[3중급] Learning Math
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| Learning Math The geomatry we learned in school isn't the only geometry. Oh, perfect I had no idea that I'd be using high school geometry to this extent. It's just about in everything that we do. In order to use a standart data form across all three-dimensional polygons, we had to begin with triangles. People don't realize how precise mathmatical language is and how imprecise …english is. Welcome back. We're going to have today's session revisit some things that we've seen before. So we're going to revisit this idea of construction that we saw with Patty paper, and what does it mean to construct something versus drawing something. At that time, you were restricting yourself to the tools of folding paper and drawing on the paper. We now have a computer tool. We're going to use Geometer's Sketch Pad, which allows you to construct things, but then also to change your construction within certain parameters. Session four has two main focus points: a section on parallel lines and a section on circles. And those two pieces are sort of historically important pieces in mathmatics, in the development of geometry. We used Geometer's Sketch Pad, which is a very nice piece of geometry software. And it allows you to create a drawing and then to manipulate pieces of it, but without just drawing you can actually create a construction. So you can create two lines that stay parallel, but then move them in a way that they stay parallel but they're moving all over the place, and even slide them apart and together. So you can look for what are called “invariants”in mathematics- things that stay the same, even while things around them are changing. We're now going to think about parallel lines. So, parallel lines are two lines in the same plane which never intersect, that is, parallel lines are everywhere equidistant. If you measure the distance between two lines along a perpendicular between them, that perpendicular distance is the same no matter where you measure. So just work with your partner through those questions on parallel lines. And it's going to ask you to do some measuring of angles and things like that. If you need help, I'll be coming around and helping with the software. The teachers are first given the task of using the computer software to construct two lines which are parallel to each other. They are then asked to draw a transversal through the parallel lines and have the software measure the resulting angles. That's an official line. So we've got to find an official parallel line. So we select that line? Okay, you're going to need another point… Another point? Off the line maybe below it or something. To make it something parallel, a place for it to be parallel through. So we select that line. Yeah. Choose it. Yeah. Hold down “shift” Choose them together, because they're working together. There we go. All right. So, there is parallel lines. There's the two parallel lines. So we need to make a transversal. The transversal line is any line that goes through the two parallel lines. Once the parallel lines and their transversal are constructed, and the resulting angles determined, the teachers are encouraged to change the orientation of the transversal and take note of the changes in angle measurements. You could actually using the cursor or the arrow with the mouse, move the trasverse back and forth So that the angle measurement that the transverse was cutting through the parallel lines would change, but you could actually, um…make a notation- the software would make a notation for you, as to what that measurement was- like, actually measure it- but as you move the line back and forth the values of the angles would change, the numbers would go up and down. But you could see by looking at it that they still added up to 180. Okay, and then Let's see if we can tug that around a little bit. So now the measurements change, but we still have corresponding angles equal, and vertical angles equal. Everything still adds up to 180. Without using the software, there is not a good way for the parallel lines to be a sort of discovery-based activity. You would have to draw lots and lots of examples and measure lots and lots of angles. And the tedium of the work would overcome any joy of discovery that you might experience. The beauty of the software is that you can draw one example, make a few measurements, and then just drag points around, and watch the measurement change in front of your eyes. And you can start to make conjectures right there without having to create so many of your own examples, or to have to think up examples to test your conjecture. We can go ahead and verify. That's the nice thing, you can… Well, that, too, here, because it's the same thing. There is a straight line, There is 39 degrees. so that should… in order to add up to a straight line, that should definitely be 39 degrees. I think everybody saw in the binder you had a diagram that looked like this with names of angles? So we've got corresponding angles, alternated interior angles, corvertical angles, and linear pairs. Can anybody summarize some of the things you found when you were working with parallel lines thinking about these kinds of angles? Yeah, Catalina. We found that corresponding angles were equal. And we usually say-just technical stuff- that angles are congruent, and their measures are equal. When a pair of parallel lines are cut by a transversal, several special pairs of angles are formed. |
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