Pam Harris  00:01

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

Kim Montague  00:07

And I'm Kim.

Pam Harris  00:07

And you found a place where math is not about memorizing, 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 mathematicians, as we co-create many together. 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 Montague  00:35

So this episode is all about a really cool connection between Proportional Reasoning and Functional Reasoning that may surprise you. So Pam, let's start off. And if I can, tell you a little bit about my experience. I know that we've talked about my mathematics before and in elementary school and middle school, and just all through Proportional Reasoning, I really was able to make sense of things. Even though my teachers weren't teaching me in a way that was really Figure-Out-Able. It was for me, it was very rote memorize practice. And that was fine for me, with me, because I didn't know any different as a teaching practice. And I was okay until about precalc, I hit a wall in precalc to be perfectly honest with you. I at that point, felt like I couldn't figure anything out anymore. And I, you know, I still, because I had some memorization skills, I still did, okay, but it fell apart in my mind. I thought for sure I had reached a point where, like, you either made sense to you or didn't, and it wasn't gonna make sense to me. And so I just shied away at that point, and said, "I don't get it anymore. This isn't for me." And I was really interested in math. And I really loved my math classes. And I just, I just didn't, I just didn't at that point.

Pam Harris  02:01

And it makes me sad hearing you talk about it. But I hear you saying at that point, math became not Figure-Out-Able anymore. 

Kim Montague  02:07

Yeah, absolutely. 

Pam Harris  02:08

It's almost like in your head. The nature of math changed.

Kim Montague  02:11

 Yeah. 

Pam Harris  02:12

You're like, "Oh, this isn't fun anymore, because it's not Figure-Out-Able anymore." 

Kim Montague  02:15

Yeah. 

Pam Harris  02:15

"Now, it's just a bunch of formulas to memorize. And I could do that. But that's not fun." 

Kim Montague  02:20

Yeah. 

Pam Harris  02:20

That's interesting. Is there a point where math is not Figure-Out-Able anymore? Is there some...? You know, like, you might be thinking teachers, "Well, you know, it's Figure-Out-Able until you have to do long division, you can't really, you know, long division, he's got to do the steps." Or is division Figure-Out-Able? You might be thinking, "Well, you know, it's Figure-Out-Able until it's division of fractions, then you just have to invert multiply, you don't, we don't know why." Or is it Figure-Out-Able? Like, is there? Do you think that there's some juncture where we really just have to do the formulas, or is math always Figure-Out-Able? So we thought we'd take today and do a little bit of some high school stuff, to illustrate a bridge that we might be able to sort of take you from some things that you know, and help you see this bridge to maybe something that you thought was kind of only memorizable, rote memorizable, but it is actually pretty Figure-Out-Able. So we're gonna have a little bit of fun today. One of the things that I want to do is mentioned that we do this wonderful work, and we've done it the podcast, we definitely do it and Building Powerful Multiplication our online workshop and an online workshop called Building Powerful Division. And in our online workshop called Building Powerful Proportional Reasoning, where we work with ratio tables, multiplicatively. And I want to build this bridge for what we could do in ratio tables, to multiply to divide, to solving proportions, and to reasoning proportionally and doing great things in ratio tables. And I want to paint this bridge of how we can then use that in an algebra class in a way that you might have not thought about. So consider, if you've never done any work in a ratio table, you might want to go back to at least one episode where we do some multiplication or division in the ratio table. You might consider that if you've done some work in a ratio table, then you will probably recognize relationships like this. So Kim, I'm just gonna throw some numbers out to you. And I want you to tell me kind of everything you're thinking about. And yeah, here we go. If I were drawing it on a board, right now, I would be putting in a ratio table. So I'm gonna say one, six, and I would put one on one side of the ratio table, and six is sort of the output. So the input one, the output six, up to 12. It's kind of like giving the ordered pairs like two comma twelve. 

Kim Montague  04:26

Yeah, ok.

Pam Harris  04:27

Three, eighteen; four, twenty-four. What do you notice? What are you thinking about?

Kim Montague  04:33

So a couple of things. It's going up by sixes in the output, but I think more importantly, the relationship is that it's from one side of the ratio table to the other from, I guess, the x to the y, it's time six, for each of those entries.

Pam Harris  04:51

So if I were to say, if you were to graph those points, what would they look like? Like, is there a pattern? Is there no pattern there? They'd be random, all over the place. 

Kim Montague  05:01

No, they're definitely, it's definitely a proportional relationship. 

Pam Harris  05:04

Okay, so it'd be a line, it'd be linear. 

Kim Montague  05:05

Yeah. 

Pam Harris  05:06

And, in fact, it's...

Kim Montague  05:07

It's gonna go up because it's increasing. 

Pam Harris  05:10

Yeah. Okay, increasing. Nice. Nice. And you said it was a proportional relationship. So you also know it goes to (0,0). Okay. You could probably, if I pushed you, you could put a context to that I bet. I bet you could think about...

Kim Montague  05:23

You know I like sticks of gum.

Pam Harris  05:24

 Yeah, so one pack of gum has...

Kim Montague  05:26

Or Jell-O's. Uh, my kids love these Jell-O's that come in a six pack.

Pam Harris  05:30

Okay, nice. All right, cool. Or six pack of soda, maybe. Well or like, whatever. So it could kind of represent all that. And what if I said that if I plotted all those points, and I connected them with a line, could you write the equation of that line? 

Kim Montague  05:43

Sure. 

Pam Harris  05:44

How do all the y's relate to all the x's?  Yeah. So y is x times six? Or I guess you would say y equals 6x? Yeah, so by convention, we usually write the coefficient first. Yeah, but I like that all the x's times six. I like that a lot. And I would totally, if a student said that to me, I would literally write y and as you said, I would say all the y's, and then you said equal bright equal and then you said x and I would write x times six, all the x's, I would write x times six. And so I would literally write y equals x times six. And then I would say, "Oh, yeah, and by convention, we usually write that as y = 6x. So I would sort of represent your thinking, and then I would write it equivalent to the convention that we usually write. Cool. So then Kim, what if I said, "Well, here's a new set of points, Maybe there's a special on the Jell-O. And so Jell-Os? Do you put an "s" on Jell-Os? I don't even

Kim Montague  06:36

Jell-O cups. I don't know.

Pam Harris  06:38

Ah, yeah. Okay, new ratio table. So (1,7), (2,13), (3,19), (4,25). I'm kind of curious what you see now. And really, I'm heading towards writing the equation.

Kim Montague  06:56

Okay. Okay. I actually really liked that you said something about the context? Because some of the numbers seem kind of random, except when I look back at the y's in the problem you just gave me. 

Pam Harris  07:11

Yeah. 

Kim Montague  07:12

They're, each time, is one more. So it's that same 6x. But it's plus one this time. So y =  6x + 1.

Pam Harris  07:27

Nice, nice. It's almost like, if you look at that second ratio table, the 7, 13, 19, 25. It's almost like if you kind of cross your eyes and sort of look double or something, kind of like I picture.

Kim Montague  07:41

 I think I know where you're going. Yeah, yeah.

Pam Harris  07:42

What just happened in your brain when I said that?

Kim Montague  07:44

So that same line, I don't know what you call it. That same line, 

Pam Harris  07:48

Yes. 

Kim Montague  07:49

Is shifted, shifted up one,

Pam Harris  07:52

It's just shifted up one, we're just taking that line, y = 6x...

Kim Montague  07:54

It's almost like I should just get, if  just get one Jell-O cup for free.

Pam Harris  07:59

Bam, that's exactly what it is. It's like a bonus. A bonus, today you get one, yeah, you get one more Jell-O cup. Nice, nice. So we have a colleague. Her name is Ann Roman. Ann, how ya doing?

Kim Montague  08:09

I love Ann.

Pam Harris  08:10

I know, right? Who one day after, she was a fifth grade teacher at the time, she since went on to work at the Dana Center. Did some amazing work. At the time, she was teaching fifth grade, taught me a lot, we worked together a lot. And she was doing so much work with multiplication, division with ratio tables. And then she went on and was doing some graduate work. And she was taking a class and they had to write the equation of a line to match data. And she said, she said to me, "Pam, I'm looking at these tables of values that I'm supposed to do all this like crazy formula work. But I just sort of, it's almost like I'm looking at an optical illusion, where it's the old woman and the young woman. And if you kind of close one eye and tip your head, you sort of see the young woman and if you kind of open your eyes, if your head the other way, you see the old woman, you know, they kind of one comes in focus and the other comes in focus." She said, "I look at these tables. And if I kind of just like, like I see a ratio table shifted. Like I can just picture a rate. I've done so much work with the ratio tables, I just see the ratio tables like screaming at me, but it's just been shifted. And so I just write down the equation for the ratio table. And then I just shifted it. Can I do that?" Y'all I was floored. I was just like, that is amazing thinking. And now I see it. Now I see tables of values. And I just look to see if I can see the shift. So let's practice that a little bit. Let me just see if I can give you some tables of values, maybe without the original ratio table this time and see if you can like tip your head and close one eye.

Kim Montague  09:37

Help you write the original. 

Pam Harris  09:38

Yeah, see if you can picture the original and then we can write the equation of the line from there. 

Kim Montague  09:41

Okay. 

Pam Harris  09:42

So what if I gave you something like (1, 6), (2, 11), (3, 16), (4, 21). And I'm just going to suggest that if you're listening to the podcast, you might write those down. pause the podcast and look at those and see if you could kind of feel a ratio table around there. So (1, 6), (2, 11), (3, 16), (4, 21). Pause. Alright, Kim, what are you thinking about?

Kim Montague  10:11

So I was looking at how essentially, how can I get from x to y? 

Pam Harris  10:16

Yeah. 

Kim Montague  10:17

And so I was thinking about what is around 6, 11, 16, 21. And I noticed that it is going to be five times x, and then just shift up one. So y = 5x + 1.

Pam Harris  10:34

Because is that 6, 11, 16, 21 is so close to 5, 10, 15, 20? 

Kim Montague  10:40

Yep. 

Pam Harris  10:41

And how close? Just one, just all just one (click), bam. So it's really important that the x's are this arithmetic sequence, that they're just upping by one. That's actually important. And I just, I have to say that because I know I have secondary teachers right now that are like, "You can't just look at the y's". And I know that. I know that. But my suggestion is, if students have been dealing with ratio tables a lot, they've already taken that into consideration. Before they look for a pattern in the y's. They're taking the x's into consideration, because they've had to in ratio tables, because the work in ratio tables has not been an order. In fact, for me, giving these points in order is almost like, "Whoa, they're in order, huh? How interesting. How droll. How immature of you that you have to have them in order?" Because they're so used to having to order them. They're so used to having them out of order in a way that makes sense for them to use that's useful for them, that it's almost odd that they're in order, not in a bad way. And uh huh. Well, that's easy. Okay. It's like almost in a way, I guess if you have to be that easy kind of way. Alright, so let's try another one. If you're ready, you're ready. 

Kim Montague  11:48

Okay. 

Pam Harris  11:49

Alright. Here we go. (1, 1), (2, 3),  (3, 5), (4 , 7). So pause the podcast. Alright, Kim. 

Kim Montague  12:05

Alright.

Pam Harris  12:05

What are you thinking about?

Kim Montague  12:06

Got it. So I think that it is y equals well, what was I gonna say? 2x minus one. Okay, so I love the fact that you giving me an answer. But I kind of hate the fact that we're not hearing you're thinking. So again, I was thinking about, you know, it's a little bit like I see the pattern, maybe a little bit quicker than I remember how to write it, like that social bit of writing it. So like, once I figured out how to write it, then I get excited. Sorry. 

Pam Harris  12:44

Sure. Okay, so that makes sense. 

Kim Montague  12:46

I'm thinking about, again, from x to y, I wanted to know what, how do I get from two to three. And at first, I was like, okay can be times one plus one. But then that doesn't work with three or four. And then I was like, let me go over. So times two would be four, then how would I get to three, would just minus one. And then I kind of confirmed it with the 3 as x and the four as x.

Pam Harris  13:13

So I really like your strategy. And I would definitely bring out your strategy., I hear you really focus on how to get from x to y, which is totally what is, the functions are based on. There is a slightly separate and I want to build your strategy. 

Kim Montague  13:28

Yeah. 

Pam Harris  13:29

So I definitely want that. And I'm glad you're, glad it's coming out. But if I was doing this with multiple students, I would expect that there would also be a different strategy that would come out that I would want to highlight and then build both strategies in students. So I might wonder if I would look at 1, 3, 5, and 7 and wonder if I know multiples that are near those. So once I've established the 1, 2, 3, 4, so I have this sequence, that's upping by one every time, once I have that on the x's, then I can kind of, I could choose to ignore the x's. You didn't, which is a great strategy. But I could also sort of look at 1, 3, 5, 7 and go, "Is that near a multiple I know? Well, I know 2, 4, 6, 8. 0h, but these are all just one less than that." And I feel like that would be kind of an Ann Roman strategy. Like sort of feel the 2, 4, 6, 8 but it's all one less than that. Does that make sense?

Kim Montague  14:27

Yeah, I'm not sure why you would go to 2, 4, 6, 8 or...

Pam Harris  14:35

It would literally only be if that one sort of pinged for you. 

Kim Montague  14:38

Okay. Alright. 

Pam Harris  14:39

If it doesn't ping for you then your strategy would be a great,

Kim Montague  14:44

 Yeah. 

Pam Harris  14:45

Almost more general one. Your strategy is more general. It's going to be, it's probably going to be, work, it's probably going to work more often. So the Ann strategy is only going to work if the multiples ping for you. I wonder if I have one that... Let's try another one. 

Kim Montague  15:01

Okay. 

Pam Harris  15:01

Okay, new one, ready? 

Kim Montague  15:03

New Post-it. 

Pam Harris  15:05

You and your Post-its. I love it. Oh, and are you writing with pen or pencil? 

Kim Montague  15:08

A pencil. 

Pam Harris  15:10

Absolutely got a pen in my hand. I love it. Okay, here we go. Ready? (1, 18) 

Kim Montague  15:15

Oh okay.

Pam Harris  15:17

(2, 35), (10, 171) 

Kim Montague  15:22

Whoa. 

Pam Harris  15:22

Oh, and then we're done. That's it. 

Kim Montague  15:24

Oh, gosh. 

Pam Harris  15:24

I just had a thought. I don't want to forget.

Kim Montague  15:26

You know what? Okay, so this, I really did see the 17, 35, 170... That's not gonna work. Oh, 17, 34, 170 sorry, get excited about 35. So then that would definitely I would see those 17, 34, 170 and know that it has something to do with 17. But then it's just shifted up one. So 17x plus one. So that when I did focus on the shift.

Pam Harris  15:58

And I wrote it on purpose, in a way, so when I write Problem Strings, right, if I'm aware that that strategy might not be coming out, 

Kim Montague  16:06

Yeah. 

Pam Harris  16:06

Then this particular problem, I put that 10 in there.

Kim Montague  16:09

Yeah. So times 17, my strategy of going from x to y times 17 is not amazing. It's not...

Pam Harris  16:17

It doesn't pop as much.

Kim Montague  16:19

Not a fun thing. 

Pam Harris  16:20

Yeah. Interesting. Interesting. 

Kim Montague  16:21

Nice. 

Pam Harris  16:22

What do you think? Yeah. 

Kim Montague  16:23

Yeah I like it. 

Pam Harris  16:24

So there, there can be this sense of if I have done a lot of ratio table work, those multiples might pop. And if they do, we can use them. If they don't, then we could go more to the Kim strategy where I'm looking at the relationship between x and y. And if that doesn't work, then I could do something a little bit more where I have to maybe find some differences and do some work to write the equation of the line. But mathematizing is not about one way to do every problem. Mathematizing is about using what you know, relationships and connections that ping for you. And if they ping for you then going down that road. And Kim, you taught me that. So something you taught me. And y'all we do stuff like this in my online workshops. And I'm so excited to debut Building Powerful Linear Functions, where we really help you think about how can we take students from really knowing nothing about the writing the equation of a line to writing the equation of a line given any information from no rote memory. And we do it in a sequence of about 13 tasks, or 13 to 15 tasks, depending on how your, how long your class periods are. And we can help you, help both you and your students learn more and more about linear functions and how they are actually really quite Figure-Out-Able. Whoa, so Building Powerful Linear Functions is now available. I am so proud to have it out there. High school teachers, your time has arrived. Sign up now so you don't miss out. Yeah, head on over to mathisFigureOutAble.com/workshop to book your spot. And you really do want to hurry because register...  Sorry, workshops, at mathisFigureOutAble.com/workshops.  Okay. Head on over to mathisFigureOutAble.com/workshops to book your spot. And you do want to hurry because registration closes this Friday. Yeah, this Friday, y'all we'd love to have you join us. Elementary, middle school teachers, remember, we also have a workshop for you. Building Powerful Addition for Young Learners, for pre K through second grade; Building Powerful Multiplication for third grade and up; Building Powerful Division for third grade and up; and Building Powerful Proportional Reasoning for sixth grade and up, though fifth grade teachers, you're really wanting to do that after you do multiplication and division; and Building Powerful Linear Functions for eighth grade and Algebra I one, but anybody higher than that, who's never really thought about how we can develop linear functions, just from student's own experience and thinking, y'all we're so excited. For the first time ever to have workshops K 12. Everyone in your district can get involved, we can really get Figure-Out-Able math happening all over the place.

Kim Montague  19:09

And I'm gonna say that one more time, so don't mess it up. You want to go to mathisFigureOutAble.com/workshops to reserve your seat.

Pam Harris  19:18

y'all 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.