 Hi everyone. Thanks for joining me today. We're here to talk about volume. Now remember what volume is? Volume is a numerical measurement of how much stuff you can fit inside of something. So what you should have done prior to joining me today is each of you should have made a box. Everyone would have started with a piece of paper eight and a half by 11 inches big, sort of like this one, and from that piece of paper you were told to cut out squares from the corners of different sizes. Some of you did squares that were really teeny tiny and only half an inch on the side. Others did ones that the square was an inch on the side, all the way up to four inches. So I made some boxes of my own. I made the first one by cutting out from each of the corners a square that was one and a half inches on a side. And when you fold up the side you can see the boxes in too deep. And so those of you who made a box kind of like this one, where the squares were one and a half inches on a side, you should look like mine. I made another box by cutting out squares that were three and a half inches on a side. So this is what it looks like after I cut out my corners. And when you fold it up you can see it's much taller than the other box. The base is smaller though. Alright and if you compare these two or look around the room at your classmates you'll see that even though you started out with a piece of paper the same size, your boxes look really different from each other. So I have a computer animation that will show you some different ones side by side compared to each other. Alright so this is the first one. This sort of looks like maybe it was a box that was created by cutting out a really one of those teeny tiny squares from the corner. So you can see that's really not too deep at all. This green one maybe that's the square that's like maybe the one I did. And this last one maybe a square that's maybe three inches or so. But you can see how different they look even though they all started out with the same size piece of paper. So what we're going to do now is you should all have a handout and we're going to work our way through that handout. And the first thing we're going to do is make sure we have all the dimensions of the boxes depending upon the size of the square that you cut out. Alright so let me make sure all of you have the same handout. I have it here on my computer to bring up. Alright and it's two pages and on the second page is a graph that we'll be getting to in a minute. Okay so in the middle you'll see a table and we're going to be filling that out in just a minute. But the first thing we need to do remember how you get volume. You get volume by doing length times width times height. So we need to figure out for all these different size squares that we would have cut out what the length, width and height of each box would be. Now if you take a look at your boxes though I bet you can figure out what the height should be. The height is always going to be the size of the square that you cut out. So for all of our heights we can easily fill those out really quickly. The trick is going to be the length and the width. Alright so let's get started. So for a square that was half an inch on the side the height of the box is going to be half an inch. The length is going to be 10 which if you think about it hopefully that makes sense because the paper was 11 inches long so if you cut out an half inch on either end of the paper what you have left in the middle is 10 inches long and a similar thing on the width. So if you cut out a half inch square on either end you have seven and a half inches remaining. Alright so that's what we're going to fill out for all of these different size squares. So the next square was one inch on a side that makes the length nine and the width six and a half and I'm just using decimals because it might be easier for our calculations later on. The next box would be one and a half inches high that makes the length eight and the width five and a half. A two inch square that's going to give us a box that's seven inches long and four and a half inches wide. Two and a half that will give us a box that is six inches long and three and a half inches wide. And you can see there's even definitely a pattern to these numbers I'm sure you've already thought of that. A three inch square is going to give us a box that's now five inches long and two and a half inches wide. Three and a half high that's going to give us a box that's four inches long and one and a half inches high. And then finally four inches that's the biggest ones any of you would have cut out. The length is going to be three and the width is going to be just a half. That's the boxes that's going to be the shortest or the tallest but the smallest base. Okay so from this we're going to calculate the volume. So remember how you get volume length times width times height. So I'll give you a few minutes on your table on your handout. You can fill out the volume calculate it on your own and complete the volumes and then you can come back and we'll make sure everybody did their calculations correct. So a volume of the first one when the square you cut out was only half an inch high would be 37.5. Hopefully that's what everybody got for the first one. Next one would have been 58.5 followed by 66 then 63 then 52.5 then 37.5 21 and then 36. So even just looking at the numbers you can notice some interesting patterns. So what we're going to do from this point on your handout you'll notice that on the second page you have a graph. So what we're going to do is take all these values and we're going to create ordered pairs. We're going to create ordered pairs and the points are going to have an x-coordinate of the height of the box which was the same as the size of the square that you cut out and the y-coordinate is going to be the volume. Alright so we're going to take a few minutes to set up our graph with the axes labeled and everything and then we're going to take a few minutes to plot our points. Alright so I do have a graph here that we can use and what I maybe would suggest doing is go every other block would be one unit. So the x-axis we're going to have that be the height of your box which again is equivalent to the size of the square that you cut out. So you can think of it either way either as the height of the box or the size of the square either one and just to make a really nice graph maybe we'll go every hash mark is a half inch so we'll make this first one the 0.5 square we would have cut out and then 1, 1.5 so on and so forth. So the y-axis is going to be the volume and remember the volume is measured in cubic inches it's always cubic dimensions for volume and it might be easiest if you take a look at the values we got for the volume maybe we'll go up in tens so this would be 10, 20, 30, up to 70 I think should be big enough if you take a look at the values that we got I think 70 should be big enough. So let's start plotting the points so the first one was 0.5, 37.5 so you can just estimate the y-coordinate obviously it's going to be a little bit past 35 so maybe right around there the next one is 1, 58.5 followed by 1.5, 66 the next one is 2, 63 then 2.5, 52.5 then 3, 37.5, 3.5, 21 and the last one remember was 4, 6 so your points should look like this so we could connect the points if we wanted to and one thing that you'll learn later on in algebra is that when we connect these points they do form a special type of shape so they're not going to be straight line segments that connect all these points it is going to be a little bit curved kind of like this and if you take a look at it the interesting thing is a lot of people when they start a project like this they just think the volume is going to get bigger and bigger and bigger when actually take a look at what happens there's actually a maximum volume that you can have when you start out with a piece of paper this size and we cut out squares from the corners of the sizes that we did if you take a look at this it looks like the maximum volume we get is up around when that size of the square was 1.5 inches and the volume ends up being 66 cubic inches and notice how all the other volumes from all the other different size squares that we cut out they are all less than that so you really do get a maximum volume that you won't get anything bigger than that and that's one of the really interesting things about this another thing you can do and talk about is well what if we wanted a box of a certain size a certain volume well we can even figure that out how large the square should be based upon our graph suppose we want a volume that would be 40 40 cubic inches all right well here's 40 all right because remember the y-coordinate is the volume so we specifically want a volume of 40 so if we were to just trace this over notice how there's two places where that volume of 40 intersects the graph it intersects right here sort of looks a little bit past half an inch and then up here maybe just shy of three inches so what that tells us is if we wanted to create a box that was 40 cubic inches big we would need to cut out two different types of squares and either one would give us a volume of 40 cubic inches we could either do a square that was just a little bit bigger than half an inch on a side or we could do one that was just a little bit smaller than three inches on a side so this is some of the things that you get into as you get older and you learn other types of math starting with algebra that's when you can get into equations that model these things and you can even then use the equations to determine what is that maximum volume what if we did want to have a volume of 40 or even 30 or 25 how big would the square have to be that we would need to cut out so when you get there hopefully I'll see you again but in the meantime thanks so much for joining us today