Math is Figure-Out-Able with Pam Harris

Ep 133: The Ratio Table, Pt 5

January 03, 2023 Pam Harris Episode 133
Math is Figure-Out-Able with Pam Harris
Ep 133: The Ratio Table, Pt 5
Show Notes Transcript

It's our wrap up episode on ratio tables. In this episode Pam and Kim answer some frequently asked questions and talk about how we can take ratio tables into higher math and beyond.
Talking Points:

  • Why teachers erase and when should students erase?
  • Do, Say, Represent: how to help students represent what's going on in their heads 
  • Ratio tables aren't a gimmick, they make thinking visiable of how mathematicians and other mathy people think for the masses
  • Using ratio tables to notice patterns and reason using mathematical relationships in order to write the equation of a line
  • Using ratio tables to reason about rational functions and transformations


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Twitter: @PWHarris
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Facebook: Pam Harris, author, mathematics education

Pam:

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

Kim:

And I'm Kim.

Pam:

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But, ya'll, it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor mathematicians, as we co-create meaning together. Not haphazardly! We're planning for it! 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:

So, today, we're going to finish up our series on the fantastic mathematical model called the ratio table. We've already talked about how to use it to multiply, to divide, to solve proportions, and to add and subtract fractions. And in this episode, we're going to finish out by clarifying some important ideas and solve an extension problem. So, Pam, let's start with some clarifications. Karen Camp, a fabulous, consistent participant in MathStratChat said, "Do kids also need to erase?"

Pam:

Yeah, totally. So, in the first episode of this series that we've just done. I don't remember the number of it. One of those numbers. A few episodes ago. So, this is the fifth one, so the first one. I should be able to do that. That's a fence post problem, Kim. I'm not even going there. So, we talked about how when you facilitate. When, teachers, when you facilitate a Problem String using a ratio table, we suggested that you strategically erase.

Kim:

Yeah.

Pam:

And Karen is asking, "But don't we want students to, like,

Kim:

Yeah.

Pam:

So, that as you, as the students are solving problems, and you're showing that they scaled times 10. So, if I scale the number of packs times 10, and I scale the number of sticks times 10. Or if I divide those in half, then I would show that scaling. Or if I add these two numbers of packs together to get these added numbers of sticks together. That when we do that there's all this notation that happens on the board during a Problem String. We suggested that there are strategic times for you to erase, so that what remains on the board are the most important, main relationships that are happening as you go to the next problem, so that when you put the next problem up, kids don't have to see how we got the last answer. They can now just focus on what relationships do we have up there that they could use to get the next one. notate their work, so we can follow what they were doing?" And the answer to that is, absolutely! So, what we didn't do at all in that first episode was parse out when are teachers erasing and when are students erasing. And, Kim, this really reminds me of when I. Early in

Kim:

Yeah. Right.

Pam:

Anyway, I know I'm kind of going off on. You're like, "Why our work together when I was diving into the research, and diving into my kids classrooms, and you were at my kids campus, and we were working with open number lines for the first time ever, and trying to model thinking, the teachers asked me. You noticed that I sometimes put arrows on open number lines. And sometimes I put plus and minus if like I was adding 20. Sometimes I would write "plus 20". And sometimes I would are you going into addition and subtraction?" write... If I was doing a subtraction problem, I would write "subtract [something]". And sometimes I wouldn't. And they were like "We can't tell what's your rhyme or reason for

Kim:

No, it's good. when you would and when you wouldn't." And we had to kind of parse that out. But then, the, strongly was this suggestion of,

Pam:

My point is that there are times where teachers are strategic about what they notate. We want students to not "But students should notate their work more." Like, there worry about that. What we want students to do is notate, like mark up your work, show us your thinking as much as possible. were times where I said "Ooh, but if I'm dealing with the And when they might go, "Oh, I did it in my head." Then. we could say, "Oh, yeah, I believe you. I believe you did it in your head. I can't see what's in your head. Can you show me?" You relationship between addition and subtraction, let's not maybe know like, "Can you help me? Help me see what's in your head? Help me see what's happening in your brain?"

Kim:

Yeah. put a plus and minus sign up there because I want students to

Pam:

Remember, Do, Say, Represent. This might be the point where a student is like, "I don't know. I don't know how to show you what's in my head." At that point, then we say "Oh, be able to see from this number line, they could write an well tell me. Tell me what's going on in your head. Here, I'll help make your thinking visible this time. Next time, addition problem, they could write a subtraction problem. you give it a go. Does this show what was happening in your head? Cool, cool. Alright, now you've seen how to put on paper what Like it's these relationships could represent, could be was in your head, how to make visible what you're thinking. Next time, you show me what's going on in your head." Because represented by many equations. But as soon as we put a that's what's important, right? It's like the relationships you're using are important. And yes, Karen, we need to see what subtraction sign on the jump. Oh, then it's a subtraction problem. students are doing as much as possible, so I'm not asking students to erase. Yeah. I'm so glad that you said that because I could hear people. The

Kim:

Yeah. part about helping students. Because I could hear people saying, "But my kids don't know how to represent quite yet." And this actually came up. We had a conversation not too long ago

Pam:

Way to go Adina! Way to go! Woohoo! with one of our stellar teachers in Journey membership site. And do you remember? She shared some work in the Facebook group with her students, and she was really excited because brilliantly, she noticed that her students were doing some great thinking with really sophisticated strategies for multiplication.

Kim:

But they didn't really represent their thinking well, so she. Like, genius. She like moved into the space, and she asked students what they were thinking about, and then she model represented for them. And that's what we would expect that the use of ratio tables would continue. We can't expect that students will see us make use of a ratio table like a few times, and then automatically know how to represent all their thinking, you know, beautifully, clearly. But in time, seeing their thinking on paper, it can become a tool for thinking as well.

Pam:

Absolutely. And then, now that it's a tool for them, then they're going to be more naturally notating they're thinking. And all of a sudden all the thinking is happening, and we're able to see it. They can make it visual because they're actually using the ratio table as a tool. They're not just, "Ooh, what happened in my head? Here, let me put that down." But they're actually like, I can't tell you the number of times that we see students now where we give them a multiplication problem, and they throw down a ratio table and start messing with values. In that way, it's actually a tool that they're using in time to think with. Yeah, that's super cool. In fact, it might be worth parsing out right now something I almost wish that I had thought of to say in the first episode on ratio tables. But hey, let's wrap up our series on ratio tables by making this point clear. What a ratio table is not is some funky, new math. "Oh, now we're going to do this different weird thing." It's actually much more like this. That what we've done is interview mathy people. This is my work. I don't know. I don't know that I've clearly said this very well on the podcast, so I'm going to be a little more clear today. Part of my work is to interview mathy people and figure out how do mathematicians actually think and reason about math? What do they actually do? Because, ya'll, they don't use algorithms most of the time. They come up with algorithms because they want to generalize relationships, but they don't actually use them when they're solving problems. So, what I do is I dive into the math, and I dive into people's brains, and I pull out the thinking, like what's actually happening as mathy people are solving problems, and then we make that accessible to the rest of the world. Then, we say, "Oh, hey, solve that proportion. Solve that multiplication. Oh, those? Those relationships? They can look like this on a ratio table. Ah, that's how you mathy people have always been thinking about division? Hmm, we could make that thinking visible on a ratio table." And then, the rest of us now have access to it.

Kim:

Yeah.

Pam:

So, the ratio table is one way that we are able to take what people have naturally done to solve math problems and represent those relationships that they have sort of naturally been using that the rest of us didn't know existed because we were just doing the steps. We're over there mimicking the steps we were told to do. But what we weren't doing was being able to see inside their brains. Well, that's what we're helping you with, with ratio tables. We're helping you go, "Oh, when sort of mathematicians, mathy people have been kind of naturally using their whatever talent to solve problems, these are the relationships they've been using." And we can make them visible in a ratio table. And then, we can help that ratio to become a tool, so that you can think the same way, you can start to use the same kinds of thinking to solve multiplication problems, the division problems, and proportions like we did in the third episode. And last week when we did in solving addition and subtraction of fractions by finding equivalent fractions. You can use ratio tables to think in those ways. And then, we can all think more and more like mathematicians. Does that make sense, Kim? Did I say that? I'm thinking I'm still parsing out how to kind of say all that.

Kim:

Yeah, yeah. No, I appreciate that what you do is really try to make sense of mathematics for the masses, in that these are things that are happening, and it's just sometimes hidden, sometimes like trapped inside people's brains because they don't have the language maybe or definitely don't have a way. That was it for me, right? I kind of stumbled my way through trying to figure out how to even talk about it, but I didn't have any way of putting stuff on paper to make sense of it for others. And so, it was just not a thing. We just didn't talk about it. We just didn't. You know, I didn't share strategies. I certainly didn't have a way of like learning more of the sophisticated strategies that I wasn't using. So, you know, it's good work.

Pam:

Yeah, super. I appreciate you saying that. Cool. So, in

Kim:

Super fun. this episode, we'd like to finish out by giving an example of where ratio tables can go in higher math. What do we do even

Pam:

Even if this is not your content, ya'll, dive in because after the work that we've done so far with solving proportions and adding and subtracting fractions? Well, first of all, let's acknowledge that if we've done the work, if we can get everybody to do the work that we've talked about so far...getting kids multiplying and dividing and solving proportions, finding equivalent fractions, ratios using ratio tables...that's a huge win because students will be thinking and reasoning proportionally. And getting them to think and reason proportionally is a super huge win because now we can build on that. Everything from rates, rates of change, to scaling to fractions and ratios, to then rational functions all are going to be based on, build on reasoning proportionally. And all of those things are going to be more of a natural outcome that we'll have more access to, if we've gotten students thinking and reasoning proportionally. And one of the super good ways to do that is using a ratio table. But there's also a fun side effect that I did not anticipate, and it's super cool the more and more that I see it as more and more people are using ratio tables to build up proportional reasoning. This super fun side effect about writing the equation of a line. And we did a little bit of this in an earlier episode where... And we're going to do it again. So, this is super fun. So, even. Kim's laughing. I think especially if you've been using ratio tables, you're going to be amazed at what's going to pop for you as we go. So, Kim, I'm going to give you a couple of points, and I'm going to ask you to consider. Well, I'm going to walk you through kind of what we're going to do, so.

Kim:

Okay.

Pam:

Here we go. If I give you the point: (1, 14). So, 1 comma 14. X is 1. Y is 14. And a point: (2, 28).

Kim:

Okay.

Pam:

Before we get into writing the equation of a line or anything, could you picture sticks of gum?

Kim:

Yeah.

Pam:

Could you tell me about that context?

Kim:

Sure. So, I'm going to. I actually (unclear) because we're talking about ratio tables. I actually put that on a ratio table, and I put 1 to 14 in the first entry. And then, I put 2 to 28 in the second. And it completely makes me think of 1 pack has 14 sticks, 2 packs has 28 sticks.

Pam:

Brilliant. So, here's a necessary step in order to get my final thing to work is that we would need to have done sticks of gum. Which obviously you're like, "Yep, clearly. Clearly, that's pinging for you right now." But we would also needed to have done a little bit of an extra step where we would graph those points as ordered pairs. So, over 1, up 14. Over 2, up 28. And we would have written the equation of the line that contains those points. So, we would have wanted to say, "If I want to find any number of sticks based on the number of packs, what equation of a line would that be? And so, what equation of a line would work for these two points?"

Kim:

So, it would be 14x.

Pam:

Yeah.

Kim:

So, y would equal 14x.

Pam:

Yeah. And if we stayed in sticks of gum, then we would say the number of sticks is equivalent to 14 times the number of packs.

Kim:

Yeah, so in mine, I actually made a third entry, and I put x in the top, and I put 14x in the bottom.

Pam:

Oh, nice. Nice. So, you can kind of see that that's the relationship, the output will be 14 times the number of sticks. Cool. So, listeners, we would want to get students to be able

Kim:

Okay. to write the equation of a proportional relation where it would fit in a sticks of gum scenario, where it includes the origin and all of the ratios are equivalent. 1 to 14 is

Pam:

And (2, 27). equivalent to 2 to 28. So, when I have a proportional relation, we'd want students to be able to write the equation of that line. In this case, y equals 14x. We would want to do that a bit, so that students kind of have this feel for proportional relations, not only operating in a ratio table but also being able to write the equation of the line that fits those points. When we do that, then I think it follows fairly naturally for me to give a set of points. I'm going to give you another set of points, Kim, and I'm going to ask you if you see any connection, or if anything pings for you. So, what if I give you the points (1, 13). 1 comma 13.

Kim:

Okay.

Pam:

Is there anything that pings for you based on....This is a separate ratio table. I should have said that. We're not adding to the first ratio table. So, new set of points. Does anything ping for you as you?

Kim:

Well, yeah. Yeah, it's like the first ratio table because instead of 1 to 14, it's 1 of the 13. Instead of 2 to 28, it's 2 to 27. So, it's like that same 14 sticks per pack, but you took away a stick. So, 14 per pack, minus 1.

Pam:

Because there's still. 13 plus 14 is still 27, right? There's still 14 sticks in between the two points. Like, if we were to look at from 13 to 27, it's still 14. So, there's still that change of 14 every time, 14x. But we've subtracted 1. And I will never forget the day that Anne Roman said to me, "I just see the 14, 28 just popping out." It's just, like, sitting there." I've dealt with ratio tables so often that I just see the 14, 28. But it's just... It's just shifted just a little. It's just off 1."

Kim:

Yeah.

Pam:

So, then, what's the equation of the line for 1, 13 and 2, 27? (unclear).

Kim:

(unclear) Minus 1. Yeah.

Pam:

Yeah, so y = 14x - 1. Let's do another one. One more example. Alright, so this time, I'm going to give you... Again, two ordered pairs. The first one is (1, -6).

Kim:

Okay. Mmm, okay.

Pam:

Second point (2,-12).

Kim:

Okay, so this one is interesting to me because you could think about, like sticks of gum 1 to 6, 2 to 12. But then, it's negative. But I also was thinking more like, one time something happened, I lost $6.00. And (unclear) happened, I lost two 6s. (unclear) like a debt, or like I owed $6.00 each time something happened.

Pam:

Sure enough. So, nice flexibility. You can kind of think of that two ways. So, what would be the equation of a line that would represent those?

Kim:

y = - 6x.

Pam:

Cool. And I'll just focus because we've been talking about ratio table. So, I like how you did the $6.00 things. But I'll focus on what you said first. That if we've got students thinking about 6 sticks of gum. 12 sticks of gum and 2 packs, then could they write the equation of line y = 6x. And they can even picture that line, sort of going through the first quadrant, and then say to themselves, "But it's the opposite. I've got to reflect everything. And so, now, it's going to go through the second and fourth quadrant. It's y = -6x." So, just a different way. We would want students to own both of the ways that you just thought about that. I'm so super glad that you do. Cool. Then, if we've got sort of that relationship kind of pinging for kids, could we give them problem points like this? (1,-5); (2,-11).

Kim:

Okay.

Pam:

So, (1,-5), (2,-11). And wonder if anything pings?

Kim:

Yeah, so it's going to be similar to the one that you just So, y e= -6x +1. gave me where it's -6x, but it's not quite as negative each time. So, it's 1 away. It's not quite as negative by 1. So, plus 1. So, -6x plus 1.

Pam:

Super. Yeah, because we've just. We can see... It's almost Yeah. like you can see the -6, -12, but it's not quite as negative. I like it. It's been shifted up 1. And I actually just sort of took that line that I had kind of drawn in the air, and I shifted it up 1 to get -6x + 1. Super cool. So, what we're suggesting is that there's this natural outcome that once students really start thinking and reasoning with ratio tables that they begin to see them in places that we might not have seen them before if we hadn't already kind of owned those relationships. And if we can write the equation of a line for proportional relation, and we can see that this non proportional relation is just a little bit off, that we decide how far off is it. Oh, bam! And then, write the equation of the line based on just those noticings. Kim, when I say in the beginning of the podcast that we can... I'm trying to go back to my script here, so I can say it correctly. When we notice patterns and reason using mathematical relationships. That's what I mean. That's what I mean when I say that. That it's about noticing the pattern like this ratio table. It's like letting it ping for you. "Oh, but it's just off a little bit." And so, I can write the equation for that proportional, and for this non proportional, I can just tweak the equation to match. It's about noticing patterns and using relationships. Woah! Super cool. And let me just note that we also have this shift happening, and so there's also this idea of shifting. And boy, I'm just. I'm reminded of two things. I'm reminded of a geometric shift that when we translate functions. And so, that's an important kind of thing that comes out naturally because of this ratio table connection, that we can use then with students as we talked about transformations of functions. There's also... This is like the hardest thing for me to talk about in higher math. When we think of that shift in transformations of functions, the vertical shift is super easy, right? Like, if you have y = x^2 + 1, you're like, "Duh," because you're taking all the y values, and you add 1, and so that parabola just shifts up 1. If it's y = x^3 - 15, then you take that x^3 function, and

Kim:

Yep. all the y values are just down 15. So, easy shift. But boy, that shift in the parentheses. It's tricky, right? It's tricky. When you replace x with x - 5, why doesn't the parabola go to the left 5?! It was x - 5. You'd think subtract, it should be left. But it's not. And I tell you, Kim, when we work with, so, I'm definitely talking to high school teachers now, or high school and above. But when we work with, well, I shouldn't just say that. Anybody who's messed with functions and transformations of functions lately, that's who I'm talking to. When we mess with this proportional relation, and how I'm encouraging us to see the ratio table. The equivalent ratio is just shifted up or down. That is the same feeling I get when I am finally now able to explain that horizontal shift for function transformations. Because for me, in my head, what's happening is I'm saying to myself, "I'm replacing x with x + 2. And when I replace x with x + 2..." So, x + 2 is in the parenthesis. And that function is going to shift to the left. "What?! But it's plus! Shouldn't it be to the right?!" It's all about me having to go in the table to where I would normally have gone the x, but I've got to go plus 2, I've got to go down the table 2, grab that y value. But then, it's got to shift back up the table to give it where that x was. See, I don't say that very well. Anybody who understands that shift just went "Yeah, yeah, yeah, yeah. That's totally right!" And anybody who doesn't is like, "What?" I can remember when we were filming for Building Powerful Linear Functions. I remember thinking, "What is going on?" Like, in this moment that you're talking about right now. I was like, you know, "Oh, it's plus 1. It goes this way. It's plus." And I remember going, "Wait a second." And like really thinking about. And then, when my oldest Luke tackled this, I was like, "Wait a second." Like, it took. Like, I had that minor experience the first time, and I was like. It was real kind of foggy for me. I was just trying to make sense of it. But then, when Luke and I kind of worked through some stuff, I was like, "Ah!" It was just one of those moments where I was like, So, I'm following you right now. Like, I have a clue what you're talking about because I've had an experience where I had to make sense of that myself.

Pam:

Super cool. Oh, I love Luke. Nice. Alright. So, why this model? Why the ratio table model? In summary, we cannot only multiply and build multiplicative reasoning with division, and throw long division away because we can use ratio tables to think and reason to division. But then, it can extend to solving proportions. It can extend to thinking about equivalent ratios and fractions and everything to do with proportional reasoning. And we can use it as a tool for reasoning in higher math like finding the equation of a line. We can use it to reason about rational functions. There's so many different places that it's going to help us reason proportionally and beyond in mathematics. And remember, it is not a strategy. It's not... When someone says, "How did you solve it?" The answer isn't, "I did a ratio table." The answer is, "Oh, I use these relationships in a ratio table." Like, "This is... I did... I was thinking about an Over strategy." Or, "I was saying about Five is Half of Ten." Or, "I was scaling up, and then." It's the idea of what were the relationships that we're using. Remember, the ratio table is a model. It's a super cool model because it can be used as a tool for reasoning.

Kim:

Yeah, Ya'll. I'm super excited because we have a challenge coming up. We are super excited.

Pam:

Woo-hoo!!

Kim:

Three times a year, we run a free challenge for math teachers everywhere. Everyone's invited. We'd love for you to join us. It's going to be January 18th through 20th.

Pam:

And I may be even more excited than you are. It is super fun! It is like a free webinar on steroids because it's not just that one hour that we get together. But we get three hours! It's three days, one hour each where we dive in. It's totally free. I teach about something very timely, something that's on my mind that I'm seeing happening as I'm working with teachers all around the world that I'm like, "Oh, we gotta dive into this thing. This is going to be super helpful for everybody." We give you short, actionable things, so that you can get quick wins and really learn a ton all at once. Things that you can try immediately with people right next to you and try immediately in your classrooms. And, ya'll, it's just a way for us to get together and do some exciting math for teaching. We always have a special guest, and we have a super fun Facebook group, and all the things. It's just a blast. Really, I want to invite you cordially to be part of our upcoming challenge.

Kim:

And you will be able to register for this coming challenge next week, January 9th, 2023.

Pam:

And if you're happening to listen to this episode after that date, you can always check mathisfigureoutable.com/change to see when our next challenge will be coming up. We would love to have you join us. And thanks for joining us today on the podcast! 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. Let's keep spreading the word that Math is Figure-Out-Able!