In this presentation, we will create custom reports related to the restricted items. In other words, we're going to be tracking the restricted items with custom reports that we will be creating. The ability to create these custom reports is a great tool within Xero and a distinguishing tool that Xero has that many other accounting softwares do not have.  Here we are in our not-for-profit organization dashboard. Let's take a quick look at Excel to see what our objective will be. You'll recall that in Excel, we're entering this information into like a trial balance type of format, which would be here and then we're also creating our financial statements from it. So if we go on over to the financial statements, then we're going to be having two columns for the statement of activities or in other words, basically the income statements. And we could break those out to items with restrictions and those without restrictions. So we saw that last time we created a custom report that does that. A worksheet report works great. Now we want to think about, well, what about the items that are restricted? We have more than one kind of restriction. Can we make another report that will support this number and tell us what types of restrictions we have? We can and let's do it here. So we're going to go back on over to Xero. Let's open up our reports. Let's go to the accounting dropdown. I'm going to open up a standard income statement. So let's open up the standard income statement. Let's see what that looks like. It's for the year 2020. So here's going to be the standard income statement with one column. This is going to be including all items that we have thus far. If I select the reporting settings dropdown, we're not restricting anything with regards to the unrestricted and restricted items here. This is our filtering field that we can use to create custom type reports and filter these reports.  I'm going to go up to the tab up top. I'm going to right click on that tab and I'm going to duplicate that tab. Then I'm going to go back to the tab to the left and we're going to then create another report. And this is going to be the income statement worksheet. Now this is a custom report we created in a prior presentation. So if you haven't seen that, you can go back. But we're going to do a similar kind of customization as we did in this presentation just for a different factor now. So this one, we took that standard income statement. We edited the layout as we will do in this presentation. And we broke out the unrestricted and the restricted items basically with those filtering fields that we had before. So now we have a report that looks much like the end result report statement of activities report except that we have these unrestricted columns over here and we're using that. That's why it's a worksheet report. So that's what we have thus far. Now we have the restricted items here. These restricted items to 108, which include two different things in the restriction. So we might want to run another report and say, well, what items are included in those restricted items? So let's do that now. To do that, I'm going to go back up top. I'm going to right click on this tab again and duplicate this tab. And then we'll start off with another basically just a fresh income statement and then we'll adjust it. So we're going to go to the accounting dropdown and let's open up another income statement. We have a clean income statement open, then the worksheet, and then we'll open up another income statement. This is the one that we will actually be changing at this point in time. Now remember the changing features we're going to use are basically if we go to the report settings on the right hand side, the items on the unrestricted and restricted. These are the segments that we have set up for the tracking features. So these are the tracking features that we have set up. We want to look at the restricted items and we basically want to be breaking out between the government and the time restrictions. So that's going to be our objective. So let's close this report. And then what we're going to do is I'm going to go down to the edits layout. We will edit the layout. Next thing we're going to do is be adding a column. So we're going to go back up top up here and say where it says columns, I'm going to add a column. So we'll say a column. We want to have a restricted items. That's going to be the tracking feature that we have been using. So a column related to the restricted items. I'm then going to choose one of the two restrictions. That's going to be let's choose the government grants education here and we'll choose that item. Then I'm going to change the name. I can't do it here for some reason, but I can do it in this blue area. So I double click in the blue area and I don't need the date. I'm just going to call it the government grants there. So we're going to save that. That's going to be the one column we want. Now let's add another column. So I'm going to say go back up top and add another column. And it's also going to be in the restricted items. And this one, we're going to say that let's pick the other, the other item, which is going to be the time. So time. So there we have that. And so that looks good. And then I'm going to change the name to just be time. So I'll double click on this item and delete the year just having the time. Now I actually don't want the total column over here. I don't need the total. What I'd like to do is total these two up and I can't do it with this column. So I don't want the whole income statement numbers.  Then we're going to add one more column for the total. So I'm going to add another column and say we want another column for the restricted items. And then I'm going to choose the government and the time. Now I'm not going to choose unassigned because the unassigned and we might go back and adjust some of the transactions related to the unassigned when we do the expense allocation. And you can think about how that works a little bit more. But because we created two separate categories for the government grants or the restricted and unrestricted, we would have to have something that would be assigned in each one. So anytime I have something that's unassigned, it's going to populate in here when we put it into the data input screen. So I'll explain that in a little bit more detail when we go through the allocation of the expenses. But at this point in time, not going to include the unassigned. There shouldn't be anything in the unassigned at this point. And if there is, if we didn't assign anything, we'll be able to check that because what we're trying to do is tie this out to what we saw on our income statement worksheet. So let's check this out. I'm going to say OK. And then let's double click on this one. I'm going to say this is the total. This is like the total. Let's do that one more time. I'm just going to call it total. And so there we have it. So now let's take a look at it. Let's say done and see what it looks like so far. So there we have it breaking out the time restriction, the government's grants, the totals. Then adding up to that 267. Now it's good to add these across because that total column is kind of calculating independently. So we can kind of, we could double check it basically by calculating across. I'll take my giant calculator here and make it a little bit smaller. And then we're going to say that if I take the totals 159000 plus 108000 we get the 267. So that looks good. Does that 267 tie out to our worksheet? So let's go over to our worksheet over here and check out. We're looking at this restricted items. The restricted items should be at the 108. So notice it's picking up that 225500 which is unrestricted. It looks like I'm going to go back on over here. You know it's got this 159. There's a problem with that number and it's actually on the worksheet report. So if I go back to the worksheet report, this 159 is in the total column but it's not allocated. It should be in the restricted items. So let's go into the editing of this report and check that out. I'm going to check out the edit of this report and see why it's not allocated to one of the columns. If I go into the restricted items here and I select the dropdown, notice in the restricted items this should be including both time and the government education. And it wasn't including this item. So I'm going to say okay. So now this restricted column is including all restricted items at this point or it should be. And so that's going to be our total. So let's go done here and see what we have. Okay so that makes sense and that's why you should want to kind of add them across here. So the restricted items should include the total. So now these all add across this way and that's going to be the 159 is now included. The restricted column now adding up to the 267,000. So now we can look at our worksheet. We say okay here's unrestricted restricted. This is not a too overwhelming worksheet. And then I can say okay if I want more detail on the restricted items, we're going to go into this report. So we'll then move over to this report where we can see that we have the items of the 267 broken out here. Now I think we made that adjustment before. I'm going to go back to this worksheet and say I think what we did not do, what I did not do is save it as a custom report. So I'm going to re-save it because I changed the formatting. I changed the layout. So I'm going to say save this report again. And so hopefully if I keep the name the same, it'll overwrite the other one. So I'm going to say save and there we have it. So hopefully we won't have to make that change again. And then if I go back to the new report we made here, now we might want to adjust the name here. So this is going to be the restricted, restricted items. So restricted items. Now note when we're running the restricted items, we can run it for the current time period, the current year in this case. And that should then tie out to the restricted items on the income statement here. If we want to also tie out to the equity section of the balance sheet, which we'll talk more about later, then we would have to run this report for the life of basically the restricted items to tie out to the equity section of the balance sheet. So just keep that in mind. We'll talk a little bit more about that as we go. But this will then be our custom report. Now let's go ahead and save this report. So we're going to say I'm going to make this a custom report. And I'm going to make it called restricted items, restricted items, and then I'll save that. And then let's just check it out. I'm going to make a new tab by right-clicking on this tab. I'm going to duplicate this tab. We don't need this open over here. And then I'm going to go back to the left tab and just check out our custom reports to make sure that it has shown up in the custom report. So we're going to go to the reports over here. See what we have in the reports. Opening up all of the reports. We're going to go into the custom report tab because we just made another custom report. And it does look like this one has been adjusted for us. And then now we have the restricted items. So again, these two are probably going to be used a lot. This one will probably be used all the time, might even be used more than the income statement if we were doing this in practice. And then this one might be used a little bit less because of the restricted items, but probably enough that we would want to put it on our star for one of our favorite reports. That's what the little star means. It's our favorite. And that means it's going to be populating if I hit this dropdown. So now I have it in the restricted reports. And now I got my balance sheet, my income statement, and my income statement worksheet, which will show us all the detail of unrestricted and restricted. Well, not all the detail. It'll show us the categories unrestricted and restricted. And then we can break out the detail of our restricted items reports. Now, later on, we're going to make one more, at least one more key report here that will break out the unrestricted items. And that will be, if I go back to our worksheet reports here for this column, but we won't do that until we apply out the expenses. So once we apply out the expenses to their categories, that's when we're really going to be in need then of seeing the more detail of the unrestricted columns. And we'll make another report for that. Now note that other software such as QuickBooks Online type of software, you could track all this with a class tracking feature, but that you're more limited with the reports, meaning you can get all the information on one report, but it's all going to be on one huge report, typically. Desktop version allows you to kind of collapse the columns, which gives you a little bit more leeway with it, but doesn't give you the flexibility that this does. It really allows us to kind of make the reports a lot more fully within the database system so that we can basically have them pretty much good and go to print out and give to somebody other than some kind of terminology things like the name of net income and whatnot. So there's more customization with the reports. Now obviously we could always still export the reports to Excel and minimize some columns. And there's pros and cons to either method because the customization feature being more flexible means that it's a little bit more complex to make and you got to make sure you organize your reports. And to have it all in the database system to me is pretty nice. So I'm liking the editing of the reports feature within Xero, a lot more flexibility with it. Anyways, that's it going to be it for now. Let's get out of here.


Again, thank you for your patience with us. So I apologize to those of you who are out in the webinar land that you couldn't hear. We just talked a little bit about kind of the overview for the day, and I was talking about Common Core State Standards, and in particular, which Common Core State Standards, in terms of the content, this webinar will hit. So if you look at the – it's basically talking about functions, but in particular about recursive functions, and we're going to think about writing arithmetic and geometric sequences from real-world problems. So I'm going to keep moving. So as we work through the math today, what I'd like for us all to think about is to think about how your students will approach these problems, to anticipate their questions, to think about possible roadblocks and how you might push them forward in relation to the particular application. Also, if you want to think about how students and the more students centered learning, we give kids an opportunity to talk to each other and come up with their own ideas, and then you as a facilitator, oftentimes, you know, you walk around, you listen to what the kids are saying, you look at the different kinds of solution methods, and ultimately when you're going to summarize the results, you have to think about how they're going to share those results out. So in terms of how we kind of wrap things up after we let kids explore ideas, how can we present the solutions in a way that helps the kids make connections both across the particular topics or those representations that they might choose, but even in a broader spectrum, other topics in the common core or even other applications. So let's start with the problem. So you guys here with me have a handout that has this problem on it. This is a pretty simple problem. It might be kind of a warm-up problem in terms of how we might think about this as a recursive problem. So we have Selena, and she's going to buy a new smartphone, so she's going to borrow some money from her parents. My daughter just did that. I have a teenager who borrowed some money from us. She wanted to borrow $250, and she's going to try to pay them back $20 per week, and we just want to know how long it'll take to pay off the loan. So it's a real simple problem, but what I want to do is just have you guys think about how your kids might approach this problem depending on where they are and also think about the different representations they might use. So I'm going to give you a minute if you want to talk to each other, those of you who are alive here with me. Again, real simple problem, but think about the different ways kids can think about representing this problem. So I'm just going to be quiet for a minute. So I think here everybody, so folks are still chatting, but I just walked around just a little bit and listened to some of the things that you guys were saying. Some of you were talking about how the kids would just pick up their calculator right away. What would your kids do if they picked up their calculator right away? Subtract what? Twenty. Twenty, just one time? No. Lots of times. So they could use the answer key, for example, on their calculator, or they could just take minus 20 and keep hitting enter. So you could just do it numerically right away, right? Did anybody think about writing equations? Anybody write an equation for them? Yeah, so we might think about creating a particular, what kind of equation does you write? A linear equation? Okay, so there are different ways to think about this. What I'd like to do is think about it kind of an in-between stage, not just taking numbers and subtracting 20 from it. We could certainly do that. Now I have an emulator on my computer, and I'll show you how to use the answer key. You can do this just on the home screen with the answer key. No problem. But if you think about it more from a more general aspect, again, this is the way we talk about it as teachers and then you think about how to implement it in your classroom, we might start with an initial balance of 250, and you're going to write an equation that says, my balance this week depends on my balance last week, minus 20, which you might be thinking is pretty much what you're doing on your calculator, but I'm abstracting it a little bit, so I can think about it in terms of an equation. So if I'm going to do this, what we might do is just, if we're just subtracting 20, you might just build a list of numbers like this, and then just keep on going until you hit zero, right? Are you going to hit zero? You don't hit zero, which is another issue, right? So there's some interesting details in here. If you think about, actually, the notation-wise, how we might represent that recursive idea, that recursive idea by recursive idea, I mean, how much I have this week depends on what I had last week. So it's not a big picture idea. It's a very local idea. What I have now depends on what I have before, minus 20. So if you think about notation, we could use function notation. I'm going to show you that. We can use subscript notation, or we can even use this idea of now versus next. Does anybody in here see that now versus next idea? Okay. And then as we think about notation, again, when we're thinking about how to present this in our classes, you need to think about what's going to be obstacles for your students, what's going to make things more easier to understand, and then which of those notations might help us build a deeper understanding of not only this context, but again, the concepts, I'm sorry, but the concepts in a broader notation or idea like in terms of function. So what I'd like to do is think about it in terms of function. So if you write a function B where B stands for balance, and I start thinking about how much Selena owes when she bars that $250 for her parents, we can say B of zero is $250, and then B of one is B of zero minus $20, or like we said before, we could say this week's balance is last week's balance minus 20, but I'm given some kind of mathematical notation here. So this is one way I see recursive equations expressed. And that's the only way. I'll use subscript notation later. And again, depending on where your kids are, you might build this from the ground up using now and next. Okay. So if we keep writing these equations, I can write B of zero is 250. B of one is B of zero minus $20. But now what I'd like to do is step out of that just a minute and think about what would happen if I start kind of generalizing some recursive equations. So before I do this, let me just escape out of here. And let's go to our calculators. I have a calculator here. Hopefully some of you guys have calculators with you. And hopefully those of you who are watching far away also have calculators. So I'm just going to go to the calculator here. And I'm just going to say we could say it's 250 minus 20 if I do it on the home screen. And then I could say, for example, if I wanted to use the answer key, I could say, second answer minus 20. So this is just using the home screen. No fancy notation. And I can just keep hitting enter. This was the idea that was presented at first, maybe not with the answer key. Or I'm going to go ahead and show you how to use sequence mode on the calculator with this simple problem. So let's hit mode. And normally you're in function mode. So I'm going to arrow down here to where it normally says function. You have four different modes. You can be on the 83 or 84 calculator. You can be in parametric mode or polar. And what we're going to do is use sequence mode. So let's hit enter. Well, if you don't have a calculator, there are a couple here in the folks in the room if you want to borrow a calculator. So what I'm going to do is talk a little bit about this notation because it looks similar to what I have on the slide. But it's got some variables. It's got this n min and u of n and u of n min. So if I go back to the slide, I said b of 0 is 250. And b of 1 is b of 0 minus 20. So if I think about these recursive equations, n min is where we start things, the starting value for the n, which is going to be 0. So I'm just going to type in a 0 and hit enter. And then now I'm going to type in, sorry for some reason I'm not in the right place. Hit y equals. That's where I am. Sorry. So if you hit your y equals, type in 0 and hit enter. That's the n min. And then here what I'm going to do is say that b of previous value minus 20.  So what I'm going to do is I'm going to use the special u on your calculator, which is second seven, the blue u. And I'm going to open a set of parentheses. And I'm going to say my value now is my previous value. So n minus 1. Now the way I got the n was I typed in. I hit the x, t, theta, n key. That key on your calculator is tied to whatever mode you're in. So those of you who are in the room, don't look at the right hand side. Look at the left hand side. I think it'll be easier if you see what I'm typing. Let me clear this out here so you can just see the screen. There you go. So I'm going to do u of n minus 1. That n, again, is tied to the x, t, theta, n key. And I close my parentheses. And that's like saying previous value minus 20. So I hit Enter. Is that OK with everybody in here? Are we good? Anybody need me to go back? I'm happy to go back. OK. Now u of n, then, is the starting value for the balance. What's the starting value for my balance? 250. So I'm going to type in 250 for that. And what I'm doing is that using that recursive idea that the value now is my previous value minus 20. Is that OK with everybody in here? We're good. So now if you hit Second, Window, which is table set, I can look at a list of numeric values. I'm going to start my table at 0 and have my delta table as 1. And the other two parameters should be auto. And now when I hit Second, Graph, which is table, I should see a list of my balances for my account. So you can scroll down. And those are the numbers that were being generated on the home screen. And I'm going to go back and talk about notation. I introduced the idea of function notation because your calculator makes you do function notation. OK. If you want to look, let's go back then to this idea, that in terms of trying to create a closed form expression, which some of you said you did, you went right to the linear expression. That might not be intuitive for the students to say, OK, I'm going to go straight to the linear expression. But this idea of generating one value based on a previous value, I think a lot of times is more intuitive. So if I think about writing my equation, I have B of 2 is B of 1 minus 20. But what I'm going to do is replace B of 1 with the entire expression B of 0 minus 20, and then subtract 20. So if I do the algebra to simplify this expression, I have B of 2 is B of 0 minus 2 times 20, and then I can do it for B of 3, and then, et cetera. So I can think about writing, turning my recursive function into a closed form function. So I get B of n would be, if I write B of n here, it's B of 0 minus n times 20, which is our linear expression, right? So we could also create a graph over time, and it could convince you that it's linear. And you could see I haven't graphed enough to see how long it'll take me to pay off Selena to pay off the loan. So this is my closed form expression, and it's probably what the expression that you guys jumped to right away. So if you think about it in terms of your linear expression, those of you who wrote linear expressions did you have something that looks like this, 250 minus 28. So we can make sense of the parameters and the problems. What does the 250 mean? Why is it negative 20x? And why is it written this way? A lot of times we write mx plus v. In this case, I wrote it as B minus mx, right? Because I thought about it the starting balance, and now I'm going to be taken off 20 each time. So my point is that we might jump straight to the linear expression, but this idea of recursion helps the kids build the linear expression. A simple example, not too complicated. Questions in here? Any questions? What are we doing out there? If you guys have questions, Carol is monitoring the screen for me. And then we haven't really answered the question, how long is it going to take Selena to pay off the loan? Now we have all these different ways to look at it. We can use a recursive equation, we can use a home screen on our calculators, you can use a closed form expression, or you can also look at a graph. So this example, I've actually graphed all those values there, it's kind of nice because we talked about how you're never going to get to zero. None of these expressions, well, that's not true. If we use the closed form expressions, you could set that equal to zero, right? And solve for x, but the problem is you're not going to get a reasonable answer. Because these are discrete values, and I'm using a continuous function to model a discrete scenario. So again, those are big ideas that kids can think about. When is the domain discrete values and the range discrete values, or when are we going to have some kind of continuous model? We're using a continuous model to represent a discrete situation. So again, I'm not going to go through all these questions because we don't have time, but I put them in there so that if folks are watching the webinar or if you guys are thinking about it later, you can think about these big ideas and how important they are in terms of the mathematics that we're doing at present and how it connects to different kinds of mathematical ideas. Okay, we got ready for another example. Selena, she decided to use her credit card. So she spent $250, but the credit card company charges her 2.1% monthly interest. Okay? And she's not going to make any payments for 12 months. So it's not this idea that she's paying off those $20 a week anymore. She doesn't have that job. She can go wait 12 months before she pays anything on her credit card bill. Okay? So again, we'd like to think about it from a recursive standpoint. So I'm changing notation. And again, I'm doing this for y'all so you can decide which notation makes more sense for your students as you do recursive problems. So instead of using function notation, I let A to then be the amount that Selena owes and month back to the beginning of the loan. A sub zero is a $250 initial balance. And then A1 is the amount she owes after one month. A2 is the amount she owes after two months, et cetera. Okay? How much does she owe in one month? Well, if you think about how much she owes in one month, she owes what she owed before. So that's that now versus next kind of idea. What she owed before plus the interest that they're going to charge her on that $250. So if I write it in subscript notation, I'll have her balance this month is that A naught or A sub zero plus .021 times A sub zero, which will be $255.25. So that's her balance after one month. Okay? So why don't you guys take a minute to write down the recursive equations. I'll let y'all have a minute to do that. Write the recursive equation for A2, for A3, for A4. The recursive notation, you can either use function notation or subscript, I don't care, and then we'll go to the calculator and do it. Okay? So we're not jumping to closed form. We're just thinking recursion. Now based on, or next based on now. I like to think about what I owe now is based on what I owe before. Those of you who are in the webinar land, if you want to either pose some answers or some questions, please feel free to do that.  Yes? No, I can just repeat it. So there's a question here about Math 1 and what kind of notation. There was a question about whether or not in Math 1 we should just stick with the next and now versus introducing these other kinds of notation and I think you're asking my opinion on it. So I think in terms of where you start next and now is a good place to start because it doesn't kind of, you know, muck up the waters with issues that kids can have in notation. But I think, again, depending on where your students are and probably if you thought about function notation, if kids understand function notation, I would probably push you to think about at least starting the function notation for the linear expressions and the linear functions because then they really think of it as then you can start talking about domains, ranges and how and really is an integer value for these particular cases depending on what our time is really measured in. Many of these are about how things change over time. So I would say at least introducing a function notation after the now next would be a good idea. That's just my opinion in terms of kind of building the foundation for later. I think if we stick too much with one particular kind of notation, I mean for starters it's good, but then later on sometimes kids get hooked on that notation and they're not as open to other kinds of expressions. Now the subscript I found can be difficult for students. So I would wait and do subscript at a higher level. But you'll see with the power of these problems that this opens up a door for all kinds of further study mathematics. And then in terms of this idea of just doing something over and over and over again that's what we're doing with recursion and there are lots of cool applications. Okay, so I think folks have had a chance to write down some of those equations and hopefully they look something like this. So if I have A sub 2 for example, I have her balance from before, which was last month A1 plus 0.021 times A1 and I've actually got values here.  And so we can figure out what Selena owes in any month where n is the month that I'm interested in based on the previous iteration, which is n minus 1. And so let's go ahead and go to sequence mode on our calculator and see if we can type in what you guys have written on your pages. So I'm going to go to sequence mode on my calculator. I'm going to clear out my last equation. You don't have to. You could actually just turn it off. Why don't we do that? I'm going to turn mine off so that if you wanted to keep that in there for later, you can. There's three places you can write recursive equations. So I'm going to go to the second set of recursive equations v of n and I'm going to say v of n is v. Remember it's the special v, so it's the blue one, which is second 8. Open a set of parentheses of n. n is on the x, t, theta, n key minus 1. So that's our previous balance. Plus, she's got to pay interest. And it's 2.1% to 021 times previous balance v of n minus 1. I haven't put the starting balance in there. The starting balance is 250. So I'm guessing that's what many of us wrote on our papers. Maybe use some other letters like v for balance. How many of you in here use subscript notation? Can I just see the rest of us use function notation? Okay. So if I want to look at the iterates over time, then what I can do is, again, let's go to second table set. I'm going to start at 0. And my delta table is 1. Again, that's the idea that this 1 is the increment for months. So when I ask my kids to write recursive equations, I tell them to put three things down on their paper. The actual recursive equation, how do you get one iterate from the previous iterate? Where do you start? Without where to start, you don't. This recursive equation makes no sense whatsoever. I actually got a question about this in Greensboro about how can I write a recursive equation if I don't know where to start? You can write how to get one iterate from the previous one, but if I don't know where to start, it doesn't make any sense. And then the third thing I always have them write down is what is your end measured in? Because if this is a 2.1% monthly interest, then I need to be measuring the end in months. Because without those three pieces of information, I can't make sense of what I'm doing with my recursive equation. And now I hit second graph, and there's my balance on the account. So I can scroll down if I want to. Again, I've just looked at numerical representation for this particular problem. And after 12 months, it looks like Selina's going to owe $320.81 on her credit card bill. Okay, she doesn't make any payments. Let's go ahead and use a, not a scatter plot, but a graph. And to do that, what we have to be very careful with is what we're going to look at in terms of the window.  And your life looks very different in sequence mode. Normally, we only set X windows and Y windows, but now you have a third kind of idea, and that's the value of N. So I start N min at zero, and if we're going to iterate 12 times and see 12 of those iterates, I'm going to scroll down and change this N max to 12. My plot start can be one. My plot step can be one. But now the X axis represents the number of months, since this is all started. I'm going to make that zero. Oops, sorry. I'm going to change that. I'm going to leave that back to one. I'm not quite used to this emulator. So I'm going to make my X min zero, and my X max 12, since along the horizontal axis, I have time. Scroll, keep going down. And now you can hit your Y window. So my Y minimum, I'm going to set to zero just because I like to be oriented in the world. You could start at 250 if you like, because we've already seen the values are going to go up from 250. And I'm going to make this maximum value something that makes sense in the context of the problem. Maybe let's try 400. And my Y scale, something that makes sense here in the context of the problem, zero to 40, it means just every 50, I'll have tick marks. And now if you hit graph, look, you can already see down there. There's my graph. So over time, I can see what her balance is going to be. I can hit trace. It actually works a little bit faster off the graphing window that it does in the table. Mine probably doesn't because the emulator's slow. But on your calculators, if your kids want to know how long will it be until the balance is some amount, a lot of times they can just put it in their graphing screen and just trace right across there. So I can see what my balance is at any given time. And what I'd like to do then is now think about how to write a closed form function for this outside of the idea of recursion. Again, building the ideas based on this idea of recursion. So let's go back to the PowerPoint. And now that we've looked at the balance over 12 months, let's develop the closed form function. Okay. So if you think about it in terms of recursive function, first of all, again, depending on what your notation is and the level of your students, you might be able to collapse the terms that a sub zero plus that point zero to one as one point zero to one. That can actually be confusing for students because they don't see that a sub zero as some parameter plus a number times a parameter. But I think if you take some time to develop the notation and help them see that there's a connection there, I can collapse those terms. So there's a one by collapsing the a zero terms. There's a two by collapsing the a one terms. But what I'd like to do now is replace, if I can use my pointer here, I'd like to replace this a one with this expression above it. So if I replace a one with this expression above it, I'll get one point zero to one times a one, which is one point zero to one times one point zero to one times a sub zero. And that low on the whole back guy right there, I can write it as a square term. And I can do it again. So again, let's do it slowly. One point zero to one times a two, it helps to use maybe different colors if you can do that. So I'm going to replace this a two with the entire expression that I have on the line above it. And I get one point zero to one cubed now times base of zero. Do that a couple times, and hopefully the kids start seeing the pattern. And what you'll do is hopefully be able to see that this turns into an exponential function. So this idea of repeated multiplication gives this exponential function. The idea of repeated addition gives this a linear function. So again, these recursive equations help us build a function and make sense of what we're doing instead of just, you know, memorizing our p, you know, our p equals our principal times, eventually exponential functions times rt or p times one plus r raised to the nth power. A lot of times kids will memorize that, and they don't really know where that comes from, but I think this helps them build some idea of where that comes from. So what we could do is we can go ahead and type this into our calculator in closed form. We're going to have to go back to regular old function mode. So let's go back to regular old function mode and type in our exponential equation.  I'm going to hit y equals. This is the life we're normally living in. Sorry about all these equations. I was doing some crazy stuff. Okay, so what I'd like to do is type in 250 times 1.021 raised to the x times. I can hit my window button, but if my window is still the same window that I used in recursive mode or the sequence mode should be fine, right? We look at this over time. If I hit graph, there's my graph. Is there anything surprising about the graph? What do you think your kids would say about the graph? Yeah, it looks linear, right? Here's this exponential function. We've been trying to distinguish between linear and exponential growth from this looks linear. Anything else they might say about it because of the packet looks linear? Yeah, it looks continuous. Again, this idea of using a continuous function to represent a discrete situation. Maybe it's not the best model. Maybe I can only look at this for some particular n values. So the domain of my function, in general, if I just hand you an exponential function, the domain of exponential functions are all real numbers. But in this context, when I've used this closed-form function to represent a discrete situation, in this case, I only want n values that are integer values because of the way I set up the function. So what can I do to help kids believe that this really isn't a linear function? What's that? Yeah, I could change the window. Other things? Yeah, exactly. Because, again, this is the idea that the repeated addition built a linear function. This is repeated multiplication. So if I look at the list of values, and I built the function so I know how it was built, right? So later on, if you're thinking about helping kids distinguish them, building this function will help, I think, later on as you give them different kinds of scenarios and different kinds of graphs, and say, this is my actually written function. Can you tell me, or even a list of values, can you tell me is this a linear growth or an exponential growth? That's a really important thing to be able to distinguish. And a lot of times we'll say, oh, when you learn about logarithms, you'll understand exponentials. That's a long time to wait, right? We want to really think about trying to help them understand it now while we're studying it, not wait six months down the road when we know more about logarithms or another year down the road when you study logarithms in a different course. So we could expand the window. Let me go ahead and do that and just look at it over a longer time span. And I might need to increase my Y values as well. If I wanted to take a little more time to do this with my kids, then what I might do is use the table feature on my calculator, excuse me, and look at more iterates over time. I'm not convinced by that one. Maybe I shouldn't have changed my Y window so much. So you can play with a window in terms of seeing that. I'm not crazy about, I mean, I know the calculators are what kids have in their hands, but it would be nice to be able to see all these representations at the same time, which they don't have the ability to do on the calculator. I think I have a graph that I produced in my PowerPoint that I use a different tool for. You can also, so there you might be more convinced that it's linear. The other thing you can do to convince students that the graph isn't linear is to slap a line on it. If I were to put a linear function on this, like, for example, do a linear regression on the data point, you'll see right away that there'll be a pattern in the residuals and you'll be able to see that this line isn't an appropriate model. And even if you haven't ever heard of residuals, if you graph it over a long period of time, you'll see the growth exponential versus linear growth. And in terms of other tools, I just wanted to mention something else too. Depending on what your kids have in their hands or what you have in your classroom, you might want to use Excel or some other spreadsheet because spreadsheets are really easy to do recursion in and then you can produce some nice graphs. Okay. So again, I just read some questions to consider as you plan to think about the different kinds of technology and the appropriate tools and to reflect on the mathematical model. We're thinking about arithmetic and geometric secrecy. Okay. So I have one more example for you and I think we're going to get on time. Here's a problem that's kind of interesting. It's about cadmium. So cadmium is a metal used to manufacture batteries also found in cigarette smoke and the way you can absorb cadmium through cigarette smoke is actually through inhaling it. So over time, what we've seen is a person who smokes one pack of cigarettes a day will absorb about 2.7 milligrams of cadmium each year from smoking and let's assume your body can eliminate about 7% of the cadmium each year.  Again, these are basic numbers that I got from... I actually got these numbers from a book by Jim Sandifer. I've got him as a reference for recursion. He's at Georgetown University and actually teaches a lot of cool high-level mathematics but has realized that some of those ideas are really important to introduce at a different level as early as Algebra 1 and he has a lot of nice problems that are real-world problems. So I got these numbers from one of his books and I want to know how much cadmium with a smoker who smokes one pack of cigarettes per day have inhibited her body after three years of smoking. So what I'd like for us to do is to write a recursive set of equations. Again, I'd say a set. I need to know what I start with, how I get the amount of cadmium for a particular year based on a previous year, and then what my end values are going to be measured in. So I'm going to give you a minute to write those equations and I've actually left a blank slide here so I can write them after you guys have a minute to think about it. Also, he suggests drawing a flow diagram if we can do it as well. So we're just writing recursive equations and then if you want to go ahead and try typing them in your calculator, you can do that too and see how much cadmium we'll have in our body after a year. So we're not trying to write first formula. This recursive formula. Okay. Okay. So it's interesting here in the room, lots of conversation going on. I'd like to go ahead and keep moving because I don't want to totally run out of time here, but I'm hearing when to put the 2.7 in and when to take the 7% out, those kinds of questions. So if I write it in terms of function, the amount of cadmium at any given time is what they had before, right? And I like to draw this flow diagram. This is one of the things that Jim Sandefur's advocate, too, is to think about what's flowing in and what's flowing out. So this is what I had before. What flows in? I'm going to add what flows in. What flows in? 2.7 milligrams of cadmium. And then I lose some amount, right? And I lose 0.7 or 7%, which is 0.07 of what I had before. So this is the amount out and this is the amount in. And I heard some people collapsing these terms. Talk about that just so that we just hushed kind of the chat in the room, which I heard people say 0.93 times A of N minus 1, which is fine. You can write that as 0.93. So if I write it as still recursive equation, I'm not talking in closed form. I say I have one of these and I subtract 0.07 of those. So I should have 0.93 times A of N minus 1 plus the 2.7. Is that okay? Questions? Any suggestions or questions about what I've done here? I wrote it as a function. I went back to function notation. You can use it as a subscript, either one. Okay, so there's a question about where to put the 0.07, I think. You're reducing what I just had before versus what I have, like you're saying I should collapse these two and then multiply by 0.07. Right, so for those of you who are far away, let me try to repeat the question. I'm going to insert a slide here just so we can go ahead and write that other representation down. So the question is where to put the 0.07. So if I say what I have now, and there's also a question about notation, if you like this notation better, this is what I had on the previous slide. I have the 0.93 times A of N minus 1 plus the 2.7. Does that answer your question? Okay. Or if you don't collapse the terms, you have what you had before minus what you lose plus what you gain. Okay, and I like to keep those of those on the slide there. Sorry about having that weird thing in the middle there. I'm just clicking and going. Okay, so the other question was when to take out the value. So for example, if I go back to your representation, I think you're saying it's A N minus 1 plus the 2.7 and then 0.93. And then where do you want another 2.7 in here? No. That's it. Okay, so this right here is really kind of where you start counting, right? I think that's what your question is if you say, let's think about starting time right now. What I have right now is what I had before minus what my body eliminated last year. That's the way I'm thinking about it. It's what I have before minus what my body eliminated last year. Right? Does that make sense? No? Okay. Plus the 2.7. I think what you're saying is it's what I had before and then at the end of the year I'm going to put the 2.7 in there and take 0.93. Okay. So it actually can be interpreted either way. It's perfectly fine to interpret it either way. We can look at both ways and see if it changes the solution of problems, okay? So you can look at it both ways. I can't see anything wrong with either one of those ways. Does anybody in here see anything wrong with either one of those ways? Yeah. There's a comment in here that says a smoker that would prefer the second way because you're losing more than the other way. Okay. So I'm going to go with my way because I'm the one with the chalk in my hand as my colleague. So I'm going to use my way but I just invite you all or whoever's interested in the other way, go ahead and put your calculator and let's see if it makes a big difference on, again, interpreting the problem and how you look at the problem. So let's go back to the calculator. Are there any questions about either one of these representations? Again, I started out with function notation but then I moved over to subscript notation. So let's go to the calculator and type this in our calculator. I'm sorry, I don't know what I'm saying. We're going to go to mode, back. I have to switch back to recursive mode which is sequence mode and hit enter. Hit Y equals. I'm going to go ahead and clear out this one so I can use the first one and then clear out this one.  So I'll help you with that just a second. I haven't come across it. Nobody's hollered about it yet. So what I want to do is go ahead and collapse my terms and type in 0.93, just in the interest of time, u of n minus 1 plus 2.7. And then now this is a question for you. How do you want to start? What starting value do you want to have? Yeah, we could say you could start at 0. I mean, there's probably a chance that there is some cadmium already in your system, right? But we don't know what that is unless they can measure it. Which they can, by the way. So I could just start at 0. Some people might say something different here. What else could you say here if we don't know how much cadmium you already have in your system? What's another alternative because of 0? Yeah, start after the first year in which case you could say the 2.7 is from that accumulated previous year. So again, either way is fine. We just need to think about kind of where we're going to start and how we're going to start counting. So if I hit second table set, that should be at 0. And then we're going to look at the table of values. And if it's just a short period of time, then again, depending on what you start with, are you going to say I'm going to start after the first year. Your answer could be the 7.54 or this 9.7, right? So there's some room for interpretation in the problem. I think that's okay. There's room for interpretation in most problems in our lives. There are very few problems that are well posed that have everything expressed for us down to the itsiest, fittiest detail that we want. And I'm adamant about this because we have to be flexible in the answers that we receive and we take unless you want to prescribe every little bitty piece of information, in which case you can say, okay, now you got to pay attention to that little bitty piece of information. I want you to do this problem based on this information. So I'm about right out of time. Of course I have other fun things for you to look at. There was one question about what tool I used to create that particular graph. That was called Fluid Math. Fluid Math is a program that recognizes your mathematics handwriting. Notice I have a stylus because I teach with a tablet. And so you can actually write mathematics on it and it recognizes your mathematics handwriting. And you can get a trial version if you have anything that takes stylus input.  So let me go back to regular life and just show you a couple of these resources and hopefully I've just kind of wedded your appetite in terms of recursion because there are lots of different applications. There's another problem here. We go back to Selena and she's got this credit card bill but she is going to make some payments. If you'll notice the last example, this is an idea of mixed recursion where we have both the geometric idea where we're multiplying by something and adding. The closed form for this is actually pretty complicated. You need to be able to write some of those terms out and to see the pattern you'll actually see a geometric series in there. But again, it opens the door to some really powerful problems whether it's paying off a loan or this idea of, for example, if you sprain your ankle and you keep taking ibuprofen, how much ibuprofen would you have in your system over a particular time? And it will reach some equilibrium level. So we have lots of materials for you. Let me share some of those with you. Sorry about kind of rushing through this. NCSSM has developed lots of recursive equation materials and even some videos on how to use them. These are brand new materials that were created over the last year. And NCTM has an illumination site that has some really nice applications. I believe one of these generates the closed form for that mixed recursion, adding and multiplying. And it shows you that you use a geometric series for that. So if you teach a higher level mathematics, maybe in pre-calculus, and you want to use a geometric series in a real-world problem, that's a great place to go. Neal Abel is a teacher at Deerfield Academy, and he has written materials that are free for his function statistics and trig courses. He uses Excel, he uses Fathom. So some of those things, you know, he's got actually little videos on how to do things. All up on his Moodle site, and you can just log in as a guest. And then Jim Sanderfer is a great resource. He's at Georgetown University. He has a lot of great materials through his hands on math. Sometimes you have to ask him permission to use his materials, but he's always happy to share his stuff with high school teachers. So our next webinar is on ratio and proportion leading to slope, and that's November 6th. I'd like to thank all of you guys that are here live and all of you guys that are out in webinar remote land for this opportunity. We're going to have probably six of these, and they'll be on various topics. This next one's on ratio and proportion, because I wanted to do something for the middle school teachers thinking about how to develop the idea of slope. The one after that I think will be on linear data analysis, the nonlinear data analysis, and then building functions for various real-world scenarios. So those are kind of the ideas for the school year. We'll probably have one more in the spring, again for some middle school teachers. But thank you guys all very much for coming. I appreciate you being here live, and I appreciate all the folks out in our remote folks. If you have any questions, send us an email. Carol is going to send you all an email asking you for, well, giving you one thing, and that'll be the site that has all our materials available for you when we get that up and running. And also Carol will send you a link to a survey to ask you how this went, what could we do better. This is all experiment. We're flying by the seat of our pants, and we're hoping that it's useful. So if you can have any ideas about how to make it more useful, please send us that information. Thank you. Thank you.


What is going on guys welcome back to the channel critical overlord here in this video I want to talk about how ghost face and our new ghost face killers in screen 5 the chance chances or possibility of them being from the dark web This is being inspired by all these videos. I watch from youtubers like llama arts and mr. Nightmare They put out some of the best Animated shorts animated horror stories or just short horror stories in general where there has to do with the dark Web or anything. I love those two channels if you want something that's I hope fictional and Something that could just scare you with with their voices and how they tell these stories I would check out those channels if you haven't already llama arts and Mr. Nightmare they put out amazing horror shorts and some of them are animated and I love those channels And I'm really interested and I love the the ones they do about the dark web So that's why I have some inspirations Coming from them now. So I've been thinking what if we have a ghost face Killer or ghost face a pair a trio of ghost face killers because again I want to have three killers in screen 5 if talk about that several times my other videos What if you have killers from the dark web and then of course in my screen 5 as you all know at this point I'm basically being told I have a heart on when it comes to Kirby I just really want Hayden pants here to get that character back. I want to see her back in that role I want us to find out that she survived her stab wounds and they were not that severe because they were able to save her in time Before she bled out but Getting into what I want to talk about. Let's say that again going back to the Kirby thing We're gonna talk about Kirby being at the center of it. Let's say Kirby Scream in screen 5. This is said a few years after after 4 and Kirby is now gone to college and she's graduated She lives outside of Woodsboro just a few towns below Woodsboro or a few towns away from Woodsboro rather and Will catch will She's she's basically living a seemingly happy life at this point She did go through some depression over the events of screen 4 and her now being in a similar position to Sydney Prescott, which is why she would be a very easy person to sympathize with in a screen 5 She's now felt what it's like to have a lover betray you from Charlie and Also the death of a best friend who also betrayed you in jail. She has that Sydney she has that Sydney aspect to her now because Sydney went through that in the first screen movie and Sydney is a sympathetic character as well so She's living a seemingly happy life. She has a boyfriend But by the end or over the course of the movie We'll see that she is acting a little odd when it comes to this killer. So What will end up finding out in the end is during her high school years are not high school years But during her time in college her stint in college for four years. She did dabble in the dark web She had a couple of friends along with herself and they dabbled in the dark web They ventured out into the into the dark web, because you know what Kirby is established as being a massive horror freak She's not afraid of anything. She's always looking to Find new ways to terrorize herself because of other things that scare things that scare other people wouldn't scare her and I could see Kirby's character going into the dark web. I could honestly see that so She ventures out into the dark web with her friends But of course that doesn't go as planned because they end up stumbling upon a Group chat or a a live stream that comes up out of the blue when they click on a link to a website and There's just somebody in a there's a shump somebody a cloaked figure Standing there and they're trying to let's say kill Somebody who's on screen there Kirby and her friends are watching in terror.  They hold down the power button. They reboot the router They reboot they reboot everything in sight that has to do with the internet Um They think that that's that's all there is to they think that's it there. They think they're safe They think it's over a few days later those two friends of Kirby's they're getting strange text messages Kirby at this point is not getting anything but Because this person that's gonna be ultimately revealed as the killer in screen five. They're strategically planning something for Kirby specifically This person here or her two friends rather now are getting strange messages two days later after they just watch someone get executed On a live stream in the dark web They're confused and they're going to Kirby and Kirby doesn't understand what they mean. She hasn't gotten any messages. So Then her two friends end up going missing and then when that happens then Kirby gets her message She gets a text message from a blocked a blocked number or just from some random let's say Skype caller or some some type of app that would just be from a random entity a fake account a fake profile or something and It's saying Your time's coming soon and then from there. That's it. That's all Kirby hears She goes through the rest of her time through college. Nothing happens to her still doesn't know what happened to her friends and She graduates college. She moves on with her life still living in fear She doesn't know what that text ever meant still does not know what happened to her friends But still she's haunted by that text your time's coming soon or your time will come something like that So flash forward a few years after college now making it let's say six or seven years after the events are screened for And a couple years after college Kirby is Now about to find out that it is her time and that that this individual is Going to decide that her time is in fact now she has a nice boyfriend She has a seemingly normal life until she gets a phone call from her From the Woodsboro PD Late in her know because again Kirby doesn't wouldn't stay in the town of which will she lives just outside of the town But she'll get a call from the Woodsboro PD letting her know that her parents have unfortunately been killed and This then sparks her to return to Woodsboro because again in my screen videos I've made mentioned that I think would be interesting if Kirby's parents were the opening kill so instead of us getting other other people we get a group of or set of parents like a reverse of the case of Becker opening in the original screen so Kirby gets a call from the Woodsboro PD that her parents have been killed this sparks her to want to return to Woodsboro But before she even goes outside of the door to go to Woodsboro and get in her car to drive She does get a message saying that your time is now So now Kirby has an interest in what's going on and now she thinks that this person who abducted her friends a couple years ago Who are still missing? She has nothing. She still has no idea who it was. She now is under the impression that this person is possibly Now the same individual who murdered her parents when Kirby arrives they disclose all the details and She finds out that her parents were murdered by someone in a ghostface mask So now this is not sitting well with Kirby at all now She has another ghostface killer on her hands And it might be that same individual who abducted her two missing friends from college because again Her two missing friends from college while there's there's there's an investigation going on. They haven't found them. They have yet to find them They still don't know where they are But everyone's still looking for them and at some point I believe they would stop looking for a missing person at some point They basically just will stop looking for them. I I've heard about things like that So I don't know how long they will go looking for a missing Individual because these people like this one been missing her two friends have been missing for a couple of years But this was spark Kirby didn't call in the help of Sydney Dewey and Gale who come into Woodsboro to help her we find out that the individual who abducted her two friends and The new killer would be her boyfriend her loving boyfriend Who would end up being the younger brother of stew mocker? And then he would go into his whole backstory and then we would go from there I would like to see him get away with it the younger brother of stew mocker who now wants revenge for his brother's demise Not only his brother's demise bow his brother's demise affected his upbringing and now he had to watch his parents suffer because they were depressed about him about it about stew's passing and that affected his life ever since he didn't have the best childhood because of that and Basically Sydney would then become a target because she's here in town now. This was all about Kirby That that would be revealed his whole motive and everything would just be revealed by the end of them into the movie And we will find out his two accomplices were actually those two missing friends they Were recruited by stew's younger brother and he's convinced them to now help him take out his vengeance on Kirby But let me know what you guys think about all this down the comment section below and I know what is his gripe with Kirby That's still a work in progress for me But let me know what you guys think about all this down the comment section below If you haven't already make sure you subscribe turn on post notifications that never miss a video in the description I have links to my social media accounts on my Facebook tour to an Instagram You can message me there to let me know if there's any movies news or views You would like me to cover in the future with all that in mind guys. I will see you in the next video


President of the Pharmaceutical Society of Nigeria, P.S. A.N. Mazi Sam Oambua has declared his interest to contest the presidential seat. Oambua said his declaration for the top seat of government has brought to a close the erroneous impression that North-South Eastern Nigerian was showing Sarah's interest and running for the office. He promised to rewrite the story of Nigeria with a vow to make appointments on the basis of geopolitical zones of the country. When I be joined via Zoom by Mazi Sam Oambua, he is the President of the Pharmaceutical Society of Nigeria, P.S. A.N. Good evening. Thank you very much. Good evening. How are you, sir? And very well. Thank you. And you? God is with us. It's interesting you've joined the Rage to Contest for the No. 1 seat of the nation. What inspired your move? Thank you. But before I take your question, let me correct that I am the immediate past President of the Pharmaceutical Society of Nigeria. My tenure ended last November, so I have a new President, Professor C. O. O. C. 4. Okay. Apologies for that. We didn't have that information. You're the immediate past President. No problem. Yeah. That's okay. Do you see part of the SQL? Yes. I'm immediate past President. I'm a member of the Nationalization Committee. Oh, I see. Okay. I now understand why that mixup came up. Okay. That's all right. All right. You're the IPP. Yes. Thank you very much. Okay. Well, I want to say that the reason I have decided to join the Raze is because like you and many Nigerians, we have gotten tired of being branded as people who are less intellectually endowed as the rest of the world. We are tired of being put down in the global sphere where each time you look at the global indices, I talk about human development index, I talk about quality of life index, I talk about life expectancy, corruption, perception index. Most of the indices in the world we are put down. We have become a country where 71% of Nigerians are poor and getting deeper in poverty. Corruption remains endemic.  We have become a nation of terrorists, bandits, and all kinds of malfecient actors. And we are acting as if we're helpless. You've just announced those who are kidnapped. I passed through the road between Lafayette this afternoon to come to Makodeh. I'm speaking from Makodeh. I use these roads and you just hear what is happening day after day as if we're overwhelmed. Together I have waited for Nigeria to change, I have preached for Nigeria to change, I have campaigned for Nigeria to change, I have retained, I have conversed, I have advocated to go to the point where I feel that I have the key to solve Nigeria's problem, to resolve Nigeria. What's that key? You have the key to solve Nigeria's problem. What's that key? The key is first to unleash Nigeria to create an enabling environment for the proliferation of productive activities. We want to make Nigeria a productive nation where everybody, able and willing, we have a means of productive activity, either you're working for people or people are working for you. We're creating, we're building, we are growing, we are creating value, we are solving problems, we are filling needs, we're exporting, we are getting opportunities for our youth and our men and women to be gainfully employed. That's the primary thing holding Nigeria down. Poverty, we are a nation of 71% poverty, 53% unemployment among the youth. Such a nation cannot be that. The kind of nation we found ourselves, beset with corruption, beset with insecurity, and then we're a nation where there's so much injustice, where people are feeling maligned, where people are feeling cheated, where people are being ignored and mistreated, where some group of people claim that they own the country and the rest of us don't own the country. Some people acquire all the wealth and the rest of us are left at the dregs of society. This society cannot be harmonious and then you're talking about unity. You can't force unity. Every union must be mutually satisfactory. So these are the things that have key, the key to resolve. Because when we resolve poverty, resolve injustice in our nation, insecurity and corruption will quickly disappear and Nigeria will become a good nation.  They sounded very convincing like in the past. So how ready are you to take up a political job of this magnitude? A political job is no more difficult than running businesses that we have run. It's no more difficult than coordinating the activities we've done. All you need as a leader is to be able to know and create a vision for our country. We don't have visionary leaders. What is the vision of the country today? Do you know where we're going? We need to fashion a leadership of the visionary, set a vision, provide the resources and the motivation, hire the right people to work with him, sell the vision to the people and push the vision to his conclusion. So not speaking is not the issue. It's look, go back and see our pedigree. What have we done before? Where have we shown competence? Where have we shown character? Where have we shown courage? It is those things that determine your ability to build on a new assignment. Leadership is applicable. If you can lead a good boy scout group and lead them successfully, you can lead a military group successfully, you can lead a nation. The principles of leadership are the same. You must be visionary. You must be able to plan. You must be able to motivate. You must be fair-minded. You must know how to measure work and reward work and motivate work and punish indolence or poor work. We need to state a country where the laws are applicable to the up, to the down, to the point of the rich. We need to create. Yes, sir. So it is not a question of people may speak things up from what is right, it's right. The question is, are you able to do what is right? Masi. Yes, sir. You've spoken very well. Now let's, I want you to do a peek, take a peek, right? Security, poverty, poor power supply, these are some of the problems bedeviling Nigeria. Which one would you address first? Well insecurity is the first because if there's no security, nothing else works. You can't build an economy. You can't build a power plant if the power plant will be vandalized the next day. So security is the first thing. So how would you go about it? Yes, sir. What exactly would you go about? Let me tell you what I'm going to go about. The first issue, sir, let me give you an example. In my community where I come from, in Arochikun, if suddenly women, people are beginning to rape women, if suddenly houses are being broken, if suddenly people going to the market are delayed, do you know what happens? The elders of the community call the leaders, the family heads together, and they sit down in the, with their working sticks in the OB. And everybody says, what do we have seen today is not what we used to see yesterday. What has come, what has overcome us? Is this, is this, is, have the gods decided to fight us or is the problem with us? So they sit down. It's at that point that one man will say, I saw Conquers son, the other day carrying a bunch of plantain with some boys, and I was wondering why they got the plantain for. I saw some, why they were sitting somewhere and they were smoking and planning. Look, this country is owned by people, they are owned by Nigerians. If there is a problem in the country that is intractable, like Mr. President said, in the area of insecurity, we are doing our best, our best is not good enough. As the admission of the president, he needs to call the owners of the country to get and say, how do we collectively solve our problem? They say I'm robbing, not be, not be spirit. I'm robbing a person, every chief, every robber, every bandit has a locality. Get the people together and let them plan with you how to solve your problem. You can be shaking, shaking people's head in the absence.


Hi, and welcome to the session. Let us discuss the following question. The question says, find the number of four digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of this will be even? Before solving this question, we should first be well versed with theorem 1. The theorem 1 states that the number of permutations in different objects, objects do not repeat, number of different objects, number of objects taken at a time. And this NPR is equal to r factorial. We will use this to solve this question. Begin with the solution. Given five digits r, 4 and 5, we have to find number of by using these digits if no digit is repeated. And also we have to find number of even four digit numbers. You should know that here, 2, 3, 4, 2, 1, are different numbers. The digit numbers with digits is repeated is equal to number of permutations in digits taken 4 at a time. We know that number of permutations of n different objects taken r at a time and objects do not repeat is NPR. Now here n different objects are five different digits and r is 4 and also the digits do not repeat. So by using theorem 1, number of permutations of five different digits taken 4 at a time is equal to 5PR. And as nPR is equal to n factorial upon n minus r factorial, therefore 5PR is equal to 5 factorial upon 5 minus 4 factorial. And this is equal to 5 factorial upon 1 factorial. Now 5 factorial is equal to 5 into 4 into 3 into 2 into 1 factorial. So we have 5 into 4 into 3 into 2 into 1 factorial upon 1 factorial. Canceling 1 factorial from both numerator and denominator we are left with 5 into 4 into 3 into 2. And this is equal to 120. Total number of four digit numbers that can be formed using the digits 1, 2, 3, 4 and 5 and no digits repeated is 120. Now we have to find number of even numbers from this four digit numbers. We know that unit place of even numbers is even. Now the five digits given to us are 1, 2, 3, 4 and 5. Out of these five digits only two that is 2 and 4 are even. So that means we have only two options for filling the unit space. If we fix 2 at the unit space of four digit numbers equal to permutations different digits 3 at a time. And we know that number of permutations of n different objects taken are at a time. And the objects to not repeat is npr. Here n different objects are four different digits and r is 3. So by using theorem 5 permutations of remaining four different digits taken 3 at a time is 4p3. We know that npr is equal to n factorial upon n minus r factorial. Here n is for an r is 3 so 4p3 is equal to 4 factorial upon 4 minus 3 factorial. And this is equal to 4 factorial upon 1 factorial. And this is equal to 4 into 3 into 2 into 1 factorial upon 1 factorial. We can cancel one factorial from both numerator and denominator. So we are left with 4 into 3 into 2 and this is equal to 24. Similarly when we fix 4 at the unit space then number of four digit numbers is equal to number of permutations of remaining four different digits taken 3 at a time that is 4p3. And on simplifying 4 to 3 we get 24. Require even numbers 24 plus 24 and this is equal to 4p8. The required answer is 120. This completes the session.


Hello everybody hooded Cobra Commander 788 here and this is a quick compare and contrast between a vintage GI Joe action figure And it's modern equivalent since I most recently reviewed shipwreck. I wanted to look at a couple other versions of shipwreck We have version one from 1985 and we have version 19 from 2015 this version 19 was from the 50th anniversary set one special thing about this modern shipwreck Action figure. I want it in a drawing from the YouTube channel comic tropes Chris Pierce from that channel has become a good friend of this channel and I've come to appreciate his insight and advice Let's start with a quick overview of version one of shipwreck from 1985 This is the version of shipwreck that most GI Joe fans would be familiar with This is how he looked in the Sunbow animated series And this is how he looked in most of his appearances in the GI Joe comic book series It's the classic Navy sailor look with his Dixie cup hat his brown beard His overall blue uniform He has his ranking Cygnia on his shoulder and he has a tattoo on his forearm As far as accessories he comes with his percussion pistols That has a little loop that can go around his wrist He has his boarding hooks with the black string Not a big fan of this accessory, but you can wind that string around the hook That is on shipwrecks left hip. Again, this is not my favorite accessory Most of the time I have this wrapped around the body of the action figure Because it can fall off of that hook really easily and then most importantly we have his parrot The green and yellow parrot named Polly Which has feet that can clip on to his wrist The original shipwreck action figure did not come with a figure stand I'm just using a generic figure stand for this one And of course this is shipwrecks iconic look This is what most of us think of when we think of shipwreck And I do like this figure. I think it's appropriate for a Navy sailor Although this is shipwrecks iconic look It is not his only look as we saw with the review of shipwreck version 2 Where he was wearing a diving suit This modern version 1950th anniversary shipwreck figure Gives us an alternative interpretation of shipwreck It's not the only shipwreck action figure that has this general overall look But it is nice to see this different take on the classic shipwreck figure This modern 50th anniversary shipwreck came in a two-pack packaged with Cobra Commander This modern shipwreck action figure has more accessories than he can carry But let's take a look at those accessories starting with his parrot He does have an updated version of Polly Instead of the parrot at rest We now have the parrot with his wings out And you can pretend that the parrot is flying with his wings out like this So that's not a bad update He also has some additional paint applications with a few more colors He has some purple on his head He has a little bit of a white paint wash on his wings He has yellow feet So this is not a bad update of Polly Not a bad update of the parrot I do have a little bit of problem with it though The feet on this parrot are very small So it's difficult to clip him onto anything I have him clipped to this loop on a shipwrecks vest I don't know that that's where it's supposed to go The feet are a flexible plastic And the vest is also a flexible plastic Which actually makes it difficult to get the feet clipped onto that loop I've seen pictures on yojo.com with the parrot Clipped onto shipwrecks hand I did try that I was not able to get that to work And the feet are too small to clip on shipwrecks wrist As the vintage parrot does So Polly only really has a few places where he can clip onto shipwrecks What I would like to see And what neither the vintage nor the modern figure have Is a Polly that can clip on shipwrecks shoulder That's what I'd like to see I want the parrot to be on shipwrecks shoulder Rather than on his vest or on his wrist I just think that's where the parrot should be Here are the two versions of Polly side by side I was always fine with the vintage Polly I thought the parrot was fine at rest I thought this was appropriate for shipwrecks partner But it is nice to get a parrot with the wings out So you can really kind of pretend that the parrot is flying But I think the feet on the modern parrot Are not an upgrade from the vintage version The vintage version doesn't clip onto the figure shoulder either Those feet are just a bit too small for the action figure shoulder They fit on the wrist But the smaller feet on the new Polly Don't clip on anywhere very well at all You might clip him on to one of shipwrecks rifles I guess he might fit on the barrel of one of the guns But I'm not sure you would want your parrot to rest there But anyway, not a bad version of Polly on the modern figure And really I kind of like both for different reasons So I think they're about even as far as I'm concerned As far as my preferences So both pretty good parrots Shipwreck has two knives One with a paint application on the blade Another unpainted silver Actually kind of a grayish knife Both are not bad But he doesn't really have a place to put them I thought he would have maybe a sheath on the vest Or on the leg or something like that to put these knives But he doesn't really have that At least not that I can find And I haven't seen pictures of him storing the knife anywhere So the knives either have to be in his hand Or just setting off to the side I do prefer to have accessories that the action figure can carry It's just a preference for me I don't like to have extra accessories That the action figure can't holster or carry Shipwreck has two assault rifles One tan with a black paint application And another one that is all black And I guess he can hold both of them at the same time But of course then he's got two extra knives that he won't hold on to Neither of them fit in the hand very well At least on my example you can kind of make them fit We kind of have to wedge them in there So not bad assault rifles But maybe more accessories than he absolutely needs In the holster on his leg he has a very tiny almost microscopic pistol And that has an even smaller suppressor that fits into the holster Not super easy to even get out But that tiny suppressor will fit on the barrel of the pistol That is really small That's a tiny accessory I guess it's nice that it's removable from the holster Rather than just sculpted in But maybe a little too small The pistol will fit in Shipwreck's hand But it does seem a bit undersized Like most modern figures This Shipwreck figure came with a figure stand With his name on it And since this is a 50th anniversary figure It has G.I. Joe in gold letters on the top The modern figure replaces Shipwreck's Donald Duck hat with a black beanie But he has his brown beard It does look like Shipwreck It's a nice look I do like it He has this black tactical vest With a loop here on the back I will remove this vest in a moment Because I want you to see what's under the vest On his shirt on the right sleeve He has a little gold star This designates him as a 50th anniversary figure He has his tattoo on his arm On his left arm He has a black band Which could mean that Shipwreck is in mourning Or maybe he is protesting the Vietnam War The war's over, Shipwreck With that vest removed We see Shipwreck is wearing a white t-shirt With navy and black letters across the chest I really like this Instead of having his blue dungarees He has green BDU trousers I has got a brown belt with a silver belt buckle He's got a brown holster on the right leg Brown pouches on the left leg He has knee pads and gray boots And I like this This is a nice update to Shipwreck It's not the same as the classic blue navy uniform look But it's not bad It's recognizable as Shipwreck But it's updated with a bit more detail And to make him maybe a bit more battle ready As you may know, if you've watched earlier videos I am not as enamored with modern articulation and sculpting As a lot of modern GI Joe collectors are I like the classic look for Shipwreck But I also do like this modern update I think the modern figure has a nice look as well So in my book, these two figures come out about even That was a quick compare and contrast Between version 1 of Shipwreck from 1985 And version 19 from 2015 I hope you enjoyed it We do full vintage GI Joe toy reviews on this channel every week So I hope you will subscribe and check back for those Thank you to my patrons for their generous support Please find me on social media on Facebook and Twitter And I have a website, HCC788.com I will see you soon with another full vintage GI Joe toy review I will see you then And until then, remember only GI Joe is GI Joe