therefore, it is helpful. But, algebra teachers, once you do the kinds of problems, the kinds of factor puzzles that we've done in the last two episodes, you can also break free from those quadratic trinomial relationships. Like, I like them, and I want them, and I want to develop them, and get students good at that, and then let that help students factor quadratic trinomials. But I also think we can then break free and do other factor puzzles to help kids get better at just multiplying variables and simplifying expressions. So, again, we're going to have these four numbers. One on the top, two in the middle, one on the bottom. Numbers. Expressions this time. So, in the top box, Kim, will you please write x^2?

Kim  6:58  
Mmhm.

Pam  6:59  
And in the bottom box, will you please write 2x, and then please tell us how you're thinking.

Kim  7:06  
I know that x^2 is x times x.

Pam  7:10  
Okay. 

Kim  7:10  
And that's convenient because I know that x plus x is 2x.

Pam  7:16  
Nice. So, you just filled in the middle two boxes with x and x.

Kim  7:20  
Mmhm.

Pam  7:21  
Cool. Alright, so everybody just think about that for a second. One of the things that kids mess up early, early when we're asking them to think about variables is the difference between x^2  and 2x.

Kim  7:32  
Mmhm.

Pam  7:32  
And x plus x and x times x. And in huge way, we just kind of like... We're not expecting students to just know this. We would do this with kids and kind of talk about the differences. Let's continue to ferret that out. So, what if, Kim, in the top box, we had 2x. And in the bottom box, we had 2 plus x. Now, what are you thinking?

Kim  7:58  
Then, I'm going to make 2x. What did you say was on the bottom? 2 plus x?

Pam  8:05  
Mmhm.

Kim  8:05  
Then I'm going to put 2 in one of the boxes and x in the other box.

Pam  8:11  
Because? 

Kim  8:11  
Because 2 plus x is 2 plus x and 2 times x is 2x. 

Pam  8:20  
Cool. And you just told us the bottom thing first, and then the top.

Kim  8:24  
Mmhm.

Pam  8:24  
Yep. So, the product of 2 and x is 2. Or 2 and x. The products of 2. How do you say that? 2 and x? I feel bad saying "and" there. It feels very additive. 

Kim  8:33  
2 times x. 

Pam  8:34  
Is 2x. And 2 plus x is 2 plus x. Okay, cool. Here's another one. Well, actually, if you look back at the two that we just did, you'll notice that we've got 2s and x's floating all around. One of them is an exponent. It's a nice way of then having kids sort of look back at these two problems and go, "Okay, wait a minute. What? What's happening? Why is this one 2x and that one's x^2? And why is this one 2 plus x and that one's 2x?" Lots of nice things that we can sort of ferret out about what's happening. And if we're not sure, you could put a number in there and have kids actually, you know like, what if x was... I wouldn't do 2 because 2 plus 2 is 4. 

Kim  9:09  
Yeah, lots of 2.

Pam  9:10  
But you might do 3 or 5 or something. Do a prime number. And then, you know, talk about does that sort of work? Okay. And then you can go back to variables. Cool. Here's another one. What if the thing in the top box is 3x^2 and the thing in the bottom box is 4x. Love hearing you're thinking out loud. 

Kim  9:29  
So, I'm going to make 3x and 1x because I know that x^2 has x times x.

Pam  9:43  
Okay.

Kim  9:44  
And so, I'm just going to make one of them 3x. Which means... So, I guess I said 1x, but
it could just be x. 

Pam  9:52  
Sure, because we've gotten lazy, and by convention, we don't put the coefficient 1 in front of x, but you could. I actually like the fact that you said 1x. Okay, so here's the...

Kim  10:03  
I will write it back just for you.

Pam  10:06  
So, 3x times x is 3x^2 and 3x plus x is 4x. Again, a nice way of sort of juxtaposing what those mean. Alright, here's another one. I'm going to also put 3x^2 in the top. 

Kim  10:18  
Mmhm.

Pam  10:20  
Can I just ask what you're thinking right now?

Kim  10:23  
I said to myself that is 3 times x^2. 

Pam  10:28  
Okay, and you don't even...

Kim  10:29  
And so I know that the two middle ones are going to be... Ooh, I guess it doesn't have to be. I was going to say I know that should be x times x. I guess, theoretically, one of them could be x^2. The other one could be 3.

Pam  10:49  
Yeah, I like the thinking that you're sort of like playing around with what are all the ways to factor 3x^2.

Kim  10:54  
Yeah, yeah. 

Pam  10:55  
It's nice. Okay, the bottom is 3x^2 plus 1.

Kim  10:58  
3x^2 plus 1. Oh, so then none of the two options that I just said. One is 3x^2. And the other box is 1. Tricky trickster

Pam  11:14  
Because 3x^2 times 1 is 3x^2. And 3x^2 plus 1 is 3x^2 plus 1. Okay, cool. The top box. You ready? You're never going to guess. Is 3x^2. The bottom box is x^2 plus 3.

Kim  11:31  
Squared. Oh, say that again. The bottom box? I automatically went to the middle box. Okay, 3x. 

Pam  11:37  
So, 3x^2 in the top.

Kim  11:39  
Mmhm.

Pam  11:39  
And x^2 plus 3 in the bottom.

Kim  11:42  
Okay, so one of the middle boxes is x^2, and the other one is 3.

Pam  11:48  
And do those indeed multiply to 3x^2? Sure enough. 

Kim  11:50  
Yep. 

Pam  11:50  
Cool. Okay, so there's one series of problems that you could use, algebra teachers. Not really. It's not about factoring quadratic trinomials at all. It's really just kind of helping kids get a sense for like when are you multiplying variables? When are you adding variables? What does that mean?

Kim  12:07  
Yeah, yeah. 

Pam  12:07  
And simplifying expressions? It's a chance to sort of juxtapose things and kind of get more solid about that. Did you want to say?

Kim  12:13  
 I was just going to say that since that is such a huge confusion for so many kids, I think this is a really nice way to put it out in front of them and have to tinker, and have discussion, and acknowledge that that's a thing. 

Pam  12:28  
Yeah, nice. Consider that when we start working with variables with kids, if we have kids that are stuck in counting strategies or additive thinking, and they really haven't have this sense of multiplication, multiplicative reasoning, then it is harder for them to think about something like 3x because if I write 3x, they are seeing a 3, and they're seeing an x. And especially if they're additive thinkers, they're like, "There's a 3 and there's an x." And they're going to see that as a 3 plus x, a 3 and an x together. It's much less for them, 3 times x, or x plus x plus x. There are ways that we can help with that. And one of our, one of my favorite strings that I've just come up with recently, we call Factor to Add or Factor to Subtract. Pretty sure we have podcast episodes on that. We'll put that link in the show notes. Where we end up with students doing problems like 9 times 7 minus 4 times 7.

Kim  13:31  
Mmhm.

Pam  13:31  
And the kids start. We pull out of them to think about nine 7s minus four 7s. And they're like, "Huh. I can just solve that problem. I know 9 things minus 4 things. That's just 5 things. So, I can think about nine 7s or 63 minus four 7s. Like, 63 minus 28. I can think about that as well 9 minus 4 of them, that's just going to be 5 of them. And I know that 5 times 7 is 35." And so, I can actually solve that subtraction problem by thinking about 9 of a thing minus 4 of a thing. And then, as algebra teachers, we could write, "Oh, it's almost like you're thinking about 9x minus 4x. Or 9..." You know like, we just put a variable in there instead of the thing. It's a way of getting kids to reify what a term is. A term is connected with multiplication. We then add and subtract terms from each other. So, terms are connected with addition and subtraction. They're created with.. the things in a term are connected with multiplication and division. And they're connected by addition and subtraction. I can tell you that, but we need kids to have experiences, so they reify this idea of what it means to have something like 3x, that 3x is not 3 plus x. So, what's the difference between 3 plus x, 3x, x plus x plus x. Even then, x times x times x, and x^3. All the differences between those we can help differentiate with experiences like this. So, Kim, will you do one more with me? 

Kim  14:58  
Sure.

Pam  14:59  
What if the top box was x times the quantity x plus 1. So, like, if you looked at it, written it was x parentheses, x plus 1, close parentheses.

Kim  15:13  
Mmhm.

Pam  15:13  
And that was for podcast listeners, not for Kim. And then the bottom box is 2x plus 1. 

Kim  15:21  
Mmhm.

Pam  15:22  
What do you got?

Kim  15:24  
I've got... I know that x times x plus 1 can be x on the left side and x plus 1 on the right. Or the other way around. So.

Pam  15:36  
Because multiplication is commutative, mmhm.

Kim  15:38  
Yeah. So, my factors are x and x plus 1.

Pam  15:43  
And do those sum to 2x plus 1? 

Kim  15:45  
Yeah. 

Pam  15:46  
Because you got x plus x plus 1. That's 2x plus 1. Nice, cool. 

Kim  15:46  
Yeah. 

Pam  15:46  
So, high school teachers, you could have a little fun with this if you wanted to. I don't know that I actually even recommend what I'm about to say, but you could. You could do this. Kim, last problem. What if...

Kim  16:01  
You said last problem on the last problem? 

Pam  16:03  
Okay, that was... I lied.

Kim  16:06  
You might not have.

Pam  16:07  
Did I really? I don't remember.

Kim  16:08  
Maybe not. That's what I heard in my head. 

Pam  16:09  
Sorry, I apologize. 

Kim  16:09  
That's okay. It's fine. No problem.

Pam  16:12  
The actual last problem. What if in the top box it was x^2 plus 7x... Well, shoot. Should I go this direction? Yeah, I will, since I started, but then remind me to go back. So, x^2 plus 7x plus 12. X^2 plus 7x plus 12.

Kim  16:28  
x^2 plus 7x plus 12. Tiny handwriting. Okay. 

Pam  16:31  
Oh, sorry. Yeah, I should have said big box. Big Box on the top. Big box on the bottom, 2x plus 7.

Kim  16:37  
2x plus 7. 

Pam  16:39  
What are the factors of x^2 plus 7x plus 12 that sum to 2x plus 7.

Kim  16:48  
Okay, so I am thinking about I want x to be in each of the boxes. Like, x on the left, x on the right. And I want two things that add to 7 but multiply to 12. 

Pam  17:05  
Okay. 

Kim  17:06  
So, I want it to be x plus 4 and x plus 3.

Pam  17:10  
And do those sum to 2x plus 7? 

Kim  17:13  
Yes. 

Pam  17:13  
Sure enough. Nice. Now, high school teachers, you might look at that and go, "We don't ever have students sum the factors, once they factored that quadratic trinomial x^2 plus 7x plus 12 into x plus 3 times x plus 4. Once we've done that, we don't ever have them add them. Okay, fine, you don't. But I wonder if you could actually give kids factor puzzles like the ones that we've done all day today and in the past two episodes, and then give them one where you give them actually the two middle numbers. Give them x plus the middle numbers, two expressions. Give them x plus 3 and x plus 4. And let them multiply them, and let them add them. And do a bunch of factor puzzles where they're actually doing the multiplication first before you then ever do the other direction. And if they're used to factor puzzles, they know they're going to be factoring because you've given them the top and bottom numbers before or top and bottom expressions before. And so, it's just this fluid flowing between am I multiplying? Are we adding? And what does it mean with variables and expressions? I think it could be a really nice, fairly quick way of... Again, do this early, and then just pepper independent work with it often. Warm ups, and exit tickets, and on homework. And anytime you can just have one or two of these factor puzzles show up. Compare, have students talk about their reasoning. You got a great routine going on. 

Kim  18:38  
Yeah. Factor puzzles good for grades three and up in lots of different ways. If you have not checked out the blog on factor puzzles, you can find that at mathisfigureoutable.com/blog.

Pam  18:52  
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. Thanks for spreading the word that Math is Figure-Out-Able!