Math is Figure-Out-Able with Pam Harris

Ep 132: The Ratio Table Model Pt 4

December 27, 2022 Pam Harris Episode 132
Math is Figure-Out-Able with Pam Harris
Ep 132: The Ratio Table Model Pt 4
Show Notes Transcript

It's time to talk equivalent ratios and fractions! In this episode Pam and Kim demonstrate how understanding equivalent ratios leads to powerful reasoning about fraction operation, without the need to memorize steps and rules or algorithms.
Talking Points:

  • Empowerment is about having choice
  • Ratio tables aren't a strategy, they're a model/representation of the strategy or strategies being used
  • If you understand fractional equivalence, you don't need rules for fraction operations
  • How to add 3/4 and 5/7 thinking and reasoning using a ratio table
  • Do non-equivalent ratios go on the same ratio table? How do you handle adding non-equivalent ratios?


Note: for more infomation about the packs and sticks of gum scenario, listen to Episode 131

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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 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 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, in the last three episodes, we've been talking about one of our very favorite mathematical models, which is the ratio table. Yeah, in Episode 129, we discussed using a ratio table to think about multiplication. In Episode 130, it was all about division with ratio tables. And then, last week in Episode 131, we had fun reasoning proportionally about solving proportions. In this episode, we're going to continue our series on the ratio table model by extending the work to equivalent ratios and fractions

Pam:

Now, if you teach younger subjects than that, you're still going to want to listen to today because it's going to impact how you and why you use ratio tables with younger stuff like multiplication, division. And really, if you're teaching multiplication, division, you are teaching some fraction stuff, so this is going to be super helpful to kind of see that extension and how it relates to all the things. And it has been super fun to see the number of people on social media who are getting so excited about using ratio tables because A, it's super cool. And B, I think it's a relatively unknown model for many of us. And so, it's been really... I mean, for me, it's been life changing to use ratio tables. So, check it out. A colleague of mine, dear friend Karen Camp, who falls and does all the things, used four strategies this last week for MathStratChat. It was super, super cool. And in one tweet, she got four strategies. And then, she followed up with another tweet and in the second tweet... It's like she couldn't fit all of them in one tweet. So, four strategies to solve the MathStratChat problem in the first tweet. And then, the second tweet, she said, "Also ratio table!" And then, in parentheses, she said, "Listen to this week's podcast" with a winky smiley face. Woah, sweet! Hey, Karen, thanks for listening to the podcast, and I love that it encouraged you to use a ratio table to solve a problem. Right after that, another colleague and dear friend, Ray O'Brien or @pimathman on Twitter. He tweeted, "Was I supposed to use ratio tables?" That was sort of his answer to Karen's, you know, "I used ratio table!" Then he's like, "Was I supposed to use ratio tables?" So, you might be interested in how I responded to Ray. And @pimathman, thanks for your question. So, I responded. I know people hate it when you answer a question with a question, but that's exactly what I did. I said, "I would answer that question with a question. Could you use a ratio table if you wanted to? If the answer is yes, and you chose not to, okay. No problem. However, if the answer is no. In other words, no you couldn't use the ratio table if you wanted to because you don't know what it is, you don't know how to use it, or you don't know what to do. If your answer is no..." I'm adding a little more in here, by the way than what was in the tweet but "If your answer is no, then I'd wish for you to own the coolness of ratio tables, so that you would have the choice. Empowerment is about choice."

Kim:

Yeah.

Pam:

Okay, so that's the end of that tweet. And Ray responded, "I can use the ratio table. I do like them. Practically, when I see the question on my phone, I want to solve the problem mentally. So, if I was on an iPad, I might be more likely to use the ratio table. I'm not gonna get out a piece of paper and pencil, but that's me."

Kim:

Sure.

Pam:

Ah, so that's a huge, interesting answer because what Ray is saying is, "Yeah, yeah, yeah. Yeah, I own that representation. I could put my thinking. I could make it visible in a ratio table. I could use the relationships that I'm thinking about, and I could put them in a ratio table. But then, I'd have to get out of paper and pencil, and I don't want to do that, so I'm going to record my thinking in a different way because I'm on Twitter, and I'm responding to MathStratChat in that way." That is a completely. Way to go, Ray! Woah! Absolutely no, don't give me a ratio table. Because it's the thinking. It's the strategy that we're interested in. Not the model. The model is a model. And on MathStratChat, it's math strategy chat. We're interested to talk about the strategies. So, then, why bring up ratio tables? Because it's such a powerful model that we can use to make the strategies visible. And as we do that, that model can then become a tool with which we actually think we use it as a tool, not just to record our thinking, but actually kind of as a tool in order to think. It can kind of nudge us to go, "Oh, well, if I've used that relationship, ooh, and it can..." We can see other relationships can appear. It can be a helpful tool in solving problems. So.

Kim:

Yeah. Yeah, so.

Pam:

(unclear).

Both Pam and Kim:

Go ahead. (laughs).

Pam:

So, if you are asking yourself, "Should I be using a ratio table?" The more important question is, do you own ratio tables.

Kim:

Yeah.

Pam:

And if you do, then it's your choice. You can use it or not. If you don't, man join in, dive in, and let us help you learn how to use ratio tables, so that then you have the choice, then you're empowered to do what works best in that problem. Whatever tool you're going to use best to help you with whatever strategy you're using, then you're empowered to do that. Do you remember what you were going to say?

Kim:

Yeah, I do actually. That's rare.

Pam:

Yay!

Kim:

You know, you and I look at the comments that people make for MathStratChat, right, and we like to talk about them and kind of one of our favorite things to do is share pictures of what people are doing. But sometimes I wonder if people scroll through MathStratChat and see other people's thinking on ratio tables and just keep scrolling, thinking that they're all the same, that everything that's on a ratio table is one strategy, which is the ratio table strategy. Which is not a strategy, right? In reality, they would have to look really carefully at what people are putting in the ratio table to understand their thinking about the problem.

Pam:

Oh, that is so well said. It's almost like the comparison could be that you could look at an open array, a rectangular area model, and you might. If the only thing you've ever seen on that area model is place value, partial products, or somebody's always splitting the numbers into the 10s and 1s, and 10s and 1s, and then they're multiplying all those parts. If that's the only thing you've ever seen on an area model rectangle, then you might be like, "Oh, they did an array," without realizing they might have cut the pieces very strategically. And they might have used very clever partial products. And that's what's worth looking at. It's the way they cut the area up. Similarly, what you're saying is in a ratio table, it depends on what relationships you use in the ratio table that's the strategy. So, yeah, we would encourage you that as you look at MathStratChat, or ratio tables anywhere else that you dive into what are the relationships that are being showcased that are being mentally... What's the word I want? The connections that are being made. That's what's cool to look at in a ratio table, not just "Oh, that person did a ratio table. Click. Let me go see if somebody did something different." No, no. Like, it's what they did in a ratio table is super important. Yeah. Yeah. Nicely said. And one of those places that's super cool that we can do with ratio tables is fraction equivalence. And you might be like "Pam fraction equivalence. That's so easy. All you do is you write down... Like, to find an equivalent fraction, there's this rule, there's this procedure that you can do. You just write down the fraction." Like, I just wrote down. I don't know. Why not? Five-sevenths because it is, and say you wanted to make it equivalent to something with 21s. And so, I've got the fraction five-sevenths equal. And then, I have the fraction bar. And then, in the denominator, I have 21 because I'm going to try to create equivalent fraction. And you might have memorized. I did as a student. 7 goes into 21 how many times? And then, you take that number, and you multiply it by the 5, and then you get the other numerator. And you might be like, "Yeah, Pam, that's what you do to find the equivalent fraction." Well, that might be a set of steps, that might be a procedure or an algorithm to find equivalent fractions, but what it's not is demonstrating understanding equivalent fractions. It's not using what you know about equivalence and building equivalence at the same time to find an equivalent fraction. I will never forget. Kim, you remember Garland Linkenhoger?

Kim:

Oh, yeah, yeah.

Pam:

So, dear friend of ours, who we have not seen lately for. Come on, Garland. We have to figure out how to meet up with her at some point. Wonderful, wonderful lady who taught us a ton. We worked really early in when you and I were working together at the school you were teaching.

Kim:

Yeah.

Pam:

She she said one day. So, this is super early in my journey in learning that Math is Figure-Out-Able and that more than I ever knew was figure-out-able. So, I'm diving into the research. I'm diving into my kids classrooms. I'm working with Garland. And she said one day to me. I'll never forget. She looked at me and she goes, "Pam, if you really understand fraction equivalence, you don't have to have any rules for fraction operations." And I said, "I'm sorry, what?" She said, "Like, all the rules to do fraction operations. You don't have to know any of those if you actually understand fraction equivalence." That idea was so new to me, and so... Like, fractions, to me we're all about which rule to do when. Don't look down on me, listeners. Not right now. Don't be like, "Pam, really? That's what you think." Yes, that is what I thought because that's what I was taught. And if the rest of you are like, "Yeah, that's what we were taught," yeah, that's what we were taught! So, if that's your sort of focus of when you think of fractions and addition, subtraction, multiplication, division, and you're like, "Yeah, which rule, and if I remember which rule to do, and the steps, and the rule," I'm going to invite you to consider if you actually understand fraction equivalence. We don't need rules to operate with fractions. So, it's important that students actually have this feel and sense and understanding, not just this sort of vague conceptual understanding, so then we can learn the rule. No, no, no, no. It's like actually really feeling fraction equivalence. In fact, do you remember? Kim, I came home from that day talking with Garland, and I said something to my personal kid. And those of you who listen to podcasts for a while know that I was hugely influenced by my oldest son, the way that he just kind of naturally mathematized was really the impetus. It was my inspiration for diving into all the math and all the things. And I went home to him, and I said something like, "I can't believe she said that. Like what? Cameron, what do you know about fraction equivalence?" I was just super, super curious. Because remember, he's my inspiration. So, I'm like, "What do you? What do you mean? What. Tell me about fraction equivalence." And he looked at me, kind of cocked his head, closed one eye just slightly, and he goes, "Well, it's like, if you cut the pieces in half, you would need to have twice as many to have the same amount of pizza. And I was like, "Say that again." Like, he really had thought about the fact that if you'd cut the pizza into 7 pieces, so you had sevenths. One-seventh of a pizza was one of those slices. If you cut them all in half, then you'd have twice as many pieces, right? But now you'd have fourteenths. And so, you'd have to take twice as many. If I took 1 piece, and I had one-seventh, and now I've cut them all in half, I would have to take two of those fourteenths in order to have the same amount of pizza as the one-seventh. And you might like, "Well, that's trivial, Pam." No, no. So

Kim:

Mmhmm. not trivial. Like, that's a generalization that he was making. That just was starting him in this idea that if you cut the pieces into even number of shares, and you make a smaller fraction, then you got to scale up that many kinds of that many. Boy, I'm saying this to (unclear). So, that's why he said it in halves, right? It was so much easier just to say, if you cut it in half, you have to twice many pieces in order to have the same amount of pizza. That idea, that sense of fraction equivalence is super important. And we can help that happen with fractions, and ratios, and with ratio tables. Ratio tables can be this model that can be one of the helps in making that happen. Yeah, so we're talking about ratio tables today. And so, let me give you a problem. And will you describe for us how you would use a ratio table to solve a problem?

Pam:

Do it. Let's do it.

Kim:

Okay. So, I'm going to give you three-fourths plus your five-sevenths that you were just talking about. You were talking about sevenths. Three-fourths plus five-sevenths.

Pam:

Okay, so to actually use a ratio table to solve this problem, I need to back up just a hair to say it's definitely not the first thing that I'm doing to build fraction understanding. So, we're going to do all of the things and to build like what does it mean to have a fourth? So, you said three-fourths. What does it mean to have one-fourth? And what does it mean to have three of those one-fourths? What does it mean to have one-seventh, and then five of those one-sevenths. So, there's lots of things that we're going to build with just fraction sense, and then we're also going to build a sense of addition and subtraction of fractions using a double open number line. So, that would be one of the models of choice first. So, it would not be a ratio table first. And I'm not going to go into that today. Maybe we'll spend a whole other podcast on that. But ratio table wouldn't be the first model that I'd use. But after I have students using double open number lines. Well, at least... Maybe I'll say it this way. After I'm helping students reason about the addition and subtraction of fractions where I'm modeling their thinking using a double open number line, and we've got that going, and simultaneously I've got students thinking and reasoning about multiplication on a ratio table. Once we have those two main pieces then it's going to be a very natural transition for students to use a ratio table to do the following for a problem like three-fourths plus five-sevenths. So, I would say that students would look at three-fourths plus five-sevenths. And they would say, "Hmm. Let's see. I need the same kinds of pieces. If I'm going to add fractions together, we need to have the same kinds of pieces to add them. Otherwise, I can picture three-fourths of something, and I can picture five-sevenths of something, and I can see that I've got more than one of a thing." That would be delightful for them to kind of. If they can picture those amounts, three-fourths is only one-fourth from one. Five-sevenths is only two-sevenths from one. So, I'm adding almost one to almost one. Gee, that's going to be more than one. So, there should be this sense already of kind of a feel of what's happening. Then, they can say, "But in order to know exactly how much I've got, I've got to have the same kind of pieces." Let's see fourths and sevenths. Now, I'm going very kind of general here. I would never start with fourths and sevenths to get students to get a feel for equivalent pieces. But now that we're there, we would have students go, "Let's see, what is fourths and sevenths? What do they have in common?" Well, in this case, four and seven are relatively prime. And so, they could think, "Well, I know I could deal with twenty-eighths. Because both of those I can make. I can cut fourths up into twenty-eighths, and I can cut sevenths up and make twenty-eighths. If I have pizzas, I can cut them both up and make twenty-eighths. Okay, cool. So, I'm gonna turn them into twenty-eighths. Now, at this point, you might be like, "Well, yeah, then I'm going to do that procedure. You said earlier 4 goes into 28. And 28..." No, no, no. Actually, what I'm going to do is I'm looking at a ratio table, and I'm going to say to myself, "Three-fourths... I've put on a horizontal ratio table. So, I have three on the top and four on the bottom. And I'm then drawing a line next to it. And then, I've drawn 28. And I'm saying to myself, "Hmm, what does it take for me to get from fourths to twenty-eighths or from 4 to 28?" Because I'm cutting those. It's like I'm thinking about fourths, and I'm thinking if I can cut them all up into pieces to make twenty-eighths. But how many pieces would I have? Or how many times I have to cut that fourth up to get twenty-eighths? So, I'd have to do it 7 times. It's kind of like Cameron said. "If you cut the..." Why did we use 4 and 7, Kim? It's alright. It's just a crazy big number. It's just like Cameron said, "If you're going to cut the pieces in half, then you get twice as many. Or you would need to have twice as many." So, if I'm going to cut each of those fourths into seven pieces, then I would have twenty-eighths. So, on the ratio table, I've just said, "Ooh, to get from 4 to 28, that's times 7. So, I've scaled from 4 to 28, and I've drawn a little arrow, and then I've written times 7. Then I would also have to scale the number of pieces that I would have. So, the 3 also has to scale times 7. Three times 7 is 21. So, now three-fourths is equivalent to 21/28 on that ratio table. Now, very importantly, I have to have a new ratio table to deal with the five-sevenths.

Kim:

Right.

Pam:

"What? Pam, why can't we put those all on the same ratio table?" I'm going to get at that in just a minute. They need a new ratio table to think about five-sevenths to get to twenty-eighths. So, I've just written a tiny ratio table next to that one. I've got 5 on the top, 7 on the bottom. I've got 28 Next to the 7. Drawn the line. 28 next to the 7. And I'm saying to myself, "How do I get from 7 to 28? Oh, yeah. That's times 4." So, then I'm going to have to scale up the 5 times 4 as well. And 5 times 4 is 20. And so, the equivalent to five-sevenths is 20/28.

Kim:

Mmhmm.

Pam:

I did those two things on ratio tables. I got the equivalence of three-fourths of 21/28. I got the equivalent of five-sevenths of 20/28. And now I can say to myself, literally could just reason, "Well, I got 21/28, and 20/28, then, I've got 21 plus 20 twenty-eights. or 41/28. So really, I'm using the ratio table to help me reason about what I need to scale the numerator, the denominator in order to get equivalent fractions that make sense in the either addition or subtraction of fraction problem.

Kim:

What I love so much about what you just described is that if you've already. You know, if you've already worked with kids a little bit with ratio tables, with multiplication, then scaling up is not a new idea. You're just bringing it to a new topic. And then, when you said 21/28 and 20/28. There was space for me in my brain. I was like, "Oh, I'm going to do a little Give and Take here and create..." There wasn't this procedure of, "I'm going to add 20/28, 21/28, then I have 41/28 that I have to pull out the whole." When I had the... I was kind of writing as you were describing, and I had 20/28 plus 21/28. And I did a little Give and Take of 7/28. Does that make sense?

Pam:

No, keep going. Why would you do that?

Kim:

Oh, so I had written down 20/28 plus 21/28.

Pam:

Oh! Okay, keep going, keep going.

Kim:

So, then I said, "Okay, that's equivalent to..." You're talking about equivalence. That's equivalent to 13/28 plus 28/28, which is 1. So, I got 1 and 13/28. So, no longer is it a little bit of like, "Do another procedure on top of that."

Pam:

That is so awesome. So, once you knew that you were thinking about 20 plus 21/28, you said, "I'm obviously over 1..." Which we'd already discussed, right? Where we knew we were over 1. And you're like, "So, I'm going to pull that 1 out. I'm not going to add them together and get 41/28, and then have to think about what's the 1 in there."

Kim:

Sure, yeah.

Pam:

"I'm going to find that 1 right off the bat." So, you're just like, "Well, I'm almost there. I'm just going to grab from the 1, grab from the 20, grab 7 to get to the 21." I know I'm re-saying it because it's so cool! So, you made. You basically made 28/28. And then you thought, well, what's leftover? Oh, the 13/28."

Kim:

Yeah, and it just makes me think about, you know, sometimes we... This is maybe off topic. Sorry, pull me back if so. But it makes me think about how... You know, sometimes we hear from teachers, "How do I get give kids some experience using addition strategies or using multiplication strategies in the midst of my content if I'm the older grade teacher?" And so, these are just. You know, right here, we've talked about some addition strategies with Give and Take and some scaling for multiplication. That (unclear).

Pam:

(unclear).

Kim:

Yeah, it's (unclear) content that they're supposed to be doing.

Pam:

Yeah, like fraction addition, subtraction. It's a super, super good. Yeah, I'm so glad you mentioned that. So, teachers that are teaching fraction operations, as you're doing that, you could consider on the day you're about to do something like... What was it? Three-fourths plus five-sevenths. You could consider doing a Give and Take addition string, and then bringing out that idea as students are messing with problems like this, or scaling with multiplication, and then bringing out that idea as students are messing with fraction addition and subtraction.

Kim:

Yeah.

Pam:

Yeah. Well said, Hey, so, Kim, it also occurs to me that there are some things that we might just want to highlight. And I mentioned, I was going to come back to it. Why did I use two different ratio tables to add three-fourths and five-sevenths? Why didn't I just? I mean, clearly, I could have just used a ratio table, wrote three-fourths. 3 on top, 4 on the bottom. And then, in the next entry, I could have put 5 on the top and 7 on the bottom. And then, I could have added those together, right? So, I could have had 3 plus 5 is 8. And then, the 4 plus 7 is 11. And the answer could be eight-elevenths. And clearly eight-elevenths is equivalent to. Or not. Like, it's not, right? It's an incorrect answer.

Kim:

Yeah.

Pam:

And you're like, "Pam, no. You can't just add fractions. You have..." Like I said earlier, you have to have the same kind of pieces, can't just add fourths and sevenths together. However, teachers of multiplication, and division, and proportional reasoning, haven't we been adding ratios in ratio tables? Consider a sticks of gum Problem String. So, if that's new to you, maybe we'll put in the show notes where you can go listen to a sticks of gum Problem String. But basically, where we have a Problem String like if I have a pack that has 27 sticks in it. And so, I just wrote down 1 to 27. And I've got 2 packs would then have 54 sticks. I did that kind of fast. So, I have two entries right now. I have 1 to 27, and 2 to 54. And then, if I said, "How many sticks would be in 3 packs? Couldn't we add the 1 plus the 2 to get 3 and the 27 plus the 54?

Kim:

Mmhmm.

Pam:

Now, I'm sorry. I'm thinking 14 things at the same time. So, 27 plus 54 is that 81? So, wouldn't it be 3 packs would have 81 sticks? Well, that's actually correct. Wait, why can I add those to that ratio 1 to 27, 2 to 54? Why can I add those together to get 3 to 81, but I can't add 3 to 4 plus 5 to 7 and get 8 to 11. And I recognize right now I'm raising kind of an interesting idea that if you've never thought about might be a little mind blowing, maybe just a little bit. And I'm hating the fact I'm about to tell, but I'm going to tell. It has everything to do with the fact that in the sticks of gum scenario, those ratios are equivalent. That 1 to 27 is equivalent to 2 to 54. So, I'm not talking about the fraction 1/27 plus the fraction 2/54 and adding those together. I'm talking about equivalent ratios. 1 pack to 27 sticks is equivalent to 2 packs to 54 sticks. Since those ratios are equivalent, you can just add the numerators and denominators in those ratios. I hate even using the names "numerator" and "denominator". You can add the parts of those ratios together and get an equivalent ratio, if the ratios are equivalent. Notice in our fraction addition problem that Kim gave me. Three-fourths plus five-sevenths. Those fractions are not equivalent. Three-fourths is not equivalent to five-sevenths, so we cannot put them on the same ratio table because by definition a ratio table is where all of the entries, all of the ratios are equivalent.

Kim:

Yeah.

Pam:

So, that's kind of an important distinction to make, that maybe now we can help you start thinking about, so that when you run into it, which you will because you are teaching fractions. If you're teaching addition of fractions, ratios are appearing in your work. If you are teaching solving proportions with ratios, addition and subtraction of fractions is appearing in places. And so, we as teachers need to parse out when are we dealing with equivalent ratios and when are we dealing with fractions, part/whole fractions that are not equivalent. And we need to help students parse out, so then what makes sense to do with those when we're dealing with those different entities, part/whole fractions that aren't equivalent versus ratios that are. Go ahead.

Kim:

I think you're bringing up such a really important point. And I think maybe for you and I, it's one of the reasons why context is so important when we're working with students on a ratio table. Because they can really feel what's happening the ratio table, and it's not just a series of digits or numbers that we're throwing at them. They really kind of own the packs and the sticks, and it makes sense to them that when you add packs of sticks together. I mean, packs... Yeah, packs, packs of gum, then you're adding the corresponding sticks as well.

Pam:

Mmhmm. Yeah. And that that makes so much sense that we can be grounded in that. So, when students are adding fractions, then we can help them go, "Wait, wait wait. Is this packs and sticks? Or are we talking about three-fourths of a candy bar plus five-sevenths of a candy bar? How much of a candy bar would we have? Ah, we're dealing with a part/whole fraction, not with ratios, especially not with the equivalent ratios.

Kim:

Yeah. Yeah. You also have some work in your Workouts that you.

Pam:

Yeah, nicely said. Yeah. So, we help you with this in my book Lessons & Activities for Building Powerful Numeracy help parse out the Oh, yeah, totally. different kinds of problems and different ways you can deal with fractions versus dealing with equivalent ratios. So, yeah. If you're interested, totally check out Lessons & Activities for Building Powerful Numeracy, which has these super cool black line masters called Workouts. And remember that we need to have fraction understanding come before what we did today with

Kim:

You do some work with that. You want to talk about that for ratio tables. And then, once kids really have the sense and a feel for fractions, what they mean, then the ratio table is a minute? sort of this natural tool to use to create equivalent fractions. Yep. We really hope this series all about the ratio table has been helpful so far. You will want to join us next week as Pam takes us to using the ratio table in higher math and we wrap up with some pro-tips you do not want to miss. Thanks for joining us each week! We are trying to grow, and we would love it if you'd support us by giving us a five star rating. Or if you think we don't deserve that, reach out and let us know how we can improve.

Pam:

Yeah, we would super appreciate that. And 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!