Pam Harris  00:02

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

Kim Montague  00:09

And I'm Kim.

Pam Harris  00:10

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  00:34

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

Pam Harris  00:48

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 54, 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 McCallum 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 numbers 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 situations, 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  06:41

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

Pam Harris  06:45

Going going going,

Kim Montague  06:45

- 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  07:09

Yes. 

Kim Montague  07:10

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? 

Pam Harris  07:20

So that's a typical multiplication question you run into in third grade.

Kim Montague  07:24

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 double 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  08:12

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

Kim Montague  08:13

Right.

Pam Harris  08:14

You could list them all. But then we want to encourage students to think in bigger chunks of numbers. 

Kim Montague  08:18

Sure. 

Pam Harris  08:18

So they could go straight from the two bags to six apples to double that two and double the six and then they have four for 12.

Kim Montague  08:26

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 3-5 teachers stress the rate part of it, we just solve the problems.

Pam Harris  09:02

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 three, four and five grades 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  09:28

Right. So what about non-unit rates?

Pam Harris  09:32

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-unit 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  10:39

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  10:49

Absolutely. So sure, I could have the multiples of seven. That'd be one way to do that. But let's conceive of one pedagogical use of ratio tables. 

Kim Montague  10:58

Okay? 

Pam Harris  10:59

What if I were to give students a lot of different sort of starting numbers and ask them to find, say, one through 11 of that number. So I could have, I could say, "Hey, you've got sevens, go and create one through 11 sevens on that table, go. And you've got eights, and you've got nines." I might do sevens, eights, and nines. Because those are facts that are harder for students. And so I might do those for some of the students that are kind of tripped up with those facts. But for other students in the class, I might say, "You do 25 and you do 36. And you do 27s." Like I might give them some crazy numbers. And again, what they're doing is they're creating tables of one 27, two 27s, three 27s, four.... So not not a very interesting table until we line them up against each other. We got this idea from Investigations in Data, Number and Space, where the kids made these different tables using different sort of rates. And they created like I said, one through 11, one through 12 of them. And then we begin to compare, we started asking questions like: Hey, what do you guys notice? What are all the twos? Oh they're gonna double? What are all the 10s? Wow, there's this zero thing showing everywhere. And it gave us a chance to kind of talk about the times 10 thing once they start seeing what happens times 10 all over the place. And you might say, "Pam, what if the kids like goof and they don't get, you know, like 27 to 10, that ten of thos 27s is 270?" Well, when most of the tables in the room have the same thing, then we could go, "Hummm, maybe we need to check out this table like should Oh, sure enough, it should have been." Like, we can kind of like find that pattern. And we kind of use it. We can also ask questions like: Can I go from the two and the four? Like, is that always double? That's interesting. Hey, how's the five related to the 10? Oh, it's just half every time that makes sense. There's lots of ways we can find patterns just looking at those tables. That's a fine way to do it. But we can also use ratio tables, guys this is our favorite way, Kim right? We can use it as a tool to solve problems. So picture a problem, a specific example. 42 times 15. Y'all if you're driving right now, you might want to just turn it off for later or, or just do the best you can to picture stuff. And then maybe later, maybe write some stuff down on paper, just kind of keep track of what's going on. But if I was going to think about a problem, like 42 times 15, could I conceive of that as fifteen 42s? Fifteen 42s, right? So if I was going to think about as fifteen 42s, my ratio table might look like this: I might say, Well, I'm going to start with one 42. So one to 42. And I'm going to think about, let's see, I need 15 of them. So I'm gonna think about 10 of them. That gets me pretty far right, that's an easy chunk to think about. So then my next entry in my table might be 10 to 420. Yeah, because I'm thinking about ten 42s. So then I've got ten 42s is 420, then I might think about, well, I need 15 of them. So I still need five of them. Ooh, five is half a 10. So the next entry in my table might be five. Now how am I going to find five 42s? Well, if I've already got ten 42 being 420, then five 42s would be half that. Is that 210? Kim, I'm doing this in the air. Is that right?

Kim Montague  14:08

Yeah.

Pam Harris  14:09

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

Kim Montague  14:36

Right.

Pam Harris  14:37

So let's see, how could I think about forty-two 15s? Well, if I'm thinking about 15s, 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 two 15s my next entry is 2 to 30, right? That's two 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 four 15s. So four 15s is going to be 60. 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'm doing that nice times 10 thing. Well, remember, now if you're doing this in the air, hang on tight, because I already had the two 15s is 30. And now I have forty 15s is 600. I just add those together, and now I have forty-two 15s. Is that 630? 

Kim Montague  15:38

Right. 

Pam Harris  15:38

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  16:09

Thanks for that.

Pam Harris  16:10

Yeah, alright. 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 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. Sure.  Come on back. Alright, Kim: 192 divided by 6.

Kim Montague  16:32

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 means to me right now, how many 6s are there in 192. And so I'm thinking my ratio table's gonna look like one to six. So I'm gonna start with what am I talking about? I'm talking about 6s, so one to six. And I'm thinking that I'm going to go ten 6s, is 60. And I'm going to 192. So I've got one 6, ten 6s is  60. So twenty 6s is 120. Oh, actually scratch that out, can I take that back? I'm going to go from ten 6s is 60. And then I'm going to scale up times three. And I'm going to say thirty 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 6s 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. Thirty 6s is 180. Two 6s is 12. And I'm just going to use the pieces that I actually need, which is thirty 6s is 180. And two more 6s is 12. And that makes thirty-two 6s to make 192.

Pam Harris  18:22

So the last entry in your table would be 32 to 192. Nice!

Kim Montague  18:29

I don't think 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  18:46

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 forty-two 15s, I found two 15s was 30. So I had the ratio 1 to 15, 2 to 30, 4 to 60, 40 to 600. And then I only used the 40 and the 2. 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  19:10

Right.

Pam Harris  19:10

So yeah, ratio tables are so handy, because I can put a lot of different things in the ratio table to help me get where I need to go. And then I can just use the ones that I actually need to finish solving the problem. Very cool. Hey, let me take that 192 divided by 6 and do one other really cool thing with a ratio table. Because we can conceive of division two ways. And you just conceived a division quotatively, because you thought about how many 6s are in 192. But we can also think partitively about that, and we can think about 192 divided by six as 192 to six or the division like if I were to write it as a fraction. So 192 sort of with that division line kind of over the six, that's just a kind of picture in your minds. It's not a mathematical relationship, but 192 that the fraction 192 sixths. Then I can actually use a ratio table to simplify that ratio. It's the ratio of 192 to six, and I can simplify it. I might think about, well, I can cut both of those in half. So if I cut 192 in half, that's 96. And if I cut six in half, that's three. So now I have two equivalent ratios, I've got 192 to 6, is equivalent to 96 to 3. And then I can think about what is 96 divided by three? Well, 90 divided by three is 30. And six divided by three is two, so that's 32. And we get the same, we get the same 100, excuse me, 192 divided by six is 32. Two different ways that we can use a ratio table. Two different ways we can conceive of that division problem. But both using this powerful tool of a ratio table to help us kind of think through the relationships that we're using. So y'all give me a multiplication or division problem and I'm gonna use a ratio table as the tool to solve it. So to be really clear, I don't know that we've said that yet on the podcast today, neither Kim or I use kind of that traditional multiplication where you line them up, the magic zero, and all that stuff. Nor do we use the long division algorithm where you do the house top and the I don't even know all those things, kids, brother, sister, mother, father, and then does McDonald's sell cheeseburger, wow, always crazy, something about monkey's smelling bad. There's all these crazy rhymes that teachers have made up - rhymes or mnemonics that teachers have made up, stories to help kids remember the steps of a long division algorithm. Because the long division algorithm is so hard to understand that nobody even tries to understand it. They just try to memorize the steps by using all these crazy things to memorize it. Or could we help kids think and use relationships by using a ratio table as that tool to help them keep track of their thinking? I know there's the Lucky Seven out there that people use to represent partial quotients. We would recommend instead, ratio tables. Ratio tables don't force kids into crazy subtraction, where they are sort of nudged into using the subtraction algorithm. Instead, they can still be thinking and reasoning, using what they know using the relationships they understand to solve those problems. Alright, so if some of what we did today was a little hard to picture, totally get it, we've got a download for you that we are going to - we're gonna listen to this. And we're gonna keep track of what we did. And we'll put it on paper. And then you could re-listen to this podcast and kind of see what we meant, as we were kind of using - what is the ratio maybe even look like? Yeah, I'll give you some examples on that. So check out the show notes, download that. And it will totally give you that link where you can look at what ratio tables actually looked like, because we know we were doing that in the air.

Kim Montague  22:40

Perfect. If you want to know more about this amazing tool that we love so much. We would love to invite you to check out the Facebook Lives and webinar that Pam is gonna be hosting later this month. And if you're listening to this after those, you can check out her YouTube channel, FiguringMath and find those lives and webinar.

Pam Harris  22:59

Yep. We're gonna be doing four Facebook Lives and a webinar that we'll repeat. You're gonna want to register for that webinar. Check out the Facebook Lives. It will be a lot of fun, and we're gonna really get down and dirty and lots of examples for the super cool tool we call a ratio table. Remember to join us on MathStratChat on Facebook, Twitter and Instagram on Wednesday evenings, where we explore problems with the world and you will see ratio tables solve multiplication, division and Proportional Reasoning problems. So if you're interested to learn more math and you want to help your students to develop as mathematicians, then don't miss the Math is Figure-Out-Able Podcast because Math is Figure-Out-Able.