 Okay. So great. It's nice to have you all with us today on this election day. Thanks for joining us. We have 60 folks registered for the webinar. So it looks like this is a topic that lots of people are interested in. I'm doing this webinar at the request of folks who have been in particular for 67th grade with a big topic that they wanted to hear about and to learn more about in terms of different kinds of tools to help students understand ratio and proportion and build understanding that would lead us to this slope. So that's why we're here today. Hopefully Alan Maloney will join us soon. He is at the Friday Institute at NC State University and he, along with his team members and Jared Comfrey, have developed materials that I'm going to highlight today and he'll be around also. Thank goodness to answer lots of questions if you have questions. So we'll try to leave some time at the end for questions and we'll go ahead and get started in. So I'm going to get out of this just real quick. So the goals for this session are to work through some mathematics. We're going to look at some examples, like I said, from the turn on ccmath.net site and from a couple of other resources. And we'll actually work through the math and think about various solution methods and representations for the problem. And we'd like to also share some web resources related to the topics. So it's a small focus in terms of, I don't want to do a ton of examples, but I want to do a couple of examples where we focus on various representations and think together what those representations might yield in terms of benefits for understanding. So I'm going to go ahead and get started with an example from the turn on ccmath.net website and we'll look at that website more in depth later. And like I said, we'll have Alan here with us to answer questions about those materials. So this is an adaptive problem that I just grabbed from their materials. It says, suppose my favorite lemonade recipe calls for eight lemons and 12 cups of sugar water. But what we want to know is how much sugar water will I need if I only have two lemons and I want to make my lemonade that tastes like my favorite recipe. So we're going to take just a minute to write down some solutions and think about the different kinds of representations. And then we'll list those representations and talk a little bit about the specific representations that you guys will use. So as you think about those, if you want to go ahead and chat some of those ideas in the chat box, Carol is monitoring the chat box for me. And think about how your students would try to answer this question. And then Carol is going to let me know what some of those are, because actually I can look at the chat window too. So feel free to chime in. I'm encouraging everybody to use the chat feature since we have to mute your microphone so that we don't get the feedback. And then later on I'll pass control. As questions come in, I'll pass control over to Alan so that he can answer some. He can answer some questions. So we have somebody here. Students just divide by four. Just divide by four. Pass the recipe then half it again. Use pictures of lemons and cups to represent the problems in the grade. Basic solutions can be found by drawing a model and circling roots. Lots of different ways. Look at this. This is awesome. 8 by 12 equals 2 over x. So somebody jumps right to the algebraic representation. And if you want to send your answers, you can send them to me privately, but you can also, under the send to box, there's the pull down menu. And if instead of choosing my name or the presenter, you can choose everyone and then that way everybody can see your responses. So I think the reason I'm not seeing those responses is because you're sending them to Carol. So the specific responses we've gotten so far are specific to this problem. Could we think about kind of representations, and some of you have mentioned these two, kind of bigger picture representations like the idea of using pictures of lemons and cups to represent the problem. A t-chart. Leah, you said there's a t-chart. I'm not familiar with a t-chart. Are you thinking about actually a table where you fill in the values for each? Okay, great. So we'll think about a table of those particular values. Any other suggestions? Okay, so let's look at some of these suggestions that we've gotten. And on the next slide, what I have is I've actually made a list. I've made a list of some of the things that you guys decided, you guys suggested. And this is what I'm going to think about just looking at the relationship from a table. I think that's what we meant by a t-chart. So if we need eight lemons, we'll need 12 cups of sugar water. That was the original idea. Somebody said have it, so we'd have four lemons to six cups of sugar water and have it again until we get our answer of three cups of sugar water. So this is what we might call, somebody said call it a t-chart or just a ratio table where you're just listing these values. And it depends on what values you think about as you move through the table, whether or not you might get your answer. So for example, if I said instead, I don't really just have two lemons, I have three lemons and I want to use all three, then this chart that I have here won't answer the question for me. If you look at this, we can think about the pattern across the columns and that's called correspondent, so the number of lemons to the cups of sugar water or across between the rows. So that's called co-variation as we think about here we doubled it from 8 to 16, so we double this 12 to 24. So I want to include something in here called a bridging standard and I'm not going to talk more specifically about the standards yet later, but what I wanted to do was include this bridging standard that's included in the turn on CPC math materials because it talks about something that we'll use later on, something known as a ratio unit versus a unit ratio. So I want to just kind of put that language on the table. The ratio unit describes the smallest whole number pair that represents the ratio, so that would be like the littlest recipe. In our case, if we look down the list here, this is our littlest recipe where these two are going to be whole numbers and then the unit ratio is a ratio that describes a relationship where one of the parts of the ratio is a one. So for example, if we have one lemon to three-half cups of water, we go down here, one lemon to three-half cups of sugar water, that's called a unit ratio, it's arbitrary as to which one I put first, right? I could talk about the number of lemons to sugar water in which case I'd say two-thirds of the lemon. I'm not sure if we could cut, I guess we could work on cutting two-thirds of the lemon and then to one cup of sugar water. So these will be important because, for example, later on we can think about, if we could get back to that unit ratio then we can solve lots of problems if we understand the unit ratio. Also, it'll come into play later on when we talk about slope and unit rate. So I wanted to put that on the table so we can go back to the problem and think about some other representations. Here's one, if I decide to graph, nobody suggests you're graphing, but if I decide to graph these on a coordinate plane where I put the number of lemons along the horizontal axis and the number of cups of sugar water on the vertical axis and then I can plot some of those points in my table. So, for example, here I have that two lemons to three cups of sugar water and then here four to six and here was the original recipe, eight lemons to 12 cups of sugar water. So if I think about this graphical representation, it's kind of interesting because when we're comparing these items in lots of these problems we're comparing two different items. So it kind of makes sense to build from that table to building this graphical representation and it'll help us later think about slope. If there are questions or comments, please stop me or just put something in the chat window and Carol will stop me. So if you think about graphing these, you'll notice that the coordinates of the first lattice point, that is if I look at my graph, this is my first lattice point. That is where the coordinates are integer values here. Two, three makes up my ratio unit. So, again, the ratio unit versus unit ratio might be a little bit confusing that language. The ratio unit is that littlest recipe, whereas the unit ratio is where one of these particular values is one. So you can see that represented on the graph. I circled that ratio unit there. And this is the graph that I took from the turn on CC math materials because it talks about how you can also see this unit ratio, either way you represent it, one to three-halves or the two-thirds to one, on the graph if you're talking about number of lemons where the number of lemons is the unit one and you're trying to figure out how much sugar water, then here's this blue one here. I got one here and this is three-halves versus if I want the cup of sugar water to be one, how many lemons would I need? That's when this value here is one. I go over here and I figure out that's two-thirds. So students also learn to associate the horizontal line segment of change with the increase in the number of lemons and the vertical line segment of change with the increase in the cups of sugar water. So when you look at these horizontal line segments, if I increase the number of lemons, what will it mean in terms of cups of sugar water? Same increase here. That increase by two, then I need to increase the number of cups of sugar water by three, which was the original question was if I had two lemons, how many cups of sugar water would I need? So that's just the different representations. We have the table. We had the graphical representation. And then somebody did set up an equation, which I'll talk a little bit about later too, but I also want to introduce the idea of double number lines or maybe not introduce it. Maybe other people are familiar with this idea. Can you let me know in the chat box if you've seen the idea of double or the representation of double number lines? Maybe you can raise your hand and then I think I can look at that and see who's raising their hand. I haven't seen that. There's a question. Okay. And there's some questions here. Forgive me for kind of stepping outside of the picture. Is there a reason for showing the steps below the line instead of above? Show me the steps. Are you talking about the double number line? Can you get back up? Who was it then? Oh, can I see it? Barry Brown. Barry. Okay. So let's go back to the previous graph. Oops, sorry. And Barry, your question is, is there a reason... Let me get back to your question, sorry. Is there a reason for showing the steps below the line instead of above the line? Oh, I see what you're saying. I don't know. I didn't create this graph. I usually think of drawing them below. Alan, do you have an answer to that one? So he's asking instead of Alan, if you can send it to all, then we can see your answer. There it is. It could be either more a matter of convention. Yeah, that's what I was thinking. I thought of it as below the line, too, because it might just be what I'm used to. If we got to go with rise over run, it would be above. Susie suggested that. If I go rise over run, yeah, I'm going up three and over two. Yep. Good question. Thanks for asking it. Okay. So let's talk a little bit about double number lines unless there's some other questions about the previous line. Okay. So for double number lines, the reason I'm talking about it is because if you look at the Common Core standards, they talk about representing ratio in terms of these different representations. In terms of the graphical representation in ratio tables, we're going to talk about something called ratio boxes, double number lines, and even something else called tape diagram. So that was one of the reasons I think folks wanted me to do this webinar is to talk about all those different representations. Alan's got another answer here. The issue of rise first always confuses many people representing triangles. Oh, yeah. So later on we'll talk about the slope triangle. Actually, I don't know that I'll talk about it today, but in the turn on CC math material. Yeah. And then in what Diana's saying, we typically go to the X value before the Y value, which creates the steps below. That's what I was thinking of. If you change the number of lemons, how will that change the sugar water? So if I go back here, I say, if I change the number of lemons here, how much sugar water will I need? That also could be a function of what question we're trying to answer. If instead I would have said, if I have this much sugar water prepared, what would I need in terms of the number of lemons, in which case I might swap the horizontal and vertical axes too. So I thought about it this way. If I move along the horizontal axes to unit, how much sugar water will I need? Okay. So let's go back, I think, to the double number line. And what I'm going to do is I've created these two lines. The top one represents the number of lemons, and the bottom one represents the cups of sugar water. And what I've marked is that my recipe, my basic recipe, the one I think is so tasty, is eight lemons to 12 cups of sugar water. Okay. So what I'm going to do is I'm going to escape out of here and go to some place I can write on. I have a one-note file here with my graphic, because I can write on it using my style. It's a little bit easier than escaping out of the PowerPoint. So what you normally do with what I've seen with these double number lines is I have these two lines straight up on top of each other, because that's my basic recipe. So if I think about if I only have two lemons, that would be the original question. How much sugar water will I need? Well, somebody said I could just divide by four. So if I divide this top length by four here, then how much would each of these? See, this would be two, right? So from here to here, I just divide by four. And I do the same operation straight under here to get my new amount, divide by four, and that would give me three lemons. Okay. So that's one way to use double number lines, not the only way. The other way I've seen it is to think about the unit ratio. I got that straight because unit is the first word in unit ratio, which means that one piece of my ratio will be one. So if I have something to something else, one of these has to be a one. So that helps me remember which is a unit ratio versus a ratio unit. So if I think about the unit ratio, instead of doing what I just did or that, what I'm going to do is I'm going to say, well, if I want to get to one lemon, so if I get to one lemon, did I do that right? Yep, this is one. To get to one from eight, I divide by eight. Now, I haven't answered my question yet, but I can at least get to what the equivalent measurement is here straight down here for my sugar water. I'm going to divide by eight, which will give me three half. But I don't want to answer how much water do I need for one lemon. I want to answer the question, so there's a one here. Sorry, I covered it up in my arrow. I want to now think about how I get to two lemons. So I'm dividing by eight on the bottom, but multiplying by two here. So I'll do the same thing on the bottom number line. I divided by eight here. That was my first operation corresponding to the top number line. And now I'm going to take this three half and multiply by two. So I get three halfs multiplied by two, and that gives me my three cups of lemon. So that's what corresponds there. So that's one way to use two ways to use the double number line. If you're fortunate enough to get the question, oh, if I have two lemons, then I can just divide both by four. Or I can get back to my unit ratio and then use the unit ratio from there. So Alan's got a bunch of information. Let's look at what he's got. Alan, do you want to answer some of these questions as they come up, or would you rather wait until the end? You can just let me know. Okay, so he said whichever. He's kind of giving me some information as I go along, thinking about how we can represent that. But it might make sense just let's go ahead and give you the microphone. So this is going to be a test. We're going to give Alan the microphone because he's got some good suggestions as we think about the problem. Okay. And then I will mute my system. I'm going to give it to you. So you've given him control? Carol's going to give you control, Alan, so you can talk. Okay. I've unmuted it at this end. Let me know when you can hear me. There we go. Okay, I can even see me if you may not want to. Yeah, I was just making the point that when you have that representation, that double line representation the way Maria just drew it, I always look at these things, these representations as a model of the situation. And so that model really gives you certain affordances. A nice way of thinking about it is that it takes whatever proportion you started with and you scale both sets of units to the same size so that when you do an equal partitioning of both quantities, you are actually partitioning the line segment exactly the same way and then you draw your vertical lines and that shows you for each number exactly where the new proportion is for the same ratio. And that's kind of nice. Now it does not show you a real true two-dimensional representation of the ratio that you're working with. So that two-dimensional representation gives you certain other affordances. I think it's always important that kids be able to reason, kids and teachers for that matter, are able to reason with multiple different representations and understand what they show and what they don't show or what they hide, what they reveal. Okay, so we're going to thank you, Alan, for that. But yeah, feel free to let us know when you want to chime in there. It's very useful. Carol's just giving me control again. She's going to pass them all. Am I sure about that something? I can, no? Okay, great. So this is just kind of an interesting representation that I hadn't seen before until somebody said, well, you know, can you help us think about the double number line? And then I have a different example that we'll look at later using the ratio table and the double number line. And then let's go back to the PowerPoint. So we have this idea of using the double number line for our recipe. And then this is an interesting idea. So these are called ratio boxes. And this, too, is from the turn-on CC mass materials. So it's another bridging standard. And for those particular standards, they always give you the rationale for introducing these bridging standards. And they talk about using a ratio box to describe the relationship and explain how to move multiplicatively between the two quantities and within the two quantities. So when we looked at our table before, we had several values listed in our table, and that might be something that the kids do. They just create various, you know, lots of different values depending on the particular question they're going to try to answer until they get to their answer. But if I lift any two consecutive rows from that previous table, I create this ratio box. But then what I'm doing is trying to think about the relationship between the two quantities here across the columns. So that puts the correspondence. So here's one way to think about it. It's not the only way to think about it. We could, how do I get from 8 to 12? I could multiply by 3. I could 24 and divide by 2. And then here's the same thing that same correspondence holds true between 16 and 24. Multiply by 3, divide by 2, and I get to 24. So that's the correspondence. If I look down from one row to the next, within the two quantities, I say, well, how do I get 16 from 8? I multiply by 2. How do I get 24 from 12? That's also true. I multiply by 2. Now, again, kind of depending on where the kids are in terms of their operations and what they can do here, I was talking to somebody else about this and they said, well, is it complicated to do two operations to multiply by 3 and then divide by 2? Can I think of that as just multiplying by 3 half? And again, it depends on the facility of the students with that type of operation. So could we multiply by 1 and a half? Yes, that's certainly true. We can multiply by 1 and a half. So this idea of ratio, the ratio boxes is interesting because it kind of, it helps you focus on just two rows of your ratio table and it identifies these relationships here. So I'm sure Alan will have more to say about that either now or later in terms of how you see ratio boxes. So for example, with ours, somebody said, well, I could divide by 2 or divide by 4 so I go from 8 and if I change this to 16 to 2, then the question would be if I have 8 and 12 and 2 here, what goes in the missing spot? So let's go back out to my one-note file and let's just draw that ratio box with the original question. Excuse me. So if I draw my ratio box here and I have lemons, I have lemons here, sugar water here. This is my original recipe. My question was if I have a 2 here, what will go here? And somebody actually answers the question setting up an equation. If I go back, let me see if I can go back to the chat window. Okay, sorry. Now somehow I muted my audio back and I apologize for that. So I was just going back to the idea that somebody wanted to set up this equation. 8 divided by 12 equals 2 divided by x. Thank you for that suggestion. But what I was thinking about was if you have the outbreak skills to solve this, this is a fine way to do this and this is what many students do at kind of the later stage. But if you think about the skills required to solve this equation, it can be tricky. Kids get confused. You have to think about how do I want to solve this? Do I want to multiply both sides by x? Do I want to cross multiply that kind of idea? So this can be a pretty sophisticated way to solve this problem. Whereas if I just think about it from a numerical standpoint, even just the list of values and trying to come up with the answer based on if I have two lemons specifically, how do I answer that question? I think that kind of removes that difficulty, that level of difficulty. So the other thing I think that makes this a little bit challenging is that it is arbitrary how I set this up. In terms of, do I say x over 2 is the same thing as 12 over 8. This equation could be much easier to solve because my unknown is in the numerator versus this one that could be a little bit trickier. But again, if we have a facility with the algebra, it's not an issue. At a younger age, I imagine this is where we're going eventually. But the table and the ratio table, I think, are a nice way to think about it. So Alan has just said one of the strengths of the ratio box with one unknown is that students quickly see that you can get to the unknown via multiplication and or division vertically or horizontally. They can also eventually recognize how to set up an equation like that and eventually why the cross multiplication even has, even as far as generating proof. So yeah, they can understand why that works, that cross multiplication in terms of generating proof. So that's a great point. In fact, once we get our answer here, again, you have the choice, right? You can say, well, how do I get from here to here? I divided by 4. I can do the same thing here for this particular example. So I get my 3. And notice when you multiply across the diagonal, you get the same thing, which is really leading us to that idea that I can say 8 divided by 12 is the same thing as 2 over x. I'm going to really eventually say this was my x. 8x equals 24. So 8x equals 24 is ultimately the equation we'll solve once we set up those ratios. And it could be a little bit harder to get to depending on which way you said that. I got another comment that the ratio box also helps them see different ways to set up the same proportion. Yeah, I really love this idea in the ratio box. I thought, oh, that's great. Okay, so I think we'll keep going. I'm going to do a different example. And Carol, are you going to show my movie for me? Actually, I've got a movie for you. It's just a really short one. So let me present the problem, and then Carol's going to take over and show the movie. This is from a Dan Meyer video. Sorry, we'll go back to those questions in a minute. Dan Meyer video that was on a website that Susan Schell shared with me. It's the live finder site, and I have that in my list of resources. And it talked about his Nana. Dan Meyer has a Nana, and she likes her chocolate milk just right. And her recipe for just right means horses chocolate to one cup of milk. That's her perfect chocolate for Nana. And then if Carol can share her screen, she's going to show you the video, although we might not be able to hear it. I can mute. You don't really know. Okay, so I just had Carol show you the video, because I like that movie. I don't know why I like that movie, but I like that movie of Dan. Maybe it's because he's making chocolate milk for his Nana. I thought that was kind of nice. And so what the question is, how do you help Dan fix the problem? And so if you think about this in the different representations we thought about, we could make a table. We could make our ratio table. And we know the perfect recipe is four scoops of chocolate to one cup of milk. And if you just start filling your ratio table, and again, I think this is what kids do sometimes. They just start thinking about, well, what if I double the number of scoops? Well, I don't know that I want to make two whole cups of milk, but if I double the number of scoops, then I know that eight scoops to two cups of milk would be one thing that would fix the problem. If I use as many as 12, then I'd make three cups of milk, and that would seem like over the top too much milk for Nana. Or if you go all the way to 12, this is what they call building up these tables in the turn-on CC math materials, you build up your table, and then you can start thinking about dividing. Now that I've got 12, what if I divided by two? Then I could say, oh, this isn't such a bad solution. If I had six scoops, since Dan made a mistake of putting five scoops in there, if I just put one more scoop in there and add another half a cup of milk, then that's not way too much milk, and it still fixes the problem. So again, this is the idea of just kind of building up your ratio table. And we'll go to the ratio box, and then Alan had a good comment too about the ratio box, because I think it might be a good time to let Alan talk about any of these things we've looked at so far. So this one's actually simpler in some ways than the lemonade problem, because I have my perfect recipe here, four scoops to one cup of milk. It's easy to get back here. All I've got to do is divide by four, so I can divide by four here. Now notice that's not one of the solutions that was in my table. I didn't say if I have five scoops. I said, well, I could put one more scoop in there and use one and a half cups of milk, which I think is the way Van Meyer actually fixes the problem of his three acts. He has three acts, several problems that have his three acts. Where he sets up the problem, there's an issue, he fixes the problem, and then he kind of sets the stage for a future question. So in this particular case, with my ratio table, I can fix the problem without adding more chocolate. All I've got to do is add a quarter cup of milk. Similarly, if I look down the rows instead, if I go from four to five, one way to get from four to five is to divide by four and multiply by five. Similarly here, if I divide by four and multiply by five, that would fill in this blank as five fours. So Alan, would it be a good time now for you to kind of, I think it would be better for you to actually say that last comment as opposed to just have us read it. Can you hear me at this point? I'm unmuted. Let's see. Can you hear me? Okay, great. Thanks. Yeah, a couple of things I was just going to say to that is, as Maria was saying, early on when kids are working through these kinds of relationships, their typical strategy was to build up or to break down any partition. So they'll multiply by two, they'll double it, then they'll double the double, and then they'll have it and that kind of thing. And then they started going for the in-betweens because lo and behold, doing the whole number multiplication years, you're likely to skip something that is the problematic that's sitting in front of you. And once they understand the multiplicative relationships between the rows of the ratio box and between the columns of the ratio box, the covariation or the correspondence relationships, they begin to understand that they have to find a way on a multiplicative joint for getting to the unknown. So as Maria was explaining, that's exactly the kind of strategies. One thing to notice in that, if you can go back to the ratio box you were showing just a moment ago, Maria, one thing you'll notice is that in going vertically from four to five and from one to the unknown, dividing by four and multiplying by five and voila, you have multiplication and division of fractions. And when they figure out when that comes into play in these ratio boxes, the multiplicative relationships of fractions, numerators and denominators, and the relationships of that as an operator to the ratio becomes a tool that students start to more readily use and reason about themselves. See any other questions that you wanted to think about? Okay, so that's all I was going to comment on that at this point. So if anybody has any particular questions, then I can see those in the chat at this point otherwise I'd turn it back and we're in. So feel free to ask questions if you want to specify to Alan that would be great. Let's get a shy away from fractions. Yeah, I see that from Patricia. And yeah, it's interesting because from our perspective, we always like to approach fractions as a subset of ratios. And we get into that discussion as well because fractions, it's all about the reference unit with fractions. You can, for instance, you can compare fractions as numbers, but there's always an implicit understanding and implicit fact that what you're referring to is a fraction of one compared to a different fraction of one and which of those is larger. When you step into the larger context of the ratio world, now you're talking more typically about either the same quantity. It could be length to length when you're comparing conversions, for example. It could be yards to feet kind of thing. Or you're comparing different quantities as the problem we had in front of us here. Scoops of chocolate to cups of milk. Then your reference units become the quantities themselves, but you don't have a single reference unit for the entire ratio. You have two different reference units, and that's what makes ratios a larger, more inclusive case of fractions. Fractions are really a subset of ratios because they refer to the same reference unit. It's always has to be a fraction of something. You have to understand what that something is that that fraction refers to. Unit rate on ratios is difficult, right? And the ratio class for algebra one and the regular geometry students. So these are what you're noticing with the experiences with your own students in your classes. And the huge part of the key of understanding ratios for kids is recognizing that you're moving multiplicatively between the rows and between the columns. And the values of those multipliers between the rows is typically different than the multiple, except in very special cases, individual values. The multiplier, the correspondence multiplier, the movement left to right is a different factor than the movement up or down. But the movement up or down on either side is always the same. If you do one thing to one side of the ratio box, you have to do the same thing to the other side of the ratio box, which is that one of the major concepts of ratio is that the 4 to 1, for example, in the ratio box, is a proportion. And that any other proportion has to be maintained through the same factor on both sides of the proportion. So if you're multiplying by 5, if you're multiplying the number of scoops by 5, you have to multiply the number of cups of milk by 5 in order to keep that ratio, those proportions, invariant in relation to, through multiplication. So this is part of the aspect of what we tend to refer as getting students to move comfortably and flexibly in multiplicative space. And it's very distinct from additive space. Just like keeping balance in equations, exactly. It's exactly that same principle. Okay. I think we'll keep moving then. Sure. Thank you, Alan. Very nice having you here. Okay. So we've looked at our chocolate milk. An example with our ratio box can use a double number line to look at that problem. So let's look at that quickly. So if I go back over here to my one-note file, here's my double number line for my shot number of chocolate scoops in my cups of milk. And so my recipe year with my perfect recipe was one cup of milk to four scoops of chocolate. But Dan put too much chocolate in there. So we actually need to go out this way further to the right of four. But we can think about, well, how do I get back to one scoop, for example? If I go back to the idea of the unit ratio, what would one scoop of chocolate correspond to? So I'm going to think about where one might live here. And to get from the four to the one, I'll divide by four. And I'll divide by four. Similarly, the unit of milk corresponds to one scoop of chocolate. But Dan made the mistake of putting too much chocolate in there. So how do I get back out all the way over here to five? I need to multiply by five. And so I'll kind of run out of room here. These are kind of tight. I'll have to multiply by five and get out here to five, four. So I'll need one cup of milk plus another quarter. So I can fix the problem without adding any more chocolate if I just put another quarter cup of milk in there. So somebody just said that they really like the double number line. Well, yeah, I think it's very interesting because, again, this gives you a different way to visualize the operation. And also, it really makes me think about how a lot of times what we want to do is want to get back to that unit ratio, so that one to a quarter is where we're headed. So let's go back to the PowerPoint because we're quickly running out of time. It's been a great conversation, though. So it's been really nice having Alan here. And again, please feel free to type in your comments. So Alan just said to me privately that it doesn't address the issue of floats. Good. Good. That's a good point. And what I'd like to do then is think about how we go back to the representation of the coordinate plane. So if we draw our coordinate planes along the horizontal axis, we put scoops of chocolate. Along the vertical axis, we put cups of milk. I apologize for this representation. It has decimals in here only because of the tool I used. But we wouldn't necessarily have the chair. We might just mark one cup of milk here, two cups of milk, and have these tick marks instead of these decimal values. So I apologize for that. Again, that was just based on the tool that I used to create the graph. So if we go back to Alan's comment about it doesn't really address the issue of slope. Now with this other representation, we can think about slope again. Because we're talking about scoops of milk. This is my perfect recipe. I can go across here four scoops to one cup of milk. And even in our table, we noticed that if that meant that if we did eight scoops, we'd need two cups of milk. Or we could notice that if we move across here, again, four scoops, then what we've done is moved up one. And again, I apologize for this representation here. Just because it's limited by the tool I used, you might not have these tick marks individually numbered. And in which case you could say, oh, well, yeah, along here, this is one cup of milk. And you might not even have these tick marks here to see, but it's just, I'm sorry, four scoops to one cup of milk. And you can see that preserved in this graph. So it gets us to the idea of slope. The other thing that somebody asked me about was constant of proportionality. So I wanted to kind of bring that up because that's a big topic. And it helps us think about how we can use these graphical representations in the plane to think about the constant of proportionality. Here's the standard. Identify the constant of proportionality, the unit rate in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. The constant of proportionality in particular context describes the ratio between two quantities, essential for students to keep track of the order of the proportional quantities. The ratio in this problem, for example, this is our chocolate problem, is the amount of milk in relation to the number of scoops of chocolate. Therefore, the constant ratio describing the amount of milk per scoop of chocolate is, and that's what I did when I did my double number line, right? I tried to figure out what my ratio here was where one of these was just one unit. And the constant of proportionality is then a quarter. That is really dependent on how you set up your, I think of it as independent and dependent variables, but you might think about your horizontal axis versus your vertical axis. It leads us to the linear equation, milk equals a quarter times the scoops of chocolate. So we have a comment here from Grayling. Constant proportionality and rate of change would be very good here, since the slope is usually more applicable to graphical representation. But yeah, this is a great way to bring this in, because then we can, let's look at the next slide. So what I've done is set up this equation, y equals k times x, where y is the cups of milk, x is the scoops of chocolate. So I think k is the constant of proportionality, and so if we go back to that idea, this constant of proportionality here is if y is one cup and x is four scoops of chocolate, then y equals, I'm sorry, yeah, one equals k times four, so k is a quarter. And again, if we go back to thinking of slope, you go up one, or if we do it the other way, I guess, you go over four, up one, but if somebody says rise over run, we say rise one unit and go across four units. And it supports the translation between symbolic equation function representation and the graphical Cartesian plan is what Alan said. So I mean, this is what ultimately we want to move towards, is this idea of being able to write equations and kind of get out of these one-dimensional representations. But the one-dimensional representations kind of helps us to look at it in different ways and helps our kids, I think, get their hands on it, kind of really internalize a little bit more. So I think we're running out of time, so I wanted to kind of flash through these. One of those slides that I kind of whipped through before was, I've put in some slides in here in case people want to look at these webinars later. Maybe you want to, maybe you're on your own now watching it, but you want to look at it in a professional learning community. So I've put in some questions that you could stop and say, well, let's stop and answer these questions here. Which representation might be more useful, and it might depend on the question that you're asking. How do these representations tie to one another? How do these representations move us forward towards these, for example, the coordinate plane representation? So Alan had a comment that K, the constant of proportionality, is the ratio, the fixed relationship between the two quantities. And again, that goes back to what he was saying was that proportional idea that K is a constant, he put constant all in capital letters. So here's our lemonade example. Five is the cups of sugar, water, X is the number of lemons. So our constant proportionality is three-hands. Again, depending on what goes along the horizontal axis and vertical axis. So these later discussion questions are for you guys just to kind of think about how you might use these in your professional learning community's materials. So I'm going to switch gears real quick just to show you this real quick example of tape diagram. I'd never heard of a tape diagram before until I started reading these materials. And this actually comes from the Common Core Tools website. If you go to this progressions document and go under ratio and proportion, this is the one Bill McCallum's team. Those resources have been built by Bill McCallum's team, I think. And it has an example of those you're making a punch where you have three cups of orange juice and two cups of cranberry juice. And that's your favorite recipe. But you want to know how many types, how many of each type of juice would you need if you need to make 15 cups for your party? So the idea of a tape diagram is that, here, let me go to the next slide. This describes it better. If your original recipe is the three parts of orange juice to two parts of cranberry juice, it means I got five parts to the whole. And then I say five parts to the whole. Well, my new whole is 15 cups. So if I think about what one part would be, it'd be 15 divided by 5. So if I think of the tape diagram here as almost compartment, I'd say, well, how much would go in each of those compartments if each of the rectangles represents three cups? So if each of the rectangles here, this one, this one, this one, et cetera, represents three cups, how do I figure out how much orange juice I use, how much cranberry juice I use? So I'd say three times three is nine of orange juice, and two times three is six of cranberry. I just want to show you that real quick because somebody said, what about tape diagram? And I really just looked up these materials. Actually, I got Lou Ann Malick at Chapel Hill was the one who pointed me to these documents in terms of the tape diagram. So Alan has one comment, and I just want to kind of move real quick and let him do his real quick comment. This is just a quick screenshot of part of the hexagonal map for the trajectories that are provided on turnonccmath.net. The teal ones are all about ration proportion. I know you can't read those, but if you click on any one of these hexagons, you'll see a page that looks like this that gives you kind of a map, an overall structure of the topic. And then if you click on any one of these particular sections, it has very, very nice descriptions of particular examples and language and just lots of great ideas on how to present these materials. Here is just a real quick, another website that I wanted to share, and I have these at the end, too. This is the live binder site, and what I did was I went under math tasks, and Dan Meyer has his own page, but there's a ton of stuff here. Really great materials. Susan Schell shared that with me, and I appreciate math, so I wanted to show you that. And then this was just the Dan Meyer site where if you look at any of these, it's tied to a particular standard. You can download the videos. He sometimes has handouts, too. What is the address? The address for these are at the very end. So, again, since we're running out of time, here are the web addresses for the resources. There's the turnonccmath.net, Common Core Tools, and what we're going to do is we're going to make this PowerPoint available to you on the website. Carol will send you that link. It'll have the archived file of this webinar, and it will have the PDF file and a handout that she sent out as well yesterday that has these resources on there. So, all of these will be shared via either PowerPoint or on the handout. I put Tad Watanabe in there as well. Somebody said they lost my audio. Alan says... So, Alan is back. Oh, Alan is back. Okay. Tad Watanabe, I'll listen to him on there, too, because he has a talk on tape diagrams, and he gave us permission to use that courses on the web, and that's where I looked up the information about tape diagrams, as well as the core materials on the Bill McCallum site. So, I think we only have three minutes. Lori Bill says she has a math part, means tomorrow we'll receive the PowerPoint by then. And Carol says, yes, as soon as we're done here, we can put the PowerPoint and the handout and the link to this site. I mean, I'm sorry, the archived version for you guys to be able to get to it. Okay. So, Alan, I think you wanted to make a couple of comments. So, if you can turn on your audio again. Okay. Can you hear me? Yes, yes, we can. Okay, great. We only have a little bit of time, but I was going to suggest that I could show you that animation on comparing ratios for different recipes, because it's kind of a nice way of thinking about, well, how do you know whether a ratio is different than another ratio? As opposed to whether two different proportions are representing the same ratio. But I don't know if you have time. Oops. Now I can't hear you. Let's go ahead and do that real quick, just because it's nice to kind of open the door at the idea of flow. Sure. Okay. Seriously. Okay. So, let's see if we can make this show up here. Can you see that? Can you see my screen now? Do you have to do the share my desktop? Yeah. Oh, thank you. Pardon. There we go. See if it'll respond. I'm not sure it wants to respond to me yet. Carol's going to give you. Oh, okay. I thought she had already. There we go. Yay. Okay. So, you can see this now? Yep. Okay, great. So, here's the question. You've got a family reunion, and several people made pictures of lemonade, and then you had an argument over whose was the strongest. So, you have to come up with a way to determine which is the strongest, and one of the ways that you can do it is to compare the ratios. So, for instance, here, your pressure memory that my own lemonade was 8 lemons to 12 cups of water. My aunt's was 9 to 15. Grandma's was 6 to 6. And grandpa's was 12 lemons to 9 cups of water. Grandpa's is sour. So, just for reasoning from the numerical relations, a lot of times you can see that, see which is the strongest. But you may need to look more closely and get more detail out of it. So, the reasoning is that grandpa's is the strongest because they're more lemons than water. So, once you cross that threshold, that's the only one that crosses the threshold of more lemons than water. Grandma's is stronger than aunt Tam's because aunt Tam would have to have 15 lemons and 15 cups of water to be the same and so on. But if you compare ratios, and I'll put this on in the animation, in the animation way because it's fun to see it that way. I think you can probably still see that when it comes up on my screen, which is, sorry, it's just taking so long. Maybe it doesn't work. It's taking a little while here. Everybody got a little bit of patience here with this? Here we go. Ah, there we go. Okay, great. So, if you look at, I don't know, I haven't done this before with an animation over a WebEx. Let's see what happens. If you're clicking, oh, sorry, clicking and you can't see anything change on the PowerPoint, click at, like, click on the PowerPoint because sometimes when you, there it is. Okay, so here's mine and my ratio table for several different proportions and we're just plotting the points on here and then you draw the line here so that you know you recognize that every proportion, no matter, you can double your amount of water and double your amount of lemons and so on and you get one line that represents that relationship for my own recipe for lemonade. And here comes Aunt Anne's. She started out, remember, with 15 cups of water and nine lemons, there's that point and there's the point of one where you actually split it by three and here's her recipe. Here's Grandma's plotting her proportions and here comes Grandma's who has the strongest lemonade we've already determined from looking at the numbers. So, when you're looking at the relationship between different recipes and, i.e., different ratios, what ends up happening is that you're rotating those rays and the proportion with the ratios are different because you've got different angles relative to one or the other of the axes. I'm going to stop you, Allen. Thank you for sharing that. There was a question from Patricia about whether or not you could share that with the participants and you can let me know if you want to share that somehow. Yeah, actually, we'll get down and let's talk and we'll figure out how to do it. Yep, and then thank you all very much. We appreciate your participation and your patience with us. I hope that this has been helpful. It's going to be a couple of things you'll get from Carol. You'll get a link to a survey as well as link to all the documents. And so, I hope, again, that this has been helpful for you all and if you have questions or something, feel free to send me an e-mail or Carol e-mail. Thank you, Allen, for being on deck. I really appreciate your helping us out. My pleasure. Thank you. Okay, bye.