Pam  00:00

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

Kim  00:08

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  00:15

This time, I didn't (unclear).

Kim  00:16

I know. I did it on purpose because last week I wasn't paying attention to you.

Pam  00:20

We know that algorithms are amazing achievements, but they're not good teaching tools because mimicking step-by-step procedures actually trap students into using less sophisticated reasoning than the problems are intended to develop. 

Kim  00:32

In this podcast, we help teach math...help you teach mathing. Ugh. 

Pam  00:37

Something somewhere.

Kim  00:37

Building relationships with your students and grappling with mathematical relationships.

Pam  00:42

Ya'll, we invite you to join us to make math more figure-out-able. Alright, Kim. 

Kim  00:47

Hi.

Pam  00:48

This one's starting out great. It's fantastic. This is going to be super, super fun.

Kim  00:55

Oh, man. 

Pam  00:58

How's it going?

Kim  00:58

It's... It's... Yeah, we're good. Okay, you know what? Shelly Gilbert sent us a message.

Pam  01:04

Hey, Shelly! 

Kim  01:05

And said, "I've been listening to you and learning from you for a while now. I'm a retired public school teacher and now teach some math content and methods classes at the University of Findlay."

Pam  01:16

Whoo!

Kim  01:17

"I pulled a book off my shelf that I used as a middle school teacher, and it was your Building Powerful Numeracy book. It took me all this time to make the connection. I've been using Problem Strings, As Close As It Gets, etc. for years, and when I stumbled onto the podcast, I thought it was wonderful to hear someone talk about these ideas." Isn't that great? "I guess you talking about your ideas is not so unusual." Isn't that amazing?

Pam  01:41

That's awesome. Oh, that is so stinkin' fun. 

Kim  01:43

Yeah. She said, "Thank you for sharing your passion with math. You've made me a better teacher, and I hope a better preparer of future teachers. Kindly, Shelly." I love that. So fun.

Pam  01:51

Shelly, you're amazing. You're exemplifying Kim and I often are doing something, and all of a sudden we're like, "Oh, wait, duh." Sure enough. 

Kim  01:59

Yeah. 

Pam  01:59

Well, I'm glad you've enjoyed Building Powerful Numeracy in your middle school class, and as you teach teachers.

Kim  02:04

Yeah.

Pam  02:05

Fantastic. Yeah, you're going to also enjoy Developing Mathematical Reasoning - Avoiding the Trap of Algorithms because...

Kim  02:11

It's so good!

Pam  02:12

That's a bit more K-12. A bit more? It is. It's totally K-12. And so, yeah, use that in your methods classes. Oh, in fact. Any university folks out there know that you can totally like give us a holler, and we'll help set you up to use our Developing Mathematical Reasoning workshop in your methods classes. We've got lots of professors using it for like assignments and asking questions about it. Totally free. We're just really trying to spread the word that Math is Figure-Out-Able. Bam! Alright, Kim. That was awesome. Thank you for sharing. That's super fun. Thanks, Shelly. So, the last few weeks, we've been doing some solving equations and creating mental maps of relationships. We've been using double open number line. Last week, we really kind of talked about a way. And you had super a lot of fun with that. A way of using a context to develop what a system of equations even is.

Kim  03:06

Yep.

Pam  03:06

And how you can add equations in a system and still get something that's in the system that has the same intersection point. And how you can kind of make sense of that. Today, I'd like to dive into a little bit of a different side of systems of equations, and I want to develop something that would be helpful for students in solving systems. So, a first thing that we would need to do before we do what we do today...I'm just going to mention. We're not actually going to do it. But I'm going to mention...is that we would need to reify...big word. I'll explain in just a minute...things like 3x. So, when a kid looks at 3x, they need to have a lot of things happening simultaneously, sort of built on each other, so that then they can use 3x as an object. So, we need things like 3x stands for x plus x plus x. An additive perspective. Ideally, kids would then also connect that to the multiplicative aspect of three times something. So, that thing plus that thing plus that thing, but it has 3 of those things. But then we also need to not be like in the weeds with that x plus x plus x or that 3 times x, and be able to use 3x as a thing, like it's an object, like we're treating it as this thing is 3x. But I need to be able to dig into it and feel the additive or the multiplicative notion of it. So, all that kind of needs to be happening in order for us to do the following. We're not talking about how we're doing that today. You can go listen to a few episodes ago where we did some of that, where we were like reflected 3x or -3x across the 0. That's in part, part of that work was happening. Anyway, alright, so moving on. Today, Kim, join me in a lovely Problem String. We are going to use a double open number line. Well, I'm going to use it. You don't have to, but I am. So, if I were to say to you... Let me just say. We have done the work of the last few weeks. So, if you haven't watched the last few weeks, go watch the last few weeks. Here we are. If I said, hey, Kim. I've got this equation x plus 3y equals 8. We might have a brief conversation about what that could mean. What... You know, is it a parabola? Is it a exponential function? Do you have any idea what kind of graph this is? 

Kim  05:33

X plus 3y?

Pam  05:34

Mmhm.

Kim  05:35

Equals 8. It's a line?

Pam  05:37

That is correct. So, with a question mark there? "It's a line?"

Kim  05:40

It's a line.

Pam  05:40

It's a line. Okay.

Kim  05:41

I was making sure I knew what your question was. 

Pam  05:43

Oh, yeah.

Kim  05:44

Yeah, yeah, yeah.

Pam  05:45

Read my mind. Read my mind. 

Kim  05:46

Well, you said it. I was just drawing. You told me what. You told me, and I started messing, and then I forgot you were still talking.

Pam  05:54

You forgot you were on a podcast, and we're recording. Okay.

Kim  05:56

Oh, my. This is going to be a mess today. I can tell you right now. It's a mess. 

Pam  06:00

Well, let's just keep messing. Let's keep messing together.

Kim  06:03

Okay, I'm with you. It's a line. 

Pam  06:04

So, if we have a line x plus 3y equals 8. Just for today. Just because I'm just sort of curious. y

Kim  06:09

Yep.

Pam  06:10

To solve for x. 

Kim  06:11

Okay.

Pam  06:12

Okay, so I'm going to use a double open number line, which you might say is totally weird. And I'm going to put, if we know x plus 3y equals 8, we also know that 3x plus... Sorry. X plus 3y is in the same location as 8.

Kim  06:25

Yep. 

Pam  06:26

So, I've got a double open number line. I've got a tick mark right in the middle. Above it, I've got x plus 3y. Below it, I've got 8.

Kim  06:32

Yep. 

Pam  06:32

I want to know where x is. If x plus 3y is there, where's x? At least in relation to that 8.

Kim  06:40

Yeah. 

Pam  06:40

What do you got?

Kim  06:41

I'm thinking that before I added 3y...

Pam  06:45

Okay.

Kim  06:45

...x is to the left.

Pam  06:48

Because then you're going to add 3y, right?

Kim  06:50

Mmhm, mmhm.

Pam  06:51

Over there to the right. I've kind of drawn x to the left of that, and I draw a little jump, and I've put 3y above that jump. Okay.

Kim  06:59

So, then below my number line I'm writing 8 minus 3y because if I jump back the same amount as I jumped the 3y on the top. So, I have now on my number line x is in the same location as 8 minus 3y.

Pam  07:15

Nice. And we can kind of use the relationships of the location to kind of think about that. Bam, nicely done. That's it. 

Kim  07:24

We're done? That's the whole string? 

Pam  07:25

With that... No with that problem. 

Kim  07:26

Oh, I was like, "Wow. I did great!"

Pam  07:30

Yes, you did...with problem number one. Number two. Alright, what if the equation of the line is x minus 3y equals 2. 

Kim  07:39

Okay. 

Pam  07:40

So, new number line. Whole new problem. Not related to the other one in anyway. X minus 3y I've now put above the number line where the tick mark is and 2 is below. And again, I want to solve for x. Where is X?

Kim  07:53

So, before I subtracted 3y, x would have been to the right. So, I'm adding 3y. Like...

Pam  08:01

Okay.

Kim  08:01

...jump of 3y.

Pam  08:02

Yep. 

Kim  08:03

And so, I'm doing that on the bottom as well, so I have 2 plus 3y is in the same location as x.

Pam  08:09

So, over on my equations over there, I could write, so if x minus 3y is 2, then x is going to be equal or in the same location to 2 plus 3y. 

Kim  08:17

Mmhm.

Pam  08:18

Nice. Next problem. What about 2x plus y equals 3, but this time I want to know where y is. So, if 2x plus y is in the same location as 3, where's just y?

Kim  08:37

So, same thing. I'm thinking about before I added y. So, I'm going to subtract y. I'm going to go to the left. 

Pam  08:45

But I want to know where y is. 

Kim  08:46

Oh, I lied. So, I'm going to subtract 2x. 

Pam  08:49

Okay. 

Kim  08:50

And that's going to put y to the left of 2x plus y. 

Pam  08:56

Okay. 

Kim  08:57

And then I'm going to subtract 2x from the 3 to kind of slide it to the same location as y.

Pam  09:03

Mmhm.

Kim  09:03

So, I have 3 minus 2x. 

Pam  09:05

So, y is in the same location as 3 minus 2x.

Kim  09:08

Mmhm.

Pam  09:08

Nice. Okay. Next question. How about 3x minus y equals 9? 

Kim  09:17

Okay.

Pam  09:17

Okay, tell us what your number line looks like.

Kim  09:20

3x minus y is on the top. 9 is at the bottom. 

Pam  09:24

Okay.

Kim  09:24

You said solve for y?

Pam  09:25

Yep, y again. 

Kim  09:27

Okay, so 3x minus y. You said solve y? Oh, okay, so... Oh.

Pam  09:39

You can do it, you can do it.

Kim  09:40

I can do it. I can do it. So, I want to like have 3x plus y somehow.

Pam  09:50

Mmhm.

Kim  09:52

So, before I subtracted y...  3x minus y. Why is this hard for me?

Pam  10:03

Uh, because it's hard.

Kim  10:09

So, I want 3x to be to the right. I mean... Yeah, to the right.

Pam  10:15

Okay. 

Kim  10:16

And then I want 3x plus y to be even further to the right.

Pam  10:20

3x plus y. 3x... Why do you want 3x to be to the right?

Kim  10:24

I don't know.

Pam  10:25

Say that again.

Kim  10:26

Like, because I was thinking about before I subtracted y, then it would be 3x plus y.

Pam  10:33

So, you're going to the right y. And you're at 3x plus y on the top. And you would be at 9 plus y on the bottom.

Kim  10:41

Mmhm.

Pam  10:42

Okay.

Kim  10:43

Is that good enough? 

Pam  10:46

Wait. But when you... Hold on a second. Subtracted y. Wouldn't you just be 3x on the top?

Kim  10:54

Before I subtract? Well, that's what I said at first. If I have 3x, and I would have...

Pam  11:02

Maybe I wrote it down wrong. What do you... Tell me what your number line has. 

Kim  11:05

I have 3x minus y on top and 9 at the bottom.

Pam  11:07

Mmhm.

Kim  11:09

But I want to think about before I subtracted y.

Pam  11:12

Mmhm.

Kim  11:15

So, then I would have 3x on the top.

Pam  11:18

Mmhm.

Kim  11:19

And 9 plus y on the bottom. 

Pam  11:21

Yeah, I heard you wrong. So, that's excellent. I like where you are. Okay. 

Kim  11:24

Okay.

Pam  11:24

Good, okay. Sorry, my bad.

Kim  11:28

And then what am I looking for? I want to know what y is?

Pam  11:30

Y. Yeah, where's y? 

Kim  11:31

Whew! Okay.

Pam  11:32

You're doing great. You now know where 9 plus y is, but we just want to know y.

Kim  11:36

I want just y, so I'm going to subtract 9...

Pam  11:40

Mmhm.

Kim  11:40

...and I'm going to go back to the left. 

Pam  11:42

Okay. 

Kim  11:43

And I've got... I'm going to do subtract 9 to the 3x also. So, I have y is 3x minus 9.

Pam  11:50

So, y is now in the same location as 3x minus 9.

Kim  11:53

Yeah, mmhm.

Pam  11:54

Yeah. What do you think? 

Kim  11:59

Hmm. 

Pam  11:59

You're done. 

Kim  12:00

Yeah.

Pam  12:01

Why would we do this string? It seems really random that we would just like solve. Well, one of the solution methods to solve a system of equations is to solve for one variable...

Kim  12:15

Mmhm. 

Pam  12:16

...and then use that in the other equation. And then you can find the value of that one variable, and then you can find the value of the other one, and you can find the intersection point. 

Kim  12:24

Mmhm.

Pam  12:25

But often, if kids are going to use this substitution method, often one of the things I found with my students that they would mess up is in the solving for one variable. So, teachers, you might consider a sequence of tasks where I would do something like we did last week in maybe a couple of contexts. One we just did with the sandwiches and the cookies where you really give students a sense of what it actually means to even be a system of equations. So, go back and listen to that one again if you didn't. But really just like a sense of this system of equations, it is a system if there is a... Well, I can't quite say it that way. I was going to say if there is an intersection point. You can have a system of equations that there is no solution or that there's an indeterminate solution. But if there is a solution, an intersection point, what does that mean? Oh, and brilliant. You know, I probably should have asked last week when we were talking about the sandwich shop. What does it mean if there wasn't an intersection? And I don't know, Kim, if you want to go there. If I had given you a price of sandwich or a number of sandwiches, number of cookies equals a price. And then a different number of sandwiches, number of cookies equals a different price. And when you graph them, there was no intersection. Do you have any ideas with that would mean?

Kim  13:36

I mean, it either means I'm at a different sandwich shop or they changed the price because they don't have a consistent price. 

Pam  13:43

Yeah. Nice.

Kim  13:44

Yeah. 

Pam  13:45

So, with all of that to develop, when you get to the point where you want kids to get a feel for solving with substitution, I think you could first just give students systems where one of the variables has a coefficient of 1. Well, maybe very first, where it's already solved. For that variable, they can just take the x. We already have what x equals. Plug it in the other one. Solve for the system. Then you could give it where the coefficient is 1. So, they still have to do a little solving, but it's just the coefficient is 1. Kind of like the first couple that I gave you.

Kim  14:21

Mmhm.

Pam  14:21

Well, actually, all of the ones I gave you today, except some of them had... One of them had a negative coefficient.

Kim  14:27

Mmhm.

Pam  14:27

And then you would want to do more solving equations work, so that they get kind of that idea of dividing if they need to divide out the coefficient before you ever ask them to then do it where they're actually solving the system where things aren't quite as complicated. It's not about making it easy to hard. It's about sort of a systemic progression where kids are thinking about this makes sense. Now, I have some great Problem Strings in Algebra Problem Strings and Advanced Algebra Problem Strings where the question then is given this system, how do you want to solve it? And then you give kids free reign, and then you have the conversation. What is it about this system that made you want to use elimination? What is it about this system that made you want to use substitution? Or use graphing? I'll just finish that. And go ahead.

Kim  15:14

Can I tell you that when my oldest... What grade is this going to be? I don't even know. But I remember him... 

Pam  15:19

This is Algebra 1 or Algebra 2. 

Kim  15:21

Okay. 

Pam  15:21

Is that what you mean?

Kim  15:22

So, it might have been Algebra 2 because I remember him having, you know, a paper that said, "You must solve these with substitution." And then, "You must solve these with elimination." And he and I are both like, "Wait, what? Why?"

Pam  15:32

(unclear). 

Kim  15:33

There were some where it... Yeah, there were some for sure...like, I'm kind of going back in the recesses of my brain...that he was like, "But it's so much better to do it the other way?" Like, "Do I have to what the directions say?"

Pam  15:44

Yeah, yeah. Nice point. And I love the fact that he knew that he should have choice. 

Kim  15:49

Yeah. 

Pam  15:49

So, teachers right now might be thinking, "Yeah, but, Pam, how do we..." Or, "Kim, how do we make sure that students can do a method?" 

Kim  15:55

Oh, oh! I know. I know. 

Pam  15:57

Alright, go, go, go, go. 

Kim  15:58

I mean, wouldn't we want to give them problems for where they would choose? Like, sometimes you want to give them problems where elimination is like the obvious, "Do that." And then sometimes you do it where the substitution was like the obvious, "Do that." And so...

Pam  16:10

And then have a conversation...

Kim  16:11

Yeah. 

Pam  16:12

...about like if kids are stuck, then you're like, "Ooh, look how nice this would have been."

Kim  16:15

Mmhm.

Pam  16:15

Gives them an opportunity to go, "Oh, yeah. That would have been nice." And your last one which was give them... I think you were going to say give them ones where it's a toss up. They can do either. And then they can choose.

Kim  16:25

Well, and if they're using the same like substitution every time, that might be a sign that like maybe they don't really own or (unclear).

Pam  16:34

Yeah. And then maybe you could do a... Sure. And then maybe you could do a Problem String where you really focus on the one that many of your students are struggling with. You could also... This is totally where I thought you were going to go. Kim, we're just planning a Live Math with Pam...help me...event with our Journey group. And one of the things that we're talking about are really cool assessment items. And one of those really cool types of assessments, I call ketchup problems. And this would be a fantastic time to do a ketchup problem. Pam, what's a ketchup problem? So, picture. You're doing your homework. And you're eating fries because, yum, fries. And you drop ketchup on your homework. And you you get up the next day. You don't realize it. You get up the next day, and you're going to turn your homework in. You're like, "Oh, crumb. I dropped ketchup." And so, you're trying to like fill in the blanks where the ketchup..." (unclear). But where the ketchup splattered all over your homework, you got to like write in what you've missed. That's just the way.... That's why I call them ketchup problems. But literally, you could give students a system, and you could solve it. So, it's a tweak on a previously saw or showing a strategy that someone else did. So, solve the system. You want kids to work on elimination? Fine, solve it using elimination, but put blanks. Put splotches and say to the kids, "Fill in what needs to be here in order for this solution to work." Vice versa, use substitution. Solve it, but put blanks in really nice places that kids have to... That's a way of instead of... Now, see if Luke got that. Or, I think it's Luke. If he got that assignment, he would have been like, "Oh, okay. My job here is to fill in the relationships that this student used to solve the problem." It isn't, "You must use this strategy to solve this problem, even though you can totally tell there's a better one right now." Right? So, it's it's a lot more agency involved, I think, when we allow students to use what they know to solve problems. But in order to help them not get stuck in a method that we know is not going to be the best one forever, we can then use a ketchup problem to kind of help them follow the thinking, and fill in sort of the relationships that were used in that thinking. Does that make sense? 

Kim  18:43

Yeah. Yeah, absolutely.

Pam  18:44

Yeah, bam. Alright. 

Kim  18:46

You know what I'm stuck on? I know you want to end, but...

Pam  18:48

It's alright.

Kim  18:48

But I'm going to pause us for a second because I'm stuck on the fact that the fourth problem...

Pam  18:53

Yeah.

Kim  18:54

...was harder for me to think about relationally. And I'm looking at the problems that you gave me.

Pam  18:59

Okay.

Kim  19:00

And I think it has everything to do with the fact that the y was being subtracted.

Pam  19:06

Absolutely.

Kim  19:06

That's a different problem. Like.

Both Pam and Kim  19:09

Yeah.

Pam  19:10

Yeah.

Kim  19:11

That's noteworthy that it's not all the same...

Pam  19:15

Difficulty level?

Kim  19:16

Yeah. It feels like it's related to like problem types for me (unclear).

Pam  19:20

Oh, I like it. I like it. I think that sounds very nice. And it would be a reason why we could do this Problem String, and we might not be done. We might need to do another Problem String where maybe there's a -x involved.

Kim  19:33

Mmhm. 

Pam  19:33

And then some discussion. A huge discussion about when do you want to use substitution, and when do you want to use elimination. Because if there's a lot of coefficients going on, how about if we don't do substitution? Like all that dividing. Especially if you end up with fraction coefficients. Like, crazy. Like, there might be a better strategy at that point to try, at least. At least to consider.

Kim  19:55

Mmhm.

Pam  19:56

And when is it crazy enough that you just want to throw it in calculator and find the intersection point?

Kim  20:00

Yeah, mmhm.

Pam  20:01

And making sure kids know that there are those choices. You know, it's funny. Kim, last, and then we're going to go. Sometimes people will look at the work that we're doing in the elementary school with numeracy, and they'll say, "Pam, Pam, Pam. Why are you giving kids all these choices? There's one and only one way." When, look. Here we are in high school math when there is... Like, this is a perfect example of where we in higher math have been demanding kids learn different strategies. We haven't done a very good job of letting them choose often. But we can. We can. We are teaching strategies. Maybe try not to teach them as procedures to memorize. And then we can give them choices and build mathematicians. Alright, ya'll, thank you 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!