Back To CourseMath 104: Calculus
14 chapters | 115 lessons | 11 flashcard sets
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Let's stop and take a minute to think about how important visualization is to everything in everyday life. Now first, a little sidetrack. I'm reading a book right now about how they're trying to find astronauts to go up to Mars, and one of the things they have astronauts do is make a thousand cranes. You know, those little origami cranes. They take a sheet of paper, they follow two pages of instructions and they make a crane. Then they repeat this a thousand times. I don't know about you, but when I try to make a crane I end up with an airplane - a paper airplane - and it never flies particularly well either. But I have a really good friend who is fantastic at making these cranes. So what's the difference? Why can she make cranes and I can't?
Well, it's because I don't follow these keys to understanding visualization problems from a sheet of paper. It's nothing more than making an origami crane. The first key is that you need to pull out the most important information. When you're following the instructions to make a crane, you might not care about the history of the crane, and certainly it's not important to actually making the crane right then and there. The second thing you want to do is to draw out the whole process. So in terms of making the crane, you want to test it out and actually try folding some things. The last big key is that you need to check your results. So in the case of origami, you need to see 'Does this actually look like a crane?' In the case of a math problem, you need to look at all of the little details and make sure that your solution fits all of those from the original problem.
Let's do an example. Say you're given the following word problem. You have a square plot of land that is surrounded by a fence on all sides. One of your next-door neighbors has a rectangular plot that is the exact same size and same dimensions as yours. He would like to put up a fence around his plot. What percentage of his fence have you already completed for him?
Let's pull out the important information. One, you have a square plot of land. The fact that it's square is important, or at least it seems like it might be important. It's surrounded by a fence on all sides. One of the keys there is 'all sides.' Another is 'your next-door neighbor' - so we are going to underline 'next-door neighbor' because that means he's right next to you. He's not across the street, there's not a lot of land between you guys, he's right next door. And he has a plot that's got the same size and dimensions as yours. Even though we called it a rectangular plot, because it's got the same size and dimensions, we know that it's actually a square. Everybody knows that's a rectangle - it's just a particular type of rectangle. What have I got so far? We have a square plot, a fence on all sides, a next-door neighbor, and the same size and dimensions. I'm also going to say that the ending question is really important. If you don't know what you're solving, you can't solve it. So I'm going to highlight 'what percentage of his fence have you completed for him already.'
The second big key is to draw it out. So let's draw this out. I'm going to draw out a square plot of land and call it mine. I'm going to surround it by a fence on all sides, with a next-door neighbor who has a rectangular plot the same size and dimensions. So here's my plot, and here's his plot. It looks just like mine, and it's right next door to mine. I'm going to circle my plot with a fence on all sides. In a different color, I'm going to put his fence, or what he would like around his entire plot. I've drawn it out, and now I just need to solve it. What percentage of his fence have I already completed for him? Let's look at what I drew out. He needs to surround his plot of land on all four sides with a fence, but one of those sides is adjacent to my plot. So I've already done the fence on that side. This means that one side of his fence is done and he needs three sides. What percentage of the fence have I completed for him? I've completed one side, and he needs four sides total. So 1 divided by 4; I've completed 25% of his fence for him. Now does this make sense? If I look at my drawing, with my 25% that I've completed for him, I see that here is my plot of land and here is his plot of land. Let's go back and re-read the question and make sure that everything in the question is what we have.
You have a square plot of land? (Check.) Surrounded by a fence on all sides? (Check.) A next-door neighbor (check) has a rectangular plot (okay, a square is a rectangle, check), and it's the same size and dimensions as mine. (Check, it looks just like mine.) He wants to put up a fence around his plot. I've got that here. What percentage of his fence have I completed for him already? Well, 25%. Okay, I match all of these. My diagram fulfills all of the requirements from here. So I'm good to go.
Let's do another one. Let's do a slightly harder one. We are given a paper rectangle of area 40 centimeters squared that has a width of 8 centimeters and a height of 5 centimeters (well, yes, a width of 8 centimeters and height of 5 centimeters will give me 40 centimeters squared, so I'm good.) I'm going to cut 1-centimeter squares from each of the four corners. Then I'm going to fold up each side to make an open box. What is the volume of the box?
This sounds a little bit more complex, but let's remember our keys. First, I'm going to pull out important information. Okay, important information: It's a rectangle. I don't know if I'm going to need this 40 centimeters squared, so I'm not exactly going to include that, but I'm going to keep in my head that I know that. So I've got a rectangle. It's got a width of 8 centimeters and a height of 5 centimeters, and I'm going to cut 1-centimeter squares from each of the corners. Let's start with that. Here's my rectangle. It's 8 centimeters wide and 5 centimeters tall. I'm going to cut 1-centimeter squares from each corner. That sounds about right, and there, the corners are gone. Hmm. If I just cut 1 centimeter off each of the corners, then I know that the whole width is still going to be 8 centimeters, but the width at the edge here is going to be 6 centimeters. Similarly, I know that the whole height is going to be 5 centimeters, but I've cut 1 centimeter from each side, and so the height right here is going to be 3 centimeters.
Now I'm going to fold up each side to make an open box. So here's my 3 by 6, and I'm going to fold up each side. I'm going to assume that means I should fold on these dotted lines here. If I fold here, then, this side is going to be 90 degrees from the bottom of the paper. This edge here is going to join this edge here, along one edge. Then I'm going to get a new edge where these two edges meet. Same with these two and same with these two. I'm not quite sure that this is going to work, so let's pull out a sheet of paper and actually do this. We'll rip out these corners, and let's try folding where my dotted lines would be on this diagram. Okay, this looks like an open box. What are the dimensions of this box? The width is going to be, now, 6 centimeters because we had 8 centimeters and we took off 1 on each side. So we've got 6 centimeters. The height from my sheet of paper is going to become the depth of the box, and that's now 3 centimeters, because we've folded up the front and the back. And the height is going to be 1 centimeter. That's from these boxes, these rectangles that we cut out. So now I have a box that's 3 centimeters by 6 centimeters by 1 centimeter. If you know how to find the volume of a box, then you would know that the whole volume of this is going to be 1 * 3 * 6. So that's 18 centimeters cubed.
We drew it out. What's more, we actually took a sheet of paper and folded it up to see. We needed some help here. It's 3-D, why not? Let's check our results. Do we actually match all of the requirements? We had a paper rectangle. There's one sitting on the floor here. That rectangle, at least in our diagram, had an area of 40 centimeters squared. It had a width of 8 centimeters and a height of 5 centimeters. All sounds great so far. I cut 1-centimeter squares from the each of the corners, yes, and I folded up each side and made an open box. I wasn't exactly sure if I knew where to fold it, but I folded up each side to make an open box, so I fulfilled the requirement. Then I found the volume of the box, and it made sense.
There are a few keys with visualization that you should always remember. One, you want to pull out all of the important information. And when you're done with the problem, you need to look at that important information and make sure that your solution satisfied all of those bits. This is like the origami. You know at the end, you need to have a head on your origami crane. If your origami crane doesn't have a head, then you probably didn't solve it correctly.
You also want to, as best as possible, draw it out. If you have something that's 3-D, drawing it out might mean that you draw it on a sheet of paper, but then you actually try to use your hands to make whatever you need. In this case, we made an open box with a sheet of paper that we had on hand.
Finally, as I mentioned before, always check your results. You need to think, does this make sense? If we had a volume that gave us -100 centimeters cubed, we might know there's a problem, because a volume should not be negative. So always check your results.
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Back To CourseMath 104: Calculus
14 chapters | 115 lessons | 11 flashcard sets