Why are ratio tables so amazing? In this episode Pam and Kim describe ratio tables and what makes it on the favorite mathematical models short list. Then they show it off with a Problem String.
Talking Points:
Online workshop registration opens in January!
Why are ratio tables so amazing? In this episode Pam and Kim describe ratio tables and what makes it on the favorite mathematical models short list. Then they show it off with a Problem String.
Talking Points:
Online workshop registration opens in January!
Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
Kim Montague:And I'm Kim.
Pam Harris:And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But y'all, 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. So, Kim.
Kim Montague:Yeah. Yeah.
Pam Harris:Here is a friendly reminder that we have online workshops that launch only three times a year, and one of those All things, get ready. times is coming up. So teachers, leaders, math learning friends, gets your funding ready, because registration opens in January for workshops that are launched at the end of the month. If you are planning to take a workshop, this coming January, this is your time to get ready. Get funding, get your permission.
Kim Montague:Super exciting.
Pam Harris:Wahoo!
Kim Montague:Yeah. So in previous episodes, we did a short series on the open number line. And that was wildly successful. So we thought that we would spend some time diving deep into one of our other favorite models, the ratio
Pam Harris:The ratio table, what in the world is a ratio table. table? Ratios, like I remember vaguely hearing about ratios in middle school, like, why are you? Oh, so if I may, why is the ratio table one of literally shortlist one of our favorite mathematical models? What is so amazing about the ratio table? So if I may, I'm gonna give kind of a little bit of my informal definition of a ratio table first. A ratio table is taking the relationships that naturally mathy people use to solve many problems, multiplication, division, proportional reasoning problems. It's taking the relationships that somehow they just kind of have a natural aptitude that they kind of use those relationships to solve problems. A ratio table is a way to make that thinking visible, it's a way to pull it and make it visible so that we can then point at it and discuss and talk. And by using it, develop that kind of reasoning in the learners head at the same time.
Kim Montague:Yeah.
Pam Harris:So I'm thinking there is this group of people out there that have this sort of natural, there's a word, I'm looking for affinity, that's the word I want, this natural affinity for seeing patterns, and then using those patterns. And the more they do the kind of more relationships and connections they build in their head. But kind of naturally, just like some kids, we've talked about this before, some kids kind of naturally pick up a basketball and start to dribble between their legs, and they can spin the ball on their finger. And not everybody has that talent. But you can learn to do that. You can watch it and you can make sense of it. And you can try your best to do that, we all have kind of natural talents that we lean towards. The ratio table is a brilliant way of taking what some people have kind of naturally done and opening it up to the rest of us.
Kim Montague:Yeah.
Pam Harris:The rest of us get a chance to reason the same way.Yeah, go ahead.
Kim Montague:Yeah. Well, I was gonna say that I think you might get into this, but I think that some of us who did kind of those mathy things and thought about patterns and relationships, you're saying it's opening it up to others so that they can make sense. I think it also made things more clear for me. So when you introduced that model to me, I was able to do even more because I had a model.
Pam Harris:Yeah, that makes a lot of sense. And for all sorts of reasons. Yeah, you had a model, we could talk about it. Like I said earlier, it was point-at-able. We can discuss it together. And the more you use it, the more you develop that reasoning and thinking.
Kim Montague:Yep.
Pam Harris:And we all just get better at it. So super cool. All right. So brass tacks, what is the ratio table? It's a paired number table. So pairs of numbers, but it's this very specific paired number table. It's not just any table where you have, you know, x's and y's or values in one column and the other column. It's very specific in that the ratios of every x to every y are equivalent, those ratios are equivalent. So if I look at any of the paired numbers in the table, the ratio of the x to the y or the one on the left hand side to the one on the right hand side is equivalent to every other ratio in the table of the left hand side to the right hand side. And so it's a very specific kind. Now that might be like, "Pam, what does that even mean?" Well, then we would definitely want to do it in context, so that it makes a whole lot more sense. And we'll do a lot of that today. Just a couple of other brass tacks things. Ratio tables can be both horizontal and vertical. When I just said, the values of the left side of the values and the right side, I was thinking about a vertical ratio table. If I was thinking about a horizontal ratio table, then I would think about the ratio of the numbers of the top row to the numbers in the bottom row. Again, it's all about those ratios being equivalent. We would suggest that as you work with ratio tables, you introduce ratio tables in context. So you have context that makes sense. And then stay in context, as much as you can, until students are really starting to get a feel for it. And then decontextualize, and let kids make the mistakes they're going to make when there's not context to lean on, and then bring back contexts to help them reason about the things that they're going to do. And that will come up a little bit later in today's episode about how to do that. A couple of other things to consider, as you work with ratio tables, specifically, as we watched people, teachers use ratio tables. Kim, I'm just aware, as I'm talking about things about ratio tables, I almost wonder if this list of things I was gonna talk about would be better after we did some work with it. I mean, I guess I could keep talking about it. We can mention it again.
Kim Montague:Yeah.
Pam Harris:Okay. Let me just mention briefly and not discuss too much. And then we'll bring it back up. One of the things that we see novice ratio table teachers do is they won't ever erase. So we'll be working in the ratio table, and they'll model the thinking, and they'll represent it making it visible. But then they leave it all up there. And there's all these marks and different things, and it just gets really muddy and crazy. And so we definitely, there are strategic times to erase. There's a brilliant inverse relationship that shows up in a ratio table between multiplication and division. So that's a really nice part about ratio tables, we definitely want students to still see multiplication and division symbols, we want them to see multiplication and division equations. We even want them to see that housetop, if we could call it that, the kind of way that long division is written. But not then have that be, "Oh, now thou shalt do the steps of a long division algorithm." We just, it's just a way of representing the relationship. So this divided into that, or it's the number under the housetop is being divided by the number sort of outside the housetop. We need students to be able to read that. It's just that with ratio tables, when they see those multiplication and division symbols or notations it means a different kind of thinking. And that kind of thinking can be done on a ratio table. Yeah. Is there anything you want to add to my little description of a ratio table?
Kim Montague:Hmm, no, I think it's pretty solid. All right.
Pam Harris:So let's get down, let's do some stuff with a ratio table. So we're gonna invite you to grab a pencil and paper. Oh, wait a pen and a paper.
Kim Montague:No pencil.
Pam Harris:I can't believe I said pencil.
Kim Montague:The right thing to grab?
Pam Harris:Oh, man. Okay, I have to just tell a quick funny story. I was recently in South Africa, and we were talking about, when you just said the right kind of thinking, they drive on the other side of the road. And I would say to my good friend who was driving, "I'm like you driving on the wrong side of the road?" And she would say, "No, I'm driving on the other side of the road, not the wrong side of the road. Who's driving on the wrong?" Anyway, so it was the right side, I mean, right is a funny word, right the correct side. Because we do drive on the right side of the road in two ways, the correct and the right hand side of the road, which she would take umbrage with. No it's the left, anyway, it made me slightly dizzy to drive on the left hand side of the road, but all as well. Okay. So the right way of thinking, here we go. The right way of writing with a pencil. Yeah. All right, Kim, I'm going to throw out a few questions. And we're going to talk about how a ratio table can be helpful in reasoning through those questions.
Kim Montague:Okay.
Pam Harris:So I'm going to start with one box of Smarties. Do you know what a Smarty is? Yes. It's a fabulous candy. Okay. It's not those, so the things in the U.S. are, they're like chalky. They're kind of like these funny disc shaped things. They're like, yeah, so I don't mean those. I mean, they're, they're kind of like M&M substitutes. And you could get them in Canada, and you can get them in South Africa. And I was just in both recently, in both of those countries. Where they were like, "Hey, you have to try Smarties." And I was like, "Or M&Ms." Anyway, so I'm just, we're just for our international listeners today, it's going to be Smarties. And you can picture whatever you want Smarties to be. They can be those chalky things if you'd like them or whatever, but we're going to picture a container that has a bunch of those Smarties in it. And I'm going to say that one of those boxes, one of those containers, has 72 Smarties in it. Yum.
Kim Montague:Okay.
Pam Harris:That's a lot of Smarties. So, on a ratio table, Kim, I'm just curious, what did you just write down?
Kim Montague:I drew a horizontal line. And I wrote one to 72. So one on top and 72 on the bottom of the line.
Pam Harris:And you do usually use horizontal ratio tables.
Kim Montague:I do. I just get more board space that way.
Pam Harris:And I just wrote a vertical ratio table, where in the left hand column I wrote "box" then one. And on the right hand column I wrote "Smarties" and 72. Alright, cool. So what if we had two boxes. How many Smarties?
Kim Montague:That would be 144. Smarties.
Pam Harris:Because if you...?
Kim Montague:Double the number of boxes, then you double the number of Smarties.
Pam Harris:And that makes sense. Cool. What if you had four boxes of Smarties?
Kim Montague:288.
Pam Harris:288 boxes? Smarties?
Kim Montague:Smarties, sorry. So I did twice the number of boxes from two to four, would be twice as many boxes, which would be twice as many Smarties. So that's 288.
Pam Harris:Because you doubled the 144. So Kim, weren't you supposed to though go back to the one to 272? Weren't you supposed to multiply the 72 by four? Like did you, were you sort of cheating there?
Kim Montague:I use a nicer relationship for sure. Because I don't want to do 72 times four. I'd rather double the two boxes, which was 144 Smarties.
Pam Harris:Yeah, why not? Right? And especially since you already had it in front of you. It'd be one thing if I just said, "Hey, one box for 72, find four." Well, then you might think about four times 72. But since we already had the two, might as well use it, cool. All right, how about...
Kim Montague:On my ratio table, you know, since we're describing kind of what my mind looked like, I actually wrote an arrow with the times two, from two to four and an arrow with a times two from 144 to 288.
Pam Harris:So did I.
Kim Montague:Okay, cool. Very good.
Pam Harris:So we're sort of keeping track, like you said, if you double the number of boxes, you double the number of Smarties. And that makes sense. We can reason about what's legal to do in a ratio table using that context. All right, what about eight boxes of Smarties?
Kim Montague:That would be 576
Pam Harris:How do you know?
Kim Montague:So if four boxes of Smarties was 288 Smarties, then again I doubled the number of boxes and doubled the number of Smarties.
Pam Harris:How did you double the 288? I'm super curious.
Kim Montague:I knew you were gonna ask me. I um, I actually doubled from left to right. Which I know is weird. And I feel like we might have talked about this at some point, that I probably theoretically could have done a nicer job and been less double checking myself if I had doubled 300, and then backed up 24. For whatever reason I was in the mode of going left to right. So I doubled the 200, double the 80, double the eight.
Pam Harris:So did I actually. I did yeah. And then I thought I could have doubled,
Kim Montague:Did you really? let me slow down your double 300. So if you're going to 720 Smarties. double 288 instead double too much, double 300. But you doubled 12 extra twice. Because 88 is, play I Have, You Need a little bit. If I have 588 you need 12. So that 12 to get to 100 to get that 300. Cool. All right. So let's see, we've got two boxes, four boxes, eight boxes. Often I will ask students right now, "What do you think will come next?" And they will often say, "Oh 2, 4, 8, 16." And I'll go, "Nope. Good try. Ten." So ten boxes of Smarties, how many Smarties?
Pam Harris:Whoa, yeah, that was kind of like, did you use the eight boxes to help you with that?
Kim Montague:No, I didn't. I actually went back to the one box which I've kind of ignored this whole time.
Pam Harris:Before you talk about that. Could we talk about a possible eight box strategy?
Kim Montague:So eight boxes was 576 Smarties, and two boxes is 144 Smarties. So I could add that eight boxes, and the two boxes and those corresponding Smarties. Which would be 720.
Pam Harris:Cool. You can add the 144, Smarties and 576. Yeah, nice. But instead you decided, I thought we decided not to go back to the one. You know earlier, we said don't go back to the one, you just double.
Kim Montague:Yeah.
Pam Harris:Just double from where you are. But so like, why are you switching up here?
Kim Montague:I think because I know a relationship between one box and 10 boxes.
Pam Harris:Which is?
Kim Montague:Times 10. And you know what I'm actually thinking about right now is I'm recording what I'm thinking about on a piece of paper. And I have that 'times two' and 'times two' on there. And I, you talked about earlier about erasing at pivotal times. And I'm trying to decide if for the sake of niceness on my paper if I want to erase those' times twos'. Because if I were in the classroom, this would be a moment where I would erase the two previous 'times twos' so that I could make a nice, going from the one to the 10, make it a nice real clear 'times 10' relationship.
Pam Harris:Because when things are clear, kids learn better? Why would you want it?
Kim Montague:Because I want to emphasize the relationship that's being used. I think it's gonna get a little muddy. I don't know how many questions you're gonna ask. And we've had the conversation of 'times two' already, so I'm ready to move forward to the new relationship.
Pam Harris:And is it also true that what's important are the number of boxes and number Smarties at this point to mess with the finding the number of Smarties in 10 boxes, the doubling that you did to get up there is no longer relevant, right? So we can get rid of the irrelevant stuff and kind of focus on times 10. And 72 times 10 is indeed 720. Nice. Fine relationship to us cool. How about nine boxes of Smarties?
Kim Montague:So nine boxes of Smarties, I could do a couple of ways. The first way I'm thinking about is I could think about eight boxes plus one box. Or the one I really want to use is 10 boxes minus a box to get to nine boxes. So that would be 720 Smarties minus 72 Smarties. And I think that is 648 Smarties.
Pam Harris:And how do you know?
Kim Montague:I actually am thinking I kind of want to check myself to be honest with you. So down below my ratio table. I wrote 720. And I want to remove 72. So I'm removing 20 to get to 700. And then I'm removing 52.
Pam Harris:The remaining 52.
Kim Montague:To get to 648.
Pam Harris:Because if I have 52. You need 48 to make 100.
Kim Montague:Yep, Yep.
Pam Harris:I agree, 648. And I actually subtracted the same way. I started to wonder about shifting it a little bit. And I didn't get a chance because you were talking. But like 72 is 28 from 100. So I could think about 720, subtract 72 as something subtract 100. If I added 28 to the 72, I'd have to add 28 to the 720. And that would be 748. So 748 minus 100 is also 648. And I don't know if anybody finds it interesting. I actually had to say that out loud in order to do it. So that was less me explaining what I was doing. And more me saying it out loud so that I could actually do the computation.
Kim Montague:Yeah.
Pam Harris:In other words, I can't talk and do it at the same time, or I can't talk about something else and do it at the same time. I have to, so anyway, I don't know if you care. Okay, so you could have done eight boxes plus the one. But only really, if we already had the eight boxes. Right?
Kim Montague:Right, right.
Pam Harris:If you're walking down the street, and I said, "Hey, Kim, one box is 72. Smarties, how many are nine?" Which strategy? The eight plus the one or the 10 minus the one do you think you would gravitate toward? You know, if you didn't have all this work that we've done before?
Kim Montague:Oh, for sure. The 10 boxes minus one box?
Pam Harris:Because?
Kim Montague:Because finding 10 times something is pretty doable.
Pam Harris:Yeah, that makes sense.
Kim Montague:I would have to walk through doubling to get to the eight.
Pam Harris:Right? Right, because you don't just know eight times 72. But ten times 72, BAM, there it is. So you might as well use it. Cool. All right. How about five boxes? Five boxes, please? Pretty please. Five boxes.
Kim Montague:Five boxes is going to be 360 Smarties.
Pam Harris:How do you know?
Kim Montague:I knew 10 boxes was 720. So half the number of boxes will be half the number of Smarties.
Pam Harris:How did you do half of 720? I'm curious.
Kim Montague:Um, hang on, I'm writing on my ratio table. I did half of 700 is 350. And half of 20 is 10. So 350 and 10 is 360.
Pam Harris:Nice. Do you wonder what I was thinking?
Kim Montague:Yeah.
Pam Harris:Half of 72 is 36.
Kim Montague:Yeah.
Pam Harris:I didn't used to know that. But I've played around with doubling and halving a lot, double. I shouldn't say double, like doubling and then halving, not the strategy Doubling and Halving, but you know, doubling a lot and then halving a lot. And so I just sorted that one, I know now that half of 72 is 36.
Kim Montague:Nice.
Pam Harris:Cool. Okay. So let's see, that was five boxes is a Smarties, you could have also used four boxes and one box if you wanted to. And we often will ask students to share that one. But then almost kind of, so we have this sort of shock value of, "Oh,but you just do half a ten." The half because it's so nice. Because five is half of 10. Yeah. How about 15 boxes of Smarties? What you got for that?
Kim Montague:1080.
Pam Harris:How do you know?
Kim Montague:Ten boxes was 725 and five boxes is 360. So that's 1080.
Pam Harris:Cool. So just like we could say five times anything is half of the 10 times that thing. How am I you verbalize your 15 times what you just did?
Kim Montague:Say that one more time.
Pam Harris:If we could generalize to say, "Oh, I could find five times anything by finding 10 times that thing and dividing it in half to get five times that thing." What if I said, "But I now I want 15 times that thing?"
Kim Montague:You could think about 10 times that thing. And half of the 10 times that thing, so another five times a thing and put those together.
Pam Harris:Add them together to get 15. Cool. And you could do that with 15 times anything.
Kim Montague:Yep.
Pam Harris:Seems kind of slick.
Kim Montague:Yeah.
Pam Harris:All right. I really, really like Smarties. So 100 boxes of Smarties, how many Smarties?
Kim Montague:One hundred. I'm actually going to go with 7200. And I know you're going to ask me how I know. I decided to go back to the 10 boxes and scale up by 10. Again. So 10 times 10 is 100. So 720 times 10 is 7200.
Pam Harris:Nice. Nice.
Kim Montague:Yeah.
Pam Harris:And I've heard lots of people say at that point that they're thinking about 100 seventy-twos and that you could, that's equivalent to 72 one hundreds. Which is this, an equivalent name for that 7200 is 72 hundred.
Kim Montague:Yeah.
Pam Harris:That's another way of thinking about that. Cool.
Kim Montague:Yep.
Pam Harris:Okay, that's ridiculous. Kim, we cannot have that many boxes of Smarties. We're only going to have 99. So how many boxes of Smarties would have nine. or how many Smarties would be in 99 boxes?
Kim Montague:7128.
Pam Harris:How do you know?
Kim Montague:One hundred boxes, you asked me about, was 7200. But they need one less box of Smarties. And that's 72 less. So 7200 minus 72 is 7128.
Pam Harris:Okay, so on your paper did you write down 7200. And underneath that 72, draw the line and start to borrow, regroup, all acrossy outies and...?
Kim Montague:Thankfully no.
Pam Harris:But that I mean, that's subtraction across zeros is often a problem. Teachers are like, "Ooo, Help us, like, how do we do that?" So how were you thinking about 7200 minus 72?
Kim Montague:So I'm in 7200, or 7200, you have 7100, and then another 100. So I kind of set aside in my head, the 7100 part.
Pam Harris:Okay.
Kim Montague:And I played a little I have you need with 72 and the extra 100. So I thought about given that extra 100 that I had broken apart from the 7200. What would I Have, You Need the first 72? And I know that that would be 28. So then I put the 28 back with the original 7100.
Pam Harris:Nice. And I just drew on my paper to make that visible, I drew a number line, and I put 7100 on the left. And I put 7200 on the right. And I kind of said, then I made a big jump between them of 100. Right? So 7100 plus 100 is 7200. And then there's that 100 you were talking about. That 100 sitting there and if we need to subtract 72, so I made a jump back of 72. I know I Have, You Need is 28. So then I made a jump forward from the 7100, a jump forward of 28. And then I can just ask myself, "Where is that? Where do those where do those meet? Oh, that's 7128." Bam!
Kim Montague:Nice.
Pam Harris:Yeah. Nice. Okay, cool. We're almost there. How about 50? Fifty boxes of Smarties, how many Smarties?
Kim Montague:I went back to the five boxes that you had asked about earlier. And I did five boxes times 10 would be 50 boxes and 360 Smarties times 10 would be 3600 Smarties. So I got 3600.
Pam Harris:So you could think about 50, based on the five. So if you needed 50 times anything you could say,"Well, I could find five of them, oh, by getting 10 of them." So 10 of them, half of that's five of them and then scale that up like you said, ten to get 50. You could also have gone from the 100. Right? We had the 100 up there. We could have cut that in half.
Kim Montague:Yeah.
Pam Harris:Cool. Nicely done. How about 49 boxes of Smarties?
Kim Montague:Well since I just did the 50 and that was 3600. Then I could do kind of like I did the 99 from the 100. I could do 49 from the 50 and just remove 72, so that would be-
Pam Harris:From?
Kim Montague:From the 3600. So remove 72 Smarties from the 3600 Smarties because it's one box less and get 3528.
Pam Harris:And you just solved a whole lot of multiplication problems, thinking and reasoning using what you know, using multiplicative relationships. And I would suggest as we do this kind of work with students, they build those relationships more and more in their minds, until it becomes sort of second nature. It's an intuitive thing that they, it's almost like they say, "Can I use what I know?" And you're like, "Well, what's up there? What do you already have? What do you?" And as we're doing a string like this, there's a bunch of things that we've already solved that they can use. So what happens when they see a problem in the wild? Like, if we literally were to say, "Hey, you've got 72, Smarties in a box and you're looking for 49 boxes?" Could they say to themselves, "Well, let's see 49. Oh, that's almost 50. Fifty that's half of 100." Or like you said, "Fifty, I can scale up from five. Ooo, if I can find 10, I can cut that 10 in half to get five, I can scale that up from five." Or even a middle school strategy could be, "Could I find half a box. Now I've got point five." And then scale that point five, up to five and then to 50, or just from point five to 50 and go that direction, and then take off to get the 49. Lots and lots of multiplicative relationships that we can do using a ratio table to help students reason the way mathematicians are reasoning about multiplication. Super cool. Anything else you want to say about ratio tables?
Kim Montague:Um, no, I don't think so. I'm looking at mine, Pam. And it's, there's a lot happening here. There's a lot happening. I think we need to have some more conversation about critical moments to erase, as we keep talking about ratio tables. Yeah.
Pam Harris:Yeah, we can also talk about what, which questions I would ask when, what strategies I would share first, why this order of problems, lots of things that are kind of embedded. We are so excited about ratio tables that we're going to continue to talk about ratio tables. So tune in next week for more on our fantastically favorite model, the ratio table. And again, a friendly reminder that we have online workshops that only launch three times a year and one of those times is coming up. So we're mentioning it today. So you have time to arrange for funding, get you're planning in order so that when registration opens January, you are ready. All right. 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 Math is Figure-Out-Able.com. Let's keep spreading the word that Math is Figure-Out-Able.