# Ep 117: Math is Figureoutable, Even In High School

September 13, 2022 Pam Harris Episode 117
Math is Figure-Out-Able with Pam Harris
Ep 117: Math is Figureoutable, Even In High School

A lot of people feel like when it comes to high school math, it's just not possible to think and reason through things anymore. Not true! In this episode Pam and Kim make the transition from Proportional Reasoning to Functional Reasoning and writing the equation of a line using relationships and what students already know.
Talking Points:

• Is math always Figure-Out-Able?
• How ratio tables prepare students for linear functions
• Example Problem Strings
• Registration for workshops closes this Friday!

Get a workshop while you can: https://www.mathisfigureoutable.com/workshops

Pam Harris:

Hey fellow mathematicians. Welcome to the podcast where math is Figure-Out-Able. I'm Pam. And you found a place where math is not about

Kim Montague:

And I'm Kim. 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. 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:

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:

Yeah, absolutely.

Pam Harris:

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

Kim Montague:

Yeah.

Pam Harris:

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

Kim Montague:

Yeah.

Pam Harris:

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

Kim Montague:

Yeah.

Pam Harris:

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:

Yeah, ok.

Pam Harris:

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

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

Pam Harris:

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

Kim Montague:

Yeah.

Pam Harris:

And, in fact, it's...

Kim Montague:

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

Pam Harris:

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:

You know I like sticks of gum.

Pam Harris:

Yeah, so one pack of gum has...

Kim Montague:

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

Pam Harris:

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:

Sure.

Pam Harris:

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:

Jell-O cups. I don't know.

Pam Harris:

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:

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:

Yeah.

Kim Montague:

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:

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:

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

Pam Harris:

What just happened in your brain when I said that?

Kim Montague:

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

Pam Harris:

Yes.

Kim Montague:

Is shifted, shifted up one,

Pam Harris:

It's just shifted up one, we're just taking that

Kim Montague:

It's almost like I should just get, if just get line, y = 6x... one Jell-O cup for free.

Pam Harris:

Bam, that's exactly what it is. It's like a bonus. A

Kim Montague:

I love Ann. bonus, today you get one, yeah, you get one more Jell-O cup.

Pam Harris:

I know, right? Who one day after, she was a fifth Nice, nice. So we have a colleague. Her name is Ann Roman. Ann, how ya doing? 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:

Pam Harris:

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

Kim Montague:

Okay.

Pam Harris:

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:

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

Pam Harris:

Yeah.

Kim Montague:

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:

Because is that 6, 11, 16, 21 is so close to 5, 10, 15, 20? And how close? Just one, just all just one (click),

Kim Montague:

Okay. Yep. 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

Pam Harris:

Alright. Here we go. (1, 1), (2, 3), (3, 5), (4 , 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, 7). So pause the podcast. Alright, Kim. so let's try another one. If you're ready, you're ready.

Kim Montague:

Alright.

Pam Harris:

What are you thinking about? Okay, so I love the fact that you giving me an answer. But I

Kim Montague:

Got it. So I think that it is y equals well, what was I gonna say? 2x minus one. 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:

Sure. Okay, so that makes sense.

Kim Montague:

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:

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. So I definitely want that. And I'm glad you're, glad

Kim Montague:

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

Pam Harris:

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

Kim Montague:

Okay. Alright.

Pam Harris:

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

Kim Montague:

Yeah.

Pam Harris:

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:

Okay.

Pam Harris:

Okay, new one, ready?

Kim Montague:

New Post-it.

Pam Harris:

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

Kim Montague:

A pencil.

Pam Harris:

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

Kim Montague:

Oh okay.

Pam Harris:

(2, 35), (10, 171)

Kim Montague:

Whoa.

Pam Harris:

Oh, and then we're done. That's it.

Kim Montague:

Oh, gosh.

Pam Harris:

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

Kim Montague:

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:

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:

Yeah.

Pam Harris:

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

Kim Montague:

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

Pam Harris:

It doesn't pop as much.

Kim Montague:

Not a fun thing.

Pam Harris:

Yeah. Interesting. Interesting.

Kim Montague:

Nice.

Pam Harris:

What do you think? Yeah.

Kim Montague:

Yeah I like it.

Pam Harris: