Ep 29: Ratio Tables!

January 05, 2021 Pam Harris Episode 29
Math is Figure-Out-Able with Pam Harris
Ep 29: Ratio Tables!
Chapters
Math is Figure-Out-Able with Pam Harris
Ep 29: Ratio Tables!
Jan 05, 2021 Episode 29
Pam Harris

In this episode Pam and Kim dive deep into their favorite model for multiplication, division and solving proportions: ratio tables! They'll discuss how ratio tables can be a powerful tool to help students develop reasoning from elementary to high school.
Talking Points:

• What are ratio tables?
• How to, and not to, use ratio tables in the classroom
• Using ratio tables to organize data
• Using ratio tables to model thinking
• Using ratio tables for computation
• Example multiplication and division problems

Use this PDF to follow along with Pam and Kim's thinking!

In this episode Pam and Kim dive deep into their favorite model for multiplication, division and solving proportions: ratio tables! They'll discuss how ratio tables can be a powerful tool to help students develop reasoning from elementary to high school.
Talking Points:

• What are ratio tables?
• How to, and not to, use ratio tables in the classroom
• Using ratio tables to organize data
• Using ratio tables to model thinking
• Using ratio tables for computation
• Example multiplication and division problems

Use this PDF to follow along with Pam and Kim's thinking!

Pam Harris:

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

Kim Montague:

And I'm Kim.

Pam Harris:

And we're here to suggest that mathematizing is not about mimicking or rote memorizing. But it's about thinking and reasoning, about creating and using mental relationships. We believe math class can be less like it has been for so many of us and more like mathematicians working together. We answer the question, if not algorithms, then what?

Kim Montague:

In today's episode, ratio tables, I'm so excited. It is our favorite tool for solving multiplication and division problems and our favorite model for building proportional reasoning and solving proportions.

Pam Harris:

We love ratio tables! Give a high five and a happy clap for ratio tables. Totally love them as tools. So let's be clear. What is a ratio table? If you're like me and Kim, we never saw ratio tables as students, especially as tools for solving problems, they just didn't even exist. I dealt with a lot of problems in high school to do functions and function relationships and all that. But what is a ratio table? And why might we be interested in using it to mathematize multiplication and division problems? So a ratio table is a paired number table. But it's a special paired number table where all of the ratios are equivalent, where all of the entries form ratios. And those ratios are equivalent. So what does that mean in layman's terms? For example, if I had, say the scenario is where I had 27 sticks of gum in a pack. We like gum, we chew gum, you can picture a pack of gum that has 27 sticks. Random kind of in that pack that I have 27 sticks of gum in a pack. I might have a table that says 1 pack to 27 sticks, is equivalent to 2 packs would have double that 54 sticks of gum. And all of those

succeeding ratios:

like I might have 10 packs would have 270 sticks of gum. So it's the ratio of 1 to 27, 2 to 254, 10 to 270, could be 20 to 540, like all of those ratios are equivalent. And if those ratios are equivalent, then that is a special paired number table that we call a ratio table. So you might find it interesting that a mathematician who works with illustrated math, Bill McCollum wrote a fine blog post the other day called ratio tables are not Elementary, which might lead you to believe that maybe we shouldn't use them in elementary school. However, we would actually agree with him that the way he describes a particular use of tables in grades three would go away and isn't particularly helpful for much. We would agree with him there. However, we conceive of ratio tables in the realm of multiplicative reasoning quite differently, Differently than just a list of say, single digit facts. So that could be a ratio table, I could have something that looks like 1 to 7, 2 to 14, 3 to 21, 4 to 28. And it can kind of be a list of what some people call multiplication tables, that would be like the table for sevens. That's a kind of limited use of ratio tables. But we offer an alternative view of how ratio tables can be used in developing and using multiplicative reasoning. I posit that ratio tables can be used first, as organizers of information to model, represent a scenario or situation. So we have a situation like this pack and sticks of gum that we could literally say, hey, let's sort of organize this information, we can kind of put it in this table, and it would represent that pack to those sticks and different members of packs to sticks, that always represented 1 pack to 27 sticks. In other words, anywhere in that table. Every one of those packs had 27 sticks in it, that makes it a ratio table and we can sort of organize information. That's kind of the first way. But secondly, we can also use ratio tables as tools to represent strategies for multiplication and division. So as we develop alternative strategies with students and students are using relationships and connections to multiply and divide, we as teachers can come in and represent their thinking using ratio tables. Now we can also represent their thinking using open arrays the area model, we can also use equations, but one of the tools that is so powerful to represent their strategies, those relationships are using for multiplication and division, is a ratio table. That is another way that we can represent the way they're thinking about solving multiplication and division problems. Third, ratio tables can be used as actual tools for solving multiplication and division problems, like they actually become the way that I begin to think multiplicatively to solve multiplication, division problems. They are actually tools to solve. So let me just say that again, one, they're kind of organizers of information, they sort of model the situation, they represent what's happening. And then we want to kind of move students, we want to help them transition to modeling their strategies, representing what they do, the relationships they use to solve problems. And then lastly, we want to transition students to actually use the ratio table as a tool to help them keep track of the relationships they're using and they actually use it to help them solve multiplication and division problems. Now, we then could go to middle school and continue to have that go and then use that ratio table as the proportional tool that it is to solve proportions using proportional reasoning. But today, we kind of want to talk about how ratio tables can be used in these multiplication and division situation, these multiplicative reasoning situations. Because as students begin to use ratio tables, they learn to scale in tandem, as I doubled the packs of gum, I doubled the sticks of gum that I have, as I multiply the packs of gum times 10, I multiply the number of sticks of gum times 10, that I'm sort of scaling in tandem. And that act of scaling in tandem, is leading toward this thing that they'll do when they are solving proportions with non unit rates.

Kim Montague:

Right. So okay, you just said a lot of really important thing. So funny, because -

Pam Harris:

Going going going,

Kim Montague:

- it's actually a conversation that I or, you know, via text I just had with one of my son's teachers and said, this is my favorite model for multiplicative relationships. And so let's break down some more of what you just said. First of all, unit and non unit rates. As a three five teacher that's something that we deal with all the time: unit rate multiplication problems.

Pam Harris:

Yes.

Kim Montague:

Let me give you an example. If we are talking about apples in a bag, we could have three apples in one bag. And the question could be how many apples in four bags. So that's

Pam Harris:

A typical multiplication question for third grade.

Kim Montague:

Sure. And so we could have students who skip count, like you just mentioned, skip count up per bag. And that would look like one bag on your ratio table, one bag for three apples. And they could say two bags would be six apples, three bags, four bags up the ratio table. But we could also have students use that as a tool to represent their thinking. And they could have one bag for three apples, or three apples for one bag. And then they could think about doubling and go doubled the number of bags would be double the number of apples so that on their ratio table, it would say two bags and six apples. And then if we're trying to figure out how many for four bags, then they could double again and say now I have four bags, and double the number of apples would be 12 apples.

Pam Harris:

In other words, they're not listing them all.

Kim Montague:

Right.

Pam Harris:

You could list them all. But then we want to encourage students to think in bigger chunks of numbers, I'm sure could go straight from the two bags to six apples to double that two and double the six and then a four for 12.

Kim Montague:

So also, they could have a problem like this, I've got, let's say 42 oranges and seven bags. How many oranges go in each bag, the answer to this is also a rate: oranges per bag. And those are third grade numbers for fourth and fifth, we'd upped the ante a little bit, maybe include some decimals. And that actually makes me think of a problem string that you do a lot with 3.2 ounces per bottle of eyedrops or whatever with a variety of numbers of bottles. We just don't as three, five teachers stress the rate part of it, we just solve the problems.

Pam Harris:

Yeah, but you absolutely are dealing with rate problems. Sometimes I hear the argument, we should not use ratio tables in third, fourth and fifth grade because students don't learn rates. And if you look at the Common Core State Standards or often a lot of the state standards don't have students really deal with rates until sixth grades. But we actually do a ton of problems in grades four and five. Dealing with rates. We just like you said, you don't stress the rate part of it, you sort of solve the problems and off the kids go.

Kim Montague:

Right. So what about non unit rates?

Pam Harris:

So non unit rates is sort of the land of the Middle School. That's where I've got problems where I start with something like Hey, I got a deal for you. You can get four slices of pizza for \$5. And then you ask students to solve for different amounts of money for different numbers of slices of pizza. Or you might say I'm in the bowl food aisle and I happened to put 2.4 pounds of flour in a bag and they charged me \$1.20 and then you could ask questions about different amounts of pounds of flour and different amounts of money. And how much flour could you get for different amounts of money? You could also ask questions like the ratio of the length of two sides of a triangle is three to five. So then can you find the missing side length of a similar triangle, if you know a corresponding side length, all of those are non union rate problems. And yes, we leave those non unit rate problems for the middle school, we really let them dig into rates and the idea of unit and non unit rates, we don't make a big deal of that in grades three-five. But in grades three, four, and five, we could use ratio tables to multiply and divide actually as tools for helping kids think through multiplication and division problems.

Kim Montague:

Yeah, and not just single digit facts. It's not just a table of multiplication facts. It's a powerful model that we use to actually do problems to compute. Want to give us an example.

Pam Harris:

Kim Montague:

Yeah.

Pam Harris:

And so then I might add those together, because now I've got 10 42s, thats 420. And I've got five being 210. And if I add those together to get 15 42 being 630. Yeah, cause I added those together. That's a way of thinking about 42 times 50. But if I may offer, I might have a student think about 15 times 42, and not be thinking about 42 at all, they might be thinking about 15.

Kim Montague:

Right.

Pam Harris:

So let's see, how could I think about 42 15s? Well, if I'm thinking about 15, that first entry in my ratio table could be one to 15. But I need 42 of them. So I think the first move I might make is 2 of them. You're like two of them, are you crazy? Well watch where I go with that. So if I got 2 15s my next entry is to 2 to 30, right? That's 2 15s. But I need 42 of them. So I need 40 of them. Ooh, I wonder if I could get four from there. So then I might say, What's 4 15s. So 4 15s is going to be 60. So so far, I've got one to 15 to 2 to 30, 4 to 60. Bam, now I can jump right to the 40. Because now I can scale times 10. Once I have that 4 to 60, now I can think of 40 to 600. Because I do that nice times 10 thing. Well, remember, now if you're doing this in the air, hang on tight, because I already had the 2 15s is 30. And now I have 40 15s is 600. I just add those together, and now I have 42 15s is that 630? Right? Okay, so that might be kind of hard for you to hang into the air. But part of our point is, it's so easy to read off of a ratio table, those relationships just jump out so clearly. The facility to do that is so easy for students to sort of keep track of the nice chunks that they're using, if they're kind of thinking in these multiplicative relationships. Alright, we mentioned earlier that we can use ratio tables for multiplication, and division. So Kim I'm gonna give you the division problem? You're welcome.

Kim Montague:

Yeah, all right.

Pam Harris:

So what about 192 divided by 6? Now, if you're listening to the podcast, y'all pause, solve that problem on your own a little bit. Well, how were you you thinking about 192 divided by 6? Because that will help you sort of think about the relationships as Kim kind of tells you what she's thinking about.

Kim Montague:

Sure.

Pam Harris:

Come on back. Come on back. All right, Kim. Okay. Well, I want a pencil, but I'm not sitting near one. So I'm going to try to picture in my head what I wish I had a piece of pencil and paper, right? Because I'm drawing a ratio table in my mind. Okay, so you said 192 divided by 6. So immediately, I'm thinking that that means to me right now, how many sixes are there in 192. And so I'm thinking my ratio tables gonna look like one to six. So I'm gonna start with what am I talking about? I'm talking about sixes, so one to six. And I'm thinking that I'm going to go 10 sixes, is 60. And I'm going to 192. So I've got 1 6, 10 sixes is 60. So 20 6s is 120. Oh, actually scratch that out, can I take that back? I'm going to go from 10 sixes is 60. And then I'm going to scale up times three. And I'm going to say 30 6s is 180. And at this point, I think I'm going to ask myself, how far away am I from my goal. So I'm at 180. And I know that I'm only 12 away from 192. And so I'm going to write down two sixes is 12. And so then on my ratio table in my mind, but on my ratio table, I would have 1 to 6. And then 10 to 60. 30 6s is 180. 2 6s is 12. And I'm just going to use the pieces that actually need, which is 30 6s is 180. And two more sixes is 12. And that makes 32 sixes to make 192. So the last entry in your table would be 32 to 192. Nice!

Kim Montague:

You mentioned this, but I think it's worth noting that we can put entries in our ratio table, when it's a tool for us that we don't actually use at the very end to piece together for our final product or final quotient, right?

Pam Harris:

Yeah, that's a really good point. So there's values that we might use, for example, in my ratio table, when I was thinking about 42 15s, I found 2 15s was 30. So I had the ratio one to 15, 2 to 30, 4 to 60, 40 to 600. And then I only used the 40 and the two I didn't use the 4, the 4 was just sort of in there because it helped me to get to the 40.

Kim Montague:

Right.

Pam Harris: