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.  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.  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.  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.


Hey what's up guys it's Josh here. Today I want to share my thoughts on three different muscle building products by the House of Vintage Muscle. Now these are all kind of topical solutions.  I guess we'll just jump right into it. My favorite one for sure of all these is definitely going to be youth and youth is very interesting because it has I think it's supposed to contain two different kinds of peptides. I think it's like TB500 and then TB500 and then BP157 I believe but basically they are different kinds of like growth hormones and things like that or like peptides from growth hormones. I just absolutely loved youth. The only problem that I had with it was that it made me tired but besides that I felt like it made my skin very very healthy and almost like it had an anti-aging effect and the only one that comes to mind is having of having a similar effect is turquoise thrown but I went through a thing about three full bottles of youth. I really really liked it. It made my skin very very healthy and almost like it had an anti-aging effect. I went through a thing about three full bottles of youth. I really really liked it. It actually does make you a little bit tired. Now I was taking a lot of stimulants at the time so I don't know you know if my body was zapped and it was the youth was kind of like me balancing my body so it just whatever I needed it was going to make me do and what I needed was sleep. I'm not really too sure. I would definitely take it before bed if you do take it and it has a very noticeable muscle building effect but it's also weird because if your body it doesn't really have an androgen style feel so you don't get as much of that rage as much as that aggression you know and I did have a problem with that and I think I believe chisel but you don't get as much of that and so it's kind of nice because if you want to stack that with another testosterone type supplements it's they're not really overlapping it's not like taking two different steroids it's like taking two different things so I like that it I gave you an extra option do you want to take both you want to take one do you want to switch off from one to the other it's kind of up to you so I really really liked um youth definitely number one and it's definitely the other two are more testosterone pro-hormone base youth is more just peptides related to growth hormone number two is chisel now this is the one that I had the most stress with I actually shot a review with it and I just really didn't like the review so I didn't edit it but there was a time when I took a vintage muscle product and people were talking shit on me like because I could only take it for like a day or two and then I gave it my effects or my summary of the product that was only in a day or two and it was because it made me so ridiculously mad and angry that there's no way that I could take it there's no way under any circumstance I kind of had a similar effect with chisel and I think I got maybe three four five days into it but it's been a long long time but I there was this basically this long rage story that I had trying to drive home where I like people were in front of me and they were going slow and then I took a different street and then like when I took that street someone else was driving slow and then I tried to pass them and then they blocked the lane on me and so it was just this long story and it was probably because I was so pissed off this chisel but basically I enjoyed the muscle building effects but it just was not worth the extra aggression that I got and yeah I could only maybe last four days on this stuff unfortunately but I think that would probably be the strongest typical androgen style supplement that is going to be on this list if you want the strongest pro hormone type supplement that I'm listening out of these three it would definitely be chisel and then the third one is Titan now Titan it seems to be a little bit more down the middle versus chisel it seems to be more of like it's just more mild that's kind of the effect that I got off it and it does seem like I can't remember exactly what they said on the side but I'm pretty sure it like turns into just testosterone and there's different forms of testosterone all the different names of steroids are basically alterations of testosterone that do different things so this one is a little bit more mild mattered it's a little bit more down the middle I actually could finish a whole bottle of this stuff if I were to stop taking it after a few days I might feel like the tiniest bit deflated or like like not really in a physical sense but just like overall energy but I was never bothered by it enough that I had to stop it and it has definite noticeable muscle building effects that I really enjoy so you know I would say like for muscle building effects I would probably rate the youth maybe like an eight you know maybe up to 8.25 you know it's good but it's not like the strongest biggest bulk supplement that you could ever try or pro hormone type supplement that you could try but it's very strong the titan you know I didn't have enough days on it but I probably do like 8.5 as far as the muscle building effects I just felt like maybe you would have a little bit more but the aggression wasn't worth it and then titan is the pretty much the one I would give titan about like a seven maybe to a 7.5 for muscle building effects very strong very noticeable but it is a pro hormone type supplement and usually any strong pro hormone gets me really really you know angry or frustrated so usually I'll go with laxogenin or turquoise throne those are kind of like my go-to's they're in the plant store right realm but anyways guys let me know what you think of these three products have you tried any of these three youth is definitely my favorite have a great day out here hopefully have a great day at home see you all in the next video peace


So you got to check out the Black Hawk behind you. What's something that you've learned that you thought was really cool about the helicopter? Everything about it.  What's it like being in the pilot city? It's crazy. It's amazing.  No. But it would be really cool.


We have, in the past, covered the astonishing mega-metropolis that has been unearthed using ground-penetrating radar beneath a nearly impenetrable rainforest that now engulfs the area. A super-civilization that not only supports our posit of there once being a number of lost civilizations that have flourished and often seemingly met an untimely demise here on our planet, but has been estimated to have been home to more than 10 million sites such as Tikal once believed to be independent separate clusters of past impressive and as yet unexplained block buildings through the use of ground-penetrating radar have been proven beyond doubt to have once been part of the same super-settlement that spanned nearly the entire rainforest that the site is now home to. Yet we have also covered the incredible ancient stone and earthworks that can be found dotting many areas of the Amazonian rainforest that, again, are indicative of a past super-civilization. Yet conveniently, since the discovery of Guatemala's super-city, funding for such penetrative studies elsewhere of said areas has dried up. The question is why? Why are we witnessing an active attempt to conceal these ruins from the world? We feel the evidence to suggest so is now beginning to mount. However, where mainstream academia won't step, many others are fortunately willing to pick up the slack, and this particular area of interest is of no exception. And as usual, the investigative researchers have turned up some astonishing characteristics of the Amazonian rainforest, features which are indicative of another super-settlement, possibly of a similar size to that of the ancient sites found within Guatemala. A group of scientists and researchers, after investigating the area, have put forward what has been pinned as the Amazonian Stonehenge. According to said researchers, they found evidence that a, quote, highly advanced ancient civilization once existed in Brazil, and although they have dated the ruins As having been built 500 years before the European colonization of the Americas began, we feel that the evidence to suggest that they were in reality far before this date will soon be realized, and that these people who once inhabited the Brazilian Amazon were possibly creating an impressive arrangement of immense towering granite blocks. As such, scientists today speculate that these massive stones were like so many other sites we have covered in the past, attributed as that of an ancient astronomical observatory. The structure consists of 27 blocks of granite, each up to 4 meters tall, standing upright in a circle measuring over 30 meters in diameter. In other words, possibly more than a thousand years ago, an ancient civilization of native peoples were flourishing in the area. According to the New York Times, radiocarbon tests and site measurements during the winter solstice shed light on the ancient civilization's abilities that inhabited the Amazon. From this, new archaeologists have realized that the people who lived in the area developed a more advanced civilization than previously thought. Who built the Amazon Stonehenge? When did they build it? Is there a lost super-settlement hidden beneath the Amazonian rainforest as that of the Guatemala rainforest? We find the evidence to suggest such highly compelling. All over the Earth, ancient structures can be found, megaliths made of quarried stones many tons in weight. They are often found to have been perched atop one another with no mortar and seemingly little effort. Stonehenge, for example, underwent a restoration effort shortly after the Second World War, a project which consisted of several cranes, diggers, trucks, and lorries. One has to wonder if one goes by currently accepted timelines, of course. Just how did such primitive people manage to build these amazing enormous structures? Structures such as the very ancient gateway in Tonka, still consisting of three 40-ton lumps of coral, the granite stones within the king's chamber at Kufu, weighing in at 80 tons, or the stone of the south at Baalbeck in Lebanon, the largest officially recorded worked monolith on Earth, weighing in at a staggering 1242 tons. There is, of course, other theories, and although to some, one in particular may not seem at all possible, it is far more of a viable option than currently attested archaeological views. Along with Stonehenge, another site we tend to unravel in the future, although a lot more modern is Coral Castle, an oolite limestone structure that was built in Florida, and in particular, Edward Leedscallen, the man who built it. With the aid of a somewhat suspect black box, often spotted atop of his confusing array of multi-ton lifting wooden contraptions. This leads us on to our main topic of the video. It seems that Edward, like a few other known individuals throughout history, may have deciphered a piece of lost knowledge, information which allowed them to create rudimentary lifting devices with the ability of lifting stones many tons in weight effortlessly. In the small pockets of the world where advanced ancient cultures lived, prospered and vanished into history, places where their ancient ruins remain untouched by all but nature. Testimonies were found describing these exact technologies. Sites such as Puma Punca, Stonehenge, and in particular, Easter Island, where not only does there still exist accounts of the huge moi statues floated along the coast by way of strange chants, the places in which the priests hid within the caves also still exist. These rituals in apparent frequency. Could this be a form of lost knowledge, or technology, which was responsible for the building of the many unexplained sites found around the world? Vibrational technology capable of making any inorganic object regardless of size weightless? This would have allowed craft, previously impossible of flight, to float across the heavens vindicating many historical accounts of Viminas. Additionally, the levitation of physical objects by tapping into their atomic resonance is currently being redeveloped. The University of Bristol in England has created a handheld device with an array of speakers that can levitate light materials with sound alone. The more time that passes and the more technologically advanced we become, the more we begin to understand these amazing ancient structures. And although, for some, the realizations that these technological epiphanies induce, for example, accepting the possibility of a past highly advanced ancient civilization, which ended with an equally as possible global cataclysm, could be seen as altering of a worldview to understand what's true is far from detrimental. Recently, an incredible story has been released by National Geographic. An enormous ancient metropolis found hidden beneath deep forests within Guatemala. Discovered using a combination of lidar mapping, drones, physical exploration, and other forms of radar. This lost city's disclosure, however, if given the publicity and understanding it deserves, brings with it some drastic, yet by some long-awaited realizations within academia, the archaeological community, the educational curriculum, and many, many authors and their so-called accurate literature. For this city's population has been estimated to have not just been thousands, but actually went far into the millions. A civilization, or rather, incredibly ancient and so far complex city, would have rivaled even the most heavily populated areas of Earth even to this day.  A drastic re-look at our currently taught understandings of history is undoubtedly required. Amazingly, this pre-Columbian civilization, hidden for untold millennia beneath the dense rainforests of Guatemala, is estimated by some to have peaked a mere 1200 years ago. Yet it is entirely possible, and much more likely, that this lost city is far, far older than this. An advanced, complex city landscape, which the most recent data suggests may have supported a population of up to 10 to 15 million people. This realized from new drone footage of the city, which spans an astonishing 2,100 square kilometers in size. The advanced infrastructure, which included advanced agricultural terracing, with elevated trade routes to prevent flooding in rainy seasons, not only has experts finally re-thinking the dimensions and complexity of past empires, but it may be the lynchpin needed in future realizations of a past civilization, a civilization which was far more enlightened than our current society. As such, upon their untimely demise, left little scars upon the earth, little evidence of their existence, apart from their sophisticated structures, as such a lost city is discovered. As predicted, modern technology will ultimately bring an end to this worldwide suppression of what should be free-flowing information regarding what once was seen as controversial, is now finally being seen as educational. And given that the newly announced Maya Megapolis is the result of only the first phase of Pakunam's LiDAR initiative, there are likely to be many more revelations about the mysterious people who built this massive urban network, clearly an amazing discovery. One which could quite possibly be pre-Diluvian. We will keep you posted. Thanks for watching, guys, and until next time, take care. This has been laid to waste, not only by future settlements and tomb robbers, but by Mother Nature herself, many of these most impressive sites having endured eons of erosion after being mysteriously abandoned, exposed to the elements. Yet there exists a number of these sites which have been somewhat protected from these forces. Although vegetation can have a catastrophic effect, uprooting the megalithic foundations of these sites, yet the actual footprint of these structures, and indeed the overall size of these once-loss settlements, can still be seen through modern penetrative radar, with one of the most incredible found in the past few years, undoubtedly the mega metropolis hidden beneath the dense forests of Guatemala. Although some clearings dotted within this landscape have been spared, somehow avoiding the suffocation of trees, it has been discovered that these sites, long argued as separate sites of habitation, were in reality once part of the same gigantic city, one of unimaginable size and complexity that was unquestionably home to not mere thousands, but was in fact a settlement that was home to more than 10 million. Yet although this reality is a compelling, supportive fact regarding our own beliefs, in regards to a far greater now-hidden and widely ignored history of mankind, there are still features of this ancient site that is still attempted to be ignored, overlooked, and hopefully concealed from the majority of the world's population, ultimately avoiding them questioning the true reality of what they have been taught, and the possible truth regarding our history, which these sites could provide to all those who gaze upon them. Although these particular megalithic blocks somehow stood on their heads have been explored and exposed for nearly a hundred years with many photographic expeditions having been made to these sites, it has now been proven that these megalithic blocks were not merely signposts made of stones in situ, but were clearly stones cut and once transported to their current location, and were actually strategically placed within one huge mega-settlement. This fact is attempted to be stifled, avoiding individuals questioning how, if indeed they were transported and cut by our more recent ancestors, the Mayans, how they actually accomplished this feat when they clearly required now lost techniques and technologies. As although they were far more primitive, technologically speaking to the modern man, with us only accomplishing such abilities within the last century, all thanks to modern technology. This is clearly an identifying feature which exposes the true capabilities of the builders of this enormous city, and the fact that although academics would like to argue that it was merely a Mayan settlement, it possesses like so many other astonishing sites on earth as yet unexplained enigmas, which not only fly in the face of this explanation for their origins, but actually suggest that they were merely re-inhabited by the Mayans, allowing archaeologists to point the finger at such a group, due to their archaeological fingerprint having been left at the location, sites which were in fact built by a now lost, yet once highly capable ancient civilization, that due to their immense age has now been lost to history, like so many of their ancient settlements, lost to the sands of time, with only the foundation of which now survive, thankfully exposed by modern technologies. Who were these ancient people? How or indeed why did they move and cut such enormous enigmatic ancient megaliths within this enormous now lost city? It is a place which we find highly compelling. Although the Olmic culture has been historically established as having flourished within the now dense jungles of Guatemala, there also exists many pre-Olmic ruins and artifacts that are still baffling researchers and historians alike. The great head of Guatemala being one of the most controversial of all. This enormous stone face indicates that not only the Olmics or indeed native Hispanic race once called Guatemala home, a gigantic, masterfully carved stone head with a face of fine features, thin lips and large nose, once engulfed in millennia of vegetation, directed to the sky as if in eternal prayer. The discovery unsurprisingly attracted a lot of attention, yet just as predictably, due to its unquestioned controversy, quickly slipped into the pages of forgotten history. The initial discovery first emerged when Dr. Oscar Raphael Padilla Lara, a doctor of philosophy, lawyer and notary, received a photograph of the head in 1987, along with a vague description. It stated that the photograph was taken in the 1950s by the owner of the land and that it was located, quote, somewhere in the jungles of Guatemala. The site was later established to have been 10 kilometers from a small village in the south of Guatemala. However, when Dr. Padilla managed to travel to the site, a short while after the discovery had been widely circulated throughout the country, he found that the site, along with the Caucasian featured stone face, had been obliterated. He stated, quote, it was destroyed by revolutionaries about 10 years ago. We had located the statue too late. It was used as target practice by rebels. This totally disfigured it, sort of like the way the Sphinx in Egypt had its nose shot off by the Turks. Only worse, the eyes, nose and mouth had been completely destroyed, end quote. The deal was able to measure its height as having been between four and six meters. Although predictably, the stone head had been destroyed due to its controversial nature, it may still shine light on who was flourishing in the jungles, far before any Olmuk had ever stepped foot there. Additionally, and fortunately, the stone head is not the only pre-Olmuk statue ever found. Named the Fat Boys, these other artifacts are another set of statues that, although not as racially controversial, possess characteristics even more so for the scientific world. These statues, retrieved and displayed, were discovered many years later to actually contain magnetic elements, which along with a number of anthropomorphic artworks from the same suspected civilization, have magnetic characteristics positioned at specific locations. On the Fat Boys, it is found at the naval, although the animal statues seemingly contain them around the faces. So, the question is obvious. How did an ancient culture located so far back within history not only know about this magnetism, but managed to create such artworks? Why did they create them? Were they attempting to tell their distant ancestors something? Regardless of the controversy surrounding their creators, they are undoubtedly highly compelling. Recently, an incredible story has been released by National Geographic. An enormous ancient metropolis found hidden beneath deep forest within Guatemala. Discovered using a combination of lidar mapping, drones, physical exploration and other forms of radar. This lost city's disclosure, however, if given the publicity and understanding it deserves, brings with it some drastic, yet by some long-awaited realizations within academia, the archaeological community, the educational curriculum, and many many authors and their so-called accurate literature. For this city's population has been estimated to have not just been thousands, but actually went far into the millions. A civilization, or rather, incredibly ancient and so far complex city, would have rivaled even the most heavily populated areas of Earth even to this day.  A drastic relook at our currently taught understandings of history is undoubtedly required. Amazingly, this pre-Columbian civilization, hidden for untold millennia beneath the dense rainforests of Guatemala, is estimated by some to have peaked a mere 1200 years ago. Yet it is entirely possible, and much more likely, that this lost city is far, far older than this. An advanced, complex city landscape, which the most recent data suggests may have supported a population of up to 10 to 15 million people. This realized from new drone footage of the city, which spans an astonishing 2,100 square kilometers in size. The advanced infrastructure, which included advanced agricultural terracing, with elevated trade routes to prevent flooding and rainy seasons, not only has experts finally rethinking the dimensions and complexity of past empires, but it may be the lynchpin needed in future realizations of a past civilization, a civilization which was far more enlightened than our current society. As such, upon their untimely demise, left little scars upon the earth, little evidence of their existence, apart from their sophisticated structures, as such a lost city is discovered. As predicted, modern technology will ultimately bring an end to this worldwide suppression of what should be free-flowing information regarding what once was seen as controversial, is now finally being seen as educational. And given that the newly announced Maya Megapolis is the result of only the first phase of Pakunam's lidar initiative, there are likely to be many more revelations about the mysterious people who built this massive urban network, clearly an amazing discovery, one which could quite possibly be pre-Diluvian. We will keep you posted. Thanks for watching guys, and until next time, take care. An effort to not only expose the truth regarding the real history of man, but to discover the actual original size of these now lost civilizations' ancient ruins. Many sites have been laid to waste, not only by future settlements and tomb robbers, but by mother nature herself, many of these most impressive sites having endured eons of erosion after being mysteriously abandoned, exposed to the elements. Yet there exists a number of these sites which have been somewhat protected from these forces, although vegetation can have a catastrophic effect, uprooting the megalithic foundations of these sites. Yet the actual footprint of these structures, and indeed the overall size of these once lost settlements, can still be seen through modern penetrative radar, with one of the most incredible found in the past few years, undoubtedly the mega metropolis hidden beneath the dense forests of Guatemala. Although some clearings dotted within this landscape have been spared, somehow avoiding the suffocation of trees, it has been discovered that these sites, long argued as separate sites of habitation, were in reality once part of the same gigantic city, one of unimaginable size and complexity that was unquestionably home to not mere thousands, but was in fact a settlement that was home to more than 10 million. Yet although this reality is a compelling, supportive fact regarding our own beliefs, in regards to a far greater now hidden and widely ignored history of mankind, there are still features of this ancient site that is still attempted to be ignored, overlooked, and hopefully concealed from the majority of the world's population, ultimately avoiding them questioning the true reality of what they have been taught, and the possible truth regarding our history, which these sites could provide to all those who gaze upon them. Although these particular megalithic blocks somehow stood on their heads, have been explored and exposed for nearly a hundred years, with many photographic expeditions having been made to these sites, it has now been proven that these megalithic blocks were not merely signposts made of stones in situ, but were clearly stones cut and once transported to their current location, and were actually strategically placed within one huge mega-settlement. This fact is attempted to be stifled, avoiding individuals questioning how, if indeed they were transported and cut by our more recent ancestors, the Mayans, how they actually accomplished this feat, when they clearly required now lost techniques and technologies. As although they were far more primitive, technologically speaking to the modern man, with us only accomplishing such abilities within the last century, all thanks to modern technology. This is clearly an identifying feature, which exposes the true capabilities of the builders of this enormous city, and the fact that although academics would like to argue that it was merely a Mayan settlement, it possesses like so many other astonishing sites on earth as yet unexplained enigmas, which not only fly in the face of this explanation for their origins, but actually suggest that they were merely re-inhabited by the Mayans, allowing archaeologists to point the finger at such a group, due to their archaeological fingerprint having been left at the location, sites which were in fact built by a now lost yet once highly capable ancient civilization. That due to their immense age has now been lost to history, like so many of their ancient settlements, lost to the sands of time, with only the foundation of which now survive, thankfully exposed by modern technologies. Who were these ancient people? How or indeed why did they move and cut such enormous enigmatic ancient megaliths within this enormous now lost city? It is a place which we find highly compelling. There are many baffling anomalous artist depictions that litter the megaliths, all over the rainforests which now submerge the super civilization or mega metropolis, which was once known as the Mayans. Pekal's tomb and the strange rocket designs we have shared before, location now unknown, the seal seemingly showing a craft of ancient high technology, indeed not to mention his choice to have a vivid green death mask. All mere coincidence? Even the stolen plaque fortunately photographed before its mysterious theft depicting a doomsday event, volcanic eruptions in the background with drowning natives which accompanied a mass submersion found and subsequently stolen from Tikal. Mere coincidence? Yet with all these compelling pieces of evidence, all these artistic accounts, whether through cover-up or lack of discovery, an accurate artistically created depiction. Of this event of a peaceful trade via ancient alien contact, what the argument needed to become unarguable may have just been discovered deep in the Mayan rainforests. Accurate depictions created in brittle yet precious jade tablets of considerable proportions, artistic interpretations of these events, and the giving of gifts actually once occurring. A statement released by Julia Levy, the Minister of Tourism for the Mexican State of Campache, Luis Augusto Garcia Rosado, the highest-ranking government official to go on record confirming the discovery of this possible extraterrestrial life, said, quote, Guatemala, like Mexico, home to the ancient yet advanced Mayan civilization, has also kept certain provocative archaeological discoveries classified, and now believes that it is time to bring forth this information. Rosado spoke of contact, quote, between the Mayans and extraterrestrials, supported by translations of certain codices which the government has kept secure in underground vaults for some time. In a telephone conversation with The Wrap, he also spoke of, quote, landing pads in the jungle that are 3,000 years old, end, quote.


I'll bitch on the riches in the state I just called you in new episode of talk of the town today. We have a special guest black jack What's happening? So tell us like how that name originated like why when did you confirm that was your name? How did that come about? Like this girl in high school Just start calling me like black is that like my real name is that so Yeah, so she just start calling me and I just ran with so since high school Yeah All right, so tell us a little bit about this South Carolina music scene like is it hard when artists like this? um You know it ain't got better But you know it definitely is like hard like you got to get out there get out the city You got to be able to travel and be willing to travel and be willing to put like money into yourself so But you know you can't really make no excuse about where you from you just got to get to where you're trying to go Yeah, I'm gonna say what will be the heart thing cuz like in New York City It's a lot of like rap beef those like that out there too, or is it more so just hating or is it like what you mean? She is seem like everybody won't be rapping.  Yeah, it's like basketball All right, so what on rapper do you feel like impacted your music career creatively? Like what's some Odyssey scene that inspired you like coming up?  Yeah Still in there just flying. It'll be like one or two people on the plane When I was still out. Yeah, that's cool. So um, so Did you ever think that you would be to where you at now? Like what is it ever a point where you want to stop rapping or like? Was what kept you like going motivated?   Okay. Yeah, so you said goals and deadlines to yourself Yeah, yeah, so all the time So what's something like that's like on your like deadline list right now? I know you got artists now He's starting up your label. So like what's something like on your deadline right now? um Ready to get no rich get the cup a company get my whole company rich like starting a real brand I've been late.  So what was like one something I stood up to you that realize you made up the hood like When this happened everybody like felt it or you felt it When I feel like I made it out here Traveling people stopping all the states like what was it like? Oh snap. This is worldwide now. I mean I feel like I made it out here like when I when I moved out But it's like I felt like I was doing something like when I go to other states and Like people know who I am and I know it's from music.  So that kept me motivated to keep going because I'm like if I can get This person on me then I know this person to get on it. It's just be a consistent grind like every day Okay, so um, so but you did sign a label before when are you signed store? Are you independent now? I'm still signing in the score SMG in the score Okay, so um, so what does it mean to like? What does it mean to the rap game to you? Like how important is music for you? I love music. I feel like the world wouldn't be the same without me So it's important and it's so much Legitimate legal money you can get and wrap so it's like why not So who's like your all-time top three dead or alive right now music artists rappers That name of top three Yeah, that's hard. I Like one I like one dude you probably don't even know like do be from Alabama. He died like He was young his name don't be okay. He was so hard like But don't be for so Without put myself in Whole it's gonna cover up the whole three cuz I'm gonna say me All right, that's not a bad answer So, um, what's something that you regret now in career or something? Could have done better Really feel like I regret nothing as far as like my career as far as awesome rap Like I know I can't say I really no regrets.  So It was a lesson learned. I ain't took a loss from it. So it was a lesson learned Okay, so how difficult is it is it difficult being a CEO in artists?  Yeah, all they care about it. Hey It's the music selling like that's just selling but I can connect with them in different way because I'm a artist too So and I came from being independent and coming up like from the bottom and having to learn that shit taking losses And keep going having to fight with myself like down. I want to stop. I want to keep going want to stop So it's like so you definitely super hands on with your artist. Yeah. Yeah so um what like you guys do artists development or just having them on road with you or Just the whole experience in itself Shit, they probably tell you like I am the artist development I'll be on anything I hear like that. I don't be in agreement. I'm gonna say something about it Whether and they gonna say something how they feel back. So you have a good communication Yeah, so a lot of us complain that you know labels not in tune or don't really connect with them So you definitely tune all your artists. That's good. So um, do you have like any upcoming features on the way or any? We should be expecting I'm about to drop we just dropped the walk down game. That's out right now Me fat Eli, we just got that make sure you go get that But yeah to pop videos is job See you tuned in I like that So that's what I'm saying. How does the video process go be out playing our job videos? Have a director or step with y'all.   So you would never shoot in the desert again.  So what some advice you would give upcoming artists That's probably want to do your same shoes be a CEO and still be an artist and managing label Like what some advice you would give them like some tips? Just whatever whatever goal you got whatever you believe in like Just do it like it's up to you. Nobody can't tell you what they what you can do what ain't work for them Yeah, it may work for you and you just gotta not be scared to invest in traveling like just Think outside the box sometime. So think outside the box very much. So, yeah Yeah, I feel like the music again again kind of saturate now For so we got like do something differently. So what's next for you? That's your artist coming got your songs coming any projects You're gonna drop a collab project I'm a I'm a drop I'm a drop album next But in the midst of me dropping who knows you might just still drop Eli fat low autumn, but Yeah Like right now we're just working on just trying to get their face You know I'm saying build them up some more so when they drop to really make sense But it's it in we still gonna be dropping like singles after singles Yeah, but as far as a project like We ain't gonna really rush that too much because we got to build that up. Yeah So everything is in the works for sure, but definitely a project this year. Yeah Okay, so tell people where to find you where to find your music how to stream your artists with their names You can follow me on all social media at blacks at BLACC ZACC Make sure you follow fat low make sure you follow for they rely on all social media Walk down gang out right now how to send the screws


Hi guys, welcome back to my channel. In today's video, I'm going to be filming a haul for you guys. I have Revolve, a lot of stuff from Revolve, and a lot of stuff from Princess Polly, and stuff from Thread Wallets.  So I wanted to share some of that stuff with you guys as well. I think you guys are really gonna like their stuff. It's really convenient. It's also really good for holiday guests if you guys are looking for that. So we will get into that.   So Thread Wallets is like very practical for me. They have really cute card cases. They have really cute lanyards. They even have like a chapstick holder, which is really helpful because one, it's obviously very convenient. And two, my dogs always find my chapstick and like ruin it and chew it up. And they're not able to when I have this holder on. Really, really recommend Thread Wallets. I love their stuff. If you guys are looking for any sort of practical, organizational things, I personally am shopping for them on Christmas to give a few things to my siblings. It's really cute, but it's also very practical. So if you guys want to use Thread Wallet, just do ThreadWallets.com slash discount slash Kinsey 15, for 15% off. I love their stuff. I think you guys will too, definitely check them out. So like I said, I have some things from them. The first thing, like I said, is just this little chapstick holder, which is so convenient, also very sleek, very simple. I have two of these. So this is the one that isn't open.  And I'll actually like throw it in a bag. There it's like a carry-on when I'm traveling or a purse. It's really helpful to keep your actual bags organized. So I love this. I have this bag in my big tote bag to keep all my chargers in to keep them organized. So it's really helpful for that. I have another lanyard here, which is great for keys. I like lanyards for a lot of reasons, obviously they're practical, but I always lose my keys in my purse somehow. They're always gone. So just having these on them, it's cute. Also this design reminds me of Dom.  I use just like a sleek black one that's downstairs. So simple and really easy, but also like really cute and stylish. So again, I will have them linked down below. Threadwallets.com slash discount slash Kenzie15 for 15% off your order. All right, guys. So getting into the clothing portion of the haul, we have a lot of things. First off, what I'm wearing right now, we're gonna start with Revolve because I'm already wearing it. It's this really long black cardigan that I just think is like so cute. I love it. I feel like I'm gonna wear it so often. I'm very into like Camryn Diaz, like the holiday style, even like my sister's keeper. Like just Camryn Diaz like slouchy cozy that's kind of in my vibe lately. And I've been wearing things like this all the time.  Another thing I have here, another black sweater I know we're shocked. It's really cute. I love the little button-up sweaters. These are just super cute with jeans. They're so cozy. I love this. This is exactly like what I am looking for when it comes to a sweater. This tank top is just like a really simple white tank top. This is more so to just wear around the house. I'm always looking for cozy, comfy things to wear around the house because I don't really leave my house often. And I wouldn't even really leave my house often if we weren't in a pandemic anyways. So I'm always looking for things like this. I got this like cute little lounge bra and I want the matching shorts and the matching cardigan. They were out of stock. So I ended up just like whatever I got it and I'm gonna eventually get the pants. But this is just so cozy, so comfy. Even wearing this with like sweatpants and like a little cardigan over it. I'm into it. I think this is super cute. Oh, I have another one of these like white little lounge sets. So love this. I have a beige one, but these are just so soft and cozy and just like the perfect outfit for lounging around the house or just even having a cozy day. You guys know me. I'm always about lounging. I'm always about being cozy. And this is freaking cute. So I've been loving sets like this and I'm really excited because I haven't had a white one. And this isn't like stark, stark white. I like this white better. You know, sometimes I feel like whites can look cheap in this material and this doesn't look like that at all. Okay, we've got this really cute black dress and you tie it up. Okay, you guys will see this better in the tryout portion of it. But I like how it's not too thick. It's great to layer over tights and boots in a coat. I think that will be really cute, but I just think it's really flattering. And then I can also honestly wear this year round. It's a really good like staple piece. I mean, look at how beautiful this is. This like come on. So freaking cute. Okay, I love this set. I've never had anything like it. I've really been getting into browns this year. It's kind of like my new thing, you know? But I love this. I think it's so nice. And then it's just absolutely beautiful. Like look at that. Okay, I finally got a new white cardigan. I've had one from Urban for like two years. It's just so soft, but I need to get rid of it. It's held up well over the years, but it needs to be, you know, tossed at this point. So I got our placement in this one. It's just so beautiful and so soft, so cozy. I really love it. I feel like it's like the perfect fit too, which is why I really, really love this piece. And I feel like this is gonna be my next cream cardigan for the next like three years because these last. That's why I'm always down to like invest in these because I just know how much wear I get out of them. I got this like brown-ish, nudish bodysuit that I think is so cute. I love the detailing right here. Like I just think that that is super fun. I also appreciate it because I have like a larger chest. So I feel like it's not as like, you know, out there. Okay, so I got just like a really basic black sweater. This one has some cool sleeves. I like a little hoof action that we've got going, but this is just something that I will definitely get a lot of wear out of. Again, I think fall is great because a cool sweater with jeans just kind of looks chic and it's like you're really, really not trying. But this like little pajama set, which you guys will see it's cozy, but like honestly guys, I haven't tried this on yet. So I can't really say if I really think this, but like I'm just not that impressed by the material. I don't feel like it's the softest thing ever, but maybe when I put it on, I will love it, you know? Okay, I got this beautiful green dress. Not that I'll be going to any holiday parties this year, but if I was going to a holiday party, this would be the perfect outfit. I just think this dress is like beautiful. I love this green. I just, oh guys, it's so cute. You can also like really dress this down. I know that you're like, what? But you really can dress this down. Astro ball piece. So this little body suit, I thought this was really cute. Really good just to have very versatile, something I'll get a lot of wear out of. I also love this just because I find it really flattering. So I'm excited for this. Out of everything in this haul, this is what I'm the most excited for. I love outerwear. It is like my guilty pleasure, my holy grail. If I could have 100 of anything, it would be like cool coats. And I have a really cool coat that I've been wearing a lot lately from this guided like last season or earlier this year, nowhere to be found. It's not on the website. I can't link anything. And this is what I found that's the most similar, but it's different colors. So I still purchased it. But if you guys are looking for a coat that was similar to the one that I have been wearing that I'm getting a lot of questions about, this is the coat for you. I just think it is so cute. The cool thing about these is that you can wear it over a really nice outfit or you can just wear a sweatsuit like a sweatshirt and put this on over it with some sneakers and automatically like you have a really cool outfit. The next thing I have is this little suit set which I am so excited for guys. I have been looking for something like this for so long. I wanted it without stripes, but like I found this on Francis Pauli for a really good deal. So I had to pick it up. And I just think it's so cute. The pants, this with a white turtleneck or something. Like guys, sneakers, my Yeezys. This is a look. Oh, I have two pairs of shoes also from Revolta. I forgot. So we're going to go back. The first pair you might feel like Kinsey, why do you have sandals? I know it's kind of weird, but like it just hasn't really been freezing here recently and these are cute to pair with jeans and a sweater. But I just love these shoes guys. I think they're so cute. And I'm like really picky with sandals. I actually don't like sandals. I'm very picky. And I love these. They're Ray, which I always love for issues, but I just thought these were so cute. I also got some Ray boots. So I needed just some black, sleek like pattern leather boots. These are pointy, which I love. My other ones are old and they're Steve Madden and they're rounded toe. And they're just like, they weren't really what I was going for anymore. So I gave them I think to my sister and then I just got these new pair because they're just like more kind of like mature. All right, so that is it for today's video. I hope you guys enjoyed. Don't forget to check out Thread Wallets. I will have it all linked down below as well as a few of my faves. Everything will be linked down below. I hope you guys enjoyed today's video. Subscribe because haven't already. Hope you're enjoying the vlogmas. I love you guys so much and I will see you tomorrow.