This eighth grade mathematics lesson focuses on two-dimensional geometry - in particular, the areas of triangles between parallel lines. The lesson is 50 minutes in duration. There are 35 students in the class.

00:00:02He says he's not filming yet. Around here (inaudible).
00:00:13Stand straight.
00:00:14Stand straight.
00:00:27Um, do you remember what we did last period?
00:00:29We did mathematics.
00:00:31Sakurai, what kind of things did we do?
00:00:34We did mathematics.
00:00:35I don't know.
00:00:38Huh? Hm. The last period?
00:00:41Um. This topic, okay?
00:00:45Is it that?
00:00:47Obtain the area of triangles which are in the places in the parallel lines.
00:00:51That's right, huh? Um, we studied the idea that parallel, parallel lines,
00:00:57the triangles between them, uhm,
00:01:02with the same base or height are all the same... like this.
00:01:08For example, here.
00:01:17This�since the height will all be the same,
00:01:21we studied that the area will also be the same.
00:01:27Uhm, we will use this this as the foundation
00:01:29for what we will be going to study today.
00:01:31Then, prepare just your notebooks. We won't need your textbooks.
00:01:34Close your textbooks.
00:01:36Just take out your notebooks. Uhm. The date. Oh.
00:01:38It's er- erased. The date is November.
00:01:42What day is it?
00:01:45The twenty-ninth.
00:01:46It's the twenty-ninth, right?
00:01:48It's the twenty-ninth.
00:02:12Uhm. Right now over here, okay? There is Bando's land.
00:02:21It's okay, huh? Okay. There is Bando's land.
00:02:24Okay? Over here is Chiba's land, okay?
00:02:26[ Laughter ]
00:02:27The teachers. It's the teachers.
00:02:31Is it okay? Let's say that there is a land like this.
00:02:36And. Is it okay? Chiba.
00:02:39And these two people's, uhm, border line is bent like this
00:02:42but we want to make it straight, okay?
00:02:47Is it okay with it around here?
00:02:51It's okay?
00:02:52[ Laughter ]
00:02:53Then, we'll end today's class, okay?
00:02:54[ Laughter ]
00:02:55Chiba, is it okay with it around here?
00:03:00Around where would you like it?
00:03:02It would be better if mine was wider.
00:03:03A lot more to the bottom?
00:03:04Huh? A little more over here?
00:03:06Around where would you like?
00:03:07Continue over more. More. More.
00:03:08More over here?
00:03:09More. More. More that way.
00:03:10[ Laughter ]
00:03:11Oh. [ Laughter ]
00:03:12Bando. Is it okay with it around here?
00:03:14It's not okay, right?
00:03:15[ Laughter ]
00:03:16Then, around where would it be good?
00:03:19Date. Around where do you think would it be good? Approximately.
00:03:22That line... well.
00:03:25Well, try doing it.
00:03:27Approximately. Estimate.
00:03:31Uhm. Take it between th- this line and this line.
00:03:34We got an estimate that says it may be okay if it's in the middle.
00:03:38I see. How about other people?
00:03:40Okay? Then, well in your notebook, okay?
00:03:43Draw a figure like this and-
00:03:45Please try thinking about it a little,
00:03:48methods of changing this shape without changing the area.
00:03:56Okay, then everybody... let's try thinking about it.
00:03:59The work time is... would you think about it for three minutes?
00:04:03First of all, please think about it individually for three minutes. Okay begin.
00:05:37Are you done?
00:05:44(inaudible) You made it straight here? Is it the same? Definitely? Is it the same?
00:05:49It's appro- approximate.
00:05:50If it's approximate. Wouldn't they fight over it if it was approximate?
00:05:52Huh? Around. It's the parallel line of this and this.
00:05:55The middle of it?
00:05:57I think it's good if you draw it exactly in the middle. I see.
00:06:05That's sharp.
00:06:08What was the thing we studied last time?
00:06:12If it's like this, for all we know they will fight.
00:06:14A method that uses the same method as last time.
00:06:17I can't solve it.
00:06:19First of all, draw a figure and-
00:06:20Draw the figure and-
00:06:22The idea from last time.
00:06:24Is there a method that uses the area of the triangles?
00:06:34That's sharp.
00:06:36You were able to make this a triangle, right? Okay?
00:06:39Then if you do what...
00:06:43Okay? Would you get triangles with the same area.
00:06:46Would you make this the base?
00:06:48The question is that somewhere there are parallel lines, okay?
00:06:51Hm. We did it like this and in the last class we did it like this...
00:06:54we get a triangle.
00:07:04Okay. Then since the three minutes are up, so... uhm...
00:07:09People who have come up with an idea for now go to
00:07:12Teacher Ebina and do it with him,
00:07:15and people who want to discuss it with his or her friends,
00:07:17discuss it with your friends.
00:07:18And for now I have placed some hint cards up here
00:07:22so people who want to refer to this can refer to it.
00:07:24Now then, uhm, in three minutes...
00:07:26Uhm, well, think about it and please try doing it
00:07:29with your friend or by yourself. Okay, begin.
00:07:38You can't bring it with you.
00:07:49You missed class the other day, huh?
00:07:51Hm, well. It's good if you make a triangle somewhere.
00:07:53Can't you make a triangle somewhere?
00:07:57Huh? It's wrong.
00:07:58I don't understand.
00:08:04Somewhere a triangle-
00:08:06if you draw one line somewhere you get a triangle.
00:08:15Make a triangle somewhere.
00:08:17Draw a line somewhere and make a triangle.
00:08:20Wait. Wait.
00:08:21Oh. I got it.
00:08:23Take a parallel line (inaudible).
00:08:27If we (inaudible), then the area of the triangle.
00:08:31I don't know.
00:08:37They're not parallel lines.
00:08:50Oh. I got it.
00:08:51Draw it.
00:08:52Teacher. That's not right. A B-
00:08:54It's different from mine.
00:08:55Hm. You see it is. You see Fujita's is.
00:08:57If these are parallel then you can say something like this.
00:09:00Oh. That's right.
00:09:01Do you understand? Since it's like this if you l-
00:09:03compare the area over here and over here,
00:09:06well it seems the one over here is smaller, right?
00:09:08Oh. It's the same, huh? If they are parallel then, okay?
00:09:10Well, I also have a feeling they will be the same. Yes.
00:09:12Oh. That's right, huh? I got it. I got it.
00:09:15But the way of thinking was good. It was quite a sharp thought, okay?
00:09:17The assumption was good.
00:09:18Can I borrow the (inaudible).
00:09:19Oh. I got it. I got it. I got it somewhat.
00:09:20Teacher, teacher, teacher.
00:09:22This okay? Went to here and here, okay?
00:09:23I got it somewhat.
00:09:26[ Laughter ]
00:09:29If it's approximate, of course it will become a fight, right?
00:09:30Oh. Did you get it?
00:09:31How do you do (inaudible)?
00:09:32Oh. Take this. You make a triangle somewhere.
00:09:35Well, here, draw one like this. This.
00:09:38You make a triangle like this. And, then you make this like that TV.
00:09:42To make the triangle you move your neck like this. Okay?
00:09:44If you do that you get a triangle, right?
00:09:46And here, here all you have to do is draw a parallel line somewhere.
00:09:50Then where would you draw a parallel line?
00:09:51A parallel is you understand? It's sideways like this.
00:09:52Oh. Like this?
00:09:53Over there isn't (inaudible).
00:09:55You don't get a triangle.
00:09:57You draw it here, huh? Hm, well where is it the same area with this triangle?
00:10:00Which is it?
00:10:02It's three (inaudible).
00:10:03Yes. And then. Yes. The area that is the same with this triangle becomes like this and it also becomes like this, okay?
00:10:09Oh. This then this is straight up like this. Here also.
00:10:11That's right, huh? Well.
00:10:12Oh. From here to here?
00:10:14That's right. That's right. That's- that's right. It's good if you do it like that.
00:10:17There aren't any complaints for that, right? If you do that.
00:10:18Oh. Well, let me try (inaudible). Teacher.
00:10:20[ Laughter ]
00:10:21It's here and here right?
00:10:22Yes. And is it just here and here?
00:10:25The more opposite side too.
00:10:27(inaudible) right? That's right, huh? Okay.
00:10:30[ Laughter ] What are you doing putti- marks?
00:10:31Don't erase it.
00:10:32Is it (inaudible)?
00:10:33[ Laughter ]
00:10:34[ Laughter ] It's okay now. Did you get it? You don't get it?
00:10:40Did you get it?
00:10:42Teacher. Would you point to me later, okay? Teacher.
00:10:45You talk too much, you stupid.
00:10:46If you were to make a triangle somewhere if you
00:10:47draw a line where would you get a triangle?
00:10:52Here, right? You get one triangle, right?
00:10:53Then where is the same area as this triangle?
00:10:57Oh. If you were to draw a line parallel to th-
00:10:59this triangle where would you draw it?
00:11:02You would draw it here, right?
00:11:04Right? Then where would you get a triangle that is
00:11:06the same area as this triangle?
00:11:09Those two.
00:11:11Ha. The two.
00:11:12You can also make one like this, you can also make one like this.
00:11:13You can make a lot, right? You can make multiple numbers of it, right?
00:11:15Th- That's what you mean.
00:11:16Oh. I have no clue what he is saying.
00:11:18Then to make this nicely into one line somewhere where should we move this?
00:11:27I got it.
00:11:29What should we do?
00:11:30Like this- like this.
00:11:31Like this.
00:11:32Okay. That's right. Okay?
00:11:38That's right. That's right. That's, that's right.
00:11:39Goto isn't good.
00:11:42Hey. I got it.
00:11:55In the end.
00:12:02Is it just this one line?
00:12:04Hm. That's right, huh? You also need the side over there, right?
00:12:07That's right, huh?
00:12:08Teacher, Teacher, Teacher.
00:12:09Yes. What is it?
00:12:11I don't really understand it but.
00:12:12Yes. Now then, okay? You got a triangle here, right?
00:12:14And they're parallel, right?
00:12:17All you have to do is look at it like this, like last period.
00:12:19Then since here it is parallel so... where is the
00:12:21area that is the same as this triangle?
00:12:23Like this.
00:12:24This and this and this.
00:12:25Yes. You can get one like this. You can get one like this.
00:12:28You can get a lot of (inaudible), huh?
00:12:31Then for this to neatly become straight where does the triangle form?
00:12:36This becomes a straight line- we want to make it straight.
00:12:39From here to here?
00:12:40Yes. Or...
00:12:41From here to here?
00:12:42That's right. That's right. That's right. That's- that's right.
00:12:45Uhm, this and this are-
00:12:47parallel, so-
00:12:49even if we move this-
00:12:50Hm. Hm. Hm.
00:12:51they are parallel bu- Since the area is the same so-
00:12:55aren't here- the obliq- oblique lines here and-
00:12:58this white part-
00:13:00the same?
00:13:01You're sharp.
00:13:04Uhm. Hamada.
00:13:07Here. Teacher. Here.
00:13:11How did you do it?
00:13:12Try explaining it.
00:13:13Since this and this are parallel so-
00:13:17if you connect them like this-
00:13:20they become equal.
00:13:21Oh, okay.
00:13:22I'm going to the teacher's one.
00:13:23Wait a second.
00:13:26Ikeda. Ikeda.
00:13:28Wait- wait a second, okay?
00:13:29Excuse me. Wai-
00:13:31Mos- have you finished most of it?
00:13:33How about the height?
00:13:36Uhm. Jimbo.
00:13:38Would you do the one you were showing just now up at the front?
00:13:39The one just now?
00:13:41Am I supposed to explain it?
00:13:42Yes. An explanation.
00:13:44Please go ahead.
00:13:45What is it?
00:13:46Huh? You got to be kidding me.
00:13:47If you draw the parallel lines and why it becomes like this why-
00:13:48Well you see.
00:13:50This is okay?
00:13:51You look at it like this and then you think about it.
00:13:52You look at it like this.
00:13:53For now please explain it one more time.
00:13:54If you do that, this will become the same, like last time.
00:13:55Oh. That's right, huh?
00:13:56Do you get it?
00:13:58If you look at it this way... you have to change the way you're thinking.
00:13:59And if you look at it like this, and if it's a triangle with parallel lines drawn then...
00:14:02Which one is the same area as this triangle?
00:14:04With which one? Tri-
00:14:05The area that's the same as this triangle?
00:14:08You won't get any numbers, right?
00:14:10And, for example, in what kind- what situations does this become a straight line?
00:14:12A straight line?
00:14:13Yes. We want to make it a straight line.
00:14:14We want to make the border line a straight line.
00:14:15A straight line?
00:14:16Should I do it like this?
00:14:17Hm. Hm?
00:14:18[ Laughter ]
00:14:19I don't understand what you are saying.
00:14:20The triangle is the one that moves, okay?
00:14:21The triangle?
00:14:22So a triangle like this becomes a triangle like this?
00:14:23Then when it goes to here does it become a straight line?
00:14:27What do you mean by a straight line? Oh.
00:14:28It's at the end of the (inaudible), right?
00:14:29I got it.
00:14:31I got it. I got it. Already. Already. I got it.
00:14:32It's okay with this.
00:14:33Is it correct with that- with that?
00:14:34But you can't (inaudible).
00:14:35With this. This- this original side.
00:14:36The triangle and the area over here are the same.
00:14:37Do you understand what I am saying?
00:14:39This triangle.
00:14:40Then doesn't it become the same as this figure?
00:14:41Yes. Over here is the same from the beginning, okay?
00:14:42It's not involved, right?
00:14:43It's the same?
00:14:44Because here. Right now only here is involved, right?
00:14:45Oh. This is a straight line- oh. That's right. That's right.
00:14:46Make it into a straight line. Did you get it?
00:14:48Jimbo hurry up.
00:14:50Please go ahead.
00:14:51Now I can't explain it since I'm not Japanese.
00:14:52Don't worry. We'll cheer for you.
00:14:53Since I'm not Japanese.
00:14:54Do it with confidence.
00:14:55[ Laughter ]
00:14:57Well, it's time, huh? Fifteen minutes.
00:15:03I'm not Japanese.
00:15:04What are you saying?
00:15:11Then Ikeda will you do it?
00:15:14Please go ahead.
00:15:15I don't know (inaudible).
00:15:16Will you do it in the two ways for us? Then, Ikeda, this side.
00:15:18Please draw this side.
00:15:20Jimbo, this side.
00:15:21I draw it from bottom to top? No. From bottom to top is (inaudible).
00:15:24[ Laughter ]
00:15:25From right to left?
00:15:26Oh. I'm sorry left to right.
00:15:27Le- le- ft.
00:15:29From left to right.
00:15:30Yes. Yes. Yes. Yes. Yes.
00:15:31From right to left.
00:15:32I want the one from the left.
00:15:33Huh? Is that from right to left?
00:15:34From the top?
00:15:35Hurry. Hurry. Hurry.
00:15:36Teacher. Please have Jimbo go first.
00:15:40Is it not okay with from the top?
00:15:41Of course, it can be said, right?
00:15:42It is okay from the top?
00:15:43It's the same even from the top, right? The area, okay? You're sharp.
00:15:46It's okay even if it's particularly from the top.
00:15:47Oh, well.
00:15:48What? You thought very hard?
00:15:49Okay, go ahead.
00:15:51Can you explain it?
00:15:52I want to explain it.
00:15:53I'm to explain it?
00:15:54The body is (inaudible).
00:15:56I see.
00:15:59I got it. I got it. I can't go back.
00:16:00What is it?
00:16:01It's one minute, right?
00:16:02Shut up.
00:16:03Hm. Wait a second. I'm sorry.
00:16:05Then I'll draw it for you so, okay? Mostly the same as the original.
00:16:09Just the base.
00:16:16(inaudible) Is either one okay?
00:16:17Whichever is fine.
00:16:18Do you want to use a ruler?
00:16:19I'll use it.
00:16:21(inaudible) but I have only two.
00:16:23It's the same as (inaudible), right?
00:16:25Teacher. Pick me. Pick me.
00:16:26Hm? Assistance, okay? Next, okay? Chiba.
00:16:28I don't understand that.
00:16:29Next, next.
00:16:31Yes. In the meantime today since Chiba, you are the main character, right?
00:16:33You're cool, okay? Huh?
00:16:34Well. Do the ones that are wrong because-
00:16:36Do the ones that are wrong?
00:16:38I don't mean that. (inaudible)
00:16:41[ Laughter ]
00:16:42No... well, okay? That is okay? The mistake is what's important.
00:16:48Because okay? If people can do it from the beginning
00:16:50then they don't have to come to school.
00:16:53Just the left (inaudible).
00:16:54It's not okay. What? Don't worry about me. It's you who is learning.
00:17:01Do it as fast as you can.
00:17:02Fas- (inaudible).
00:17:03Anyway do it fast.
00:17:04Either one is fine.
00:17:06Do it like as you did it. Your own method.
00:17:12You shifted th- this to this side like this, huh? (inaudible)
00:17:15Up to this side, right? Yes. That.
00:17:21And make the triangle and parallel.
00:17:24Oh. Oh.
00:17:29Peace to that.
00:17:30I won't do it.
00:17:36What's up? Kada. What's up? Were you able to do it? Oh. You're sharp.
00:17:41That's right. Lai, right? You got it right? Good.
00:17:44Okay. Is this just this side?
00:17:47It's also over here.
00:17:48Yes. Right?
00:17:51I can't do it even with this.
00:17:52This is okay? Is it okay?
00:17:53This isn't parallel, right? It's bent, right?
00:17:58It's bent, right? That means that for example
00:18:01when you have drawn it like this, okay?
00:18:04When you have drawn it like this, of the area over here and
00:18:06the area over here, which seems to be bi- bigger?
00:18:07Since it's bent like this, okay?
00:18:11The area over here and over here.
00:18:12Over here.
00:18:13It seems to be sma- smaller, huh?
00:18:14If it's between over here and over here which seems to be smaller?
00:18:18Doesn't it seem somewhat that they are the same, huh? Seems somewhat?
00:18:20I'll give up A B.
00:18:23It doesn't seem like it? If this is parallel then
00:18:25don't you think it's okay with this? I think it's okay- okay but.
00:18:29They aren't parallel, right? They are bent like this, right?
00:18:31All you have to do is draw it more exaggerated. In a figure like this, okay?
00:18:34Um. For example, if there is one like this and okay?
00:18:36When we cut it like this and did it...
00:18:37Now. It's okay.
00:18:39The base is the same.
00:18:41How is that?
00:18:42Are the areas of the triangles the same?
00:18:44Over here and over here are the same.
00:18:46Over here and over here are also the same.
00:18:47I think it's better if we don't let him/her do it.
00:18:48What? [ Laughter ]
00:18:49The end and the end are the same and-
00:18:51It's okay if this and this are the same, right?
00:18:53This and this are the same and...
00:18:54this and this are the same- ahh, which is it?
00:18:56Oh. It's this. It's okay if these are the same but. The same- oh, it's wrong.
00:18:59It's over here. I'm sorry. It's okay if this and this are the same, right?
00:19:01It's not the same.
00:19:02Well, I don't understand something, but- Hm.
00:19:06Oh. I'm sorry.
00:19:08Then- oh, it took a little bit of time, huh? Okay.
00:19:09I'm saying you can't draw it.
00:19:10Okay. Good.
00:19:11Stop it.
00:19:12Stop it, okay?
00:19:13Stop it, okay?
00:19:14Okay, then. Please return to your seats.
00:19:15Teacher, he's drawing extra things.
00:19:16Don't write it.
00:19:17Teacher. This.
00:19:18Okay, then the explanation. It'll take too much time.
00:19:20Okay then. Everyone please return once to your seats.
00:19:23Since they are equal, so-
00:19:25It's okay. Good luck.
00:19:28Okay, then.
00:19:29Am I doing it?
00:19:30Yes. Good luck.
00:19:31I don't understand it very well. Teacher, please listen to me.
00:19:35Now it's time to listen.
00:19:37Pay attention, okay? Okay.
00:19:38He's not listening.
00:19:40Go ahead.
00:19:41Is Chiba going to do it?
00:19:42It's not me, right?
00:19:43Okay. Hurry. Please do it.
00:19:45This is okay?
00:19:47Hm. First of all we make a triangle, okay?
00:19:50[ Laughter ]
00:19:52What are you saying?
00:19:53You talk too much.
00:19:54[ Laughter ]
00:19:55You make a triangle, right? And then here...
00:19:57His face is completely in the camera.
00:20:01Draw a line para- para- parallel over here also and...
00:20:03we make over here as the base.
00:20:08As the base. Here.
00:20:11And then we make it the height and this triangle and...
00:20:15That's the height? (inaudible)
00:20:18Which is it?
00:20:19I think it's not that.
00:20:20This triangle and a tri- somewhere.
00:20:24[ Laughter ]
00:20:25Uhm. The red triangle.
00:20:26Oh. It's this, right?
00:20:28The red triangle is-
00:20:30The area is...
00:20:31Over here. Over here. Over here.
00:20:32What is it?
00:20:34Well, they are the same, okay?
00:20:35[ Laughter ]
00:20:36The tri- over here.
00:20:37Actually and the triangle over here-
00:20:41The fact is that the areas are the same, okay?
00:20:43Since the base and the height are the same so.
00:20:46So� first of all... the fact is, we can draw a line here.
00:20:50Yes, yes. Well, I don't know what I am saying, but-
00:20:54[ Laughter ]
00:20:55No. We can understand enough, right?
00:20:56Oh. You understand?
00:20:57Is there anybody who does not understand?
00:20:58[ Laughter ]
00:20:59Oh. You don't understand?
00:21:01I also don't understand.
00:21:02You don't understand? Then one more time then. With this side.
00:21:05This time please explain it, Ikeda. You're saying that it was a good explanation.
00:21:07Okay, then applause. Wonderful.
00:21:10[ Laughter ]
00:21:11People who didn't understand it please listen to it closely, okay?
00:21:12First of all...
00:21:13[ Laughter ]
00:21:14After this. The place that has a weird place like this.
00:21:15I got it.
00:21:20In the place that is below this, like this draw a line and,
00:21:24uhm, we make it into a triangle, okay? Ha- then this-
00:21:29This. Something that has this side as a base?
00:21:33That has it as a base?
00:21:36Draw a line that is parallel to this base by
00:21:40fitting it with this apex. And then the parallel line... hm?
00:21:51Since the length of base doesn't change so...
00:21:54and in between the parallel lines always-
00:21:57The height is e-
00:21:58The height doesn't change so... like this. Up to wherever you draw it, anywhere you draw it,
00:22:05the area doesn't change with this, this triangle that we got first.
00:22:11So, uhm, it's just that even if we draw up to this best place's- end,
00:22:17for this triangle and this yellow, this triangle, the areas don't change.
00:22:24Okay. Thank you. Wonderful.
00:22:30Okay. Uhm, for now, okay? Since it's hard to see we will make it clearer.
00:22:34The areas of this triangle and the red triangle over here,
00:22:37and this triangle and the yellow triangle over here are the same areas.
00:22:40So we want to do it so this here becomes straight like this, okay?
00:22:43So that the corner here is gone. The angle, okay?
00:22:45Then we were able to draw a straight line here.
00:22:47That's what we can say. People who were able to d- who did it like this?
00:22:50People who can say I drew it like this over here?
00:22:51People who say I was able to draw it.
00:22:53People who say I was able to do it like this way? People who say they were able to do both?
00:22:56Okay. That's good. Uhm, then, okay? Next, uhm,
00:22:59using this as a foundation, okay?
00:23:03Oh. I don't know if it's using it as a foundation or not but- A quadrilateral.
00:23:05Oh. Of course it can be crooked like this.
00:23:07Well for now, um, (inaudible) quadrilateral.
00:23:12Taking this quadrilateral...
00:23:15without changing the area... make it into a triangle.
00:23:24This shape of the quadrilateral,
00:23:26without changing the area, please try making it into a triangle.
00:23:32Okay, then... please think for three minutes
00:23:36and try doing it in your own way. Okay begin.
00:23:38We can't open our textbooks?
00:23:41This (inaudible).
00:23:43Teacher, can we open our textbooks, yet?
00:23:45Hm? The textbooks? First try thinking about it by yourselves.
00:23:49You know that it was in the textbook, huh?
00:24:05You're sharp, huh?
00:24:32You're sharp.
00:24:40We- well Maehata.
00:24:47You can't do it with it just as it is, right?
00:24:48I don't change this?
00:24:49Uh huh. It's a quadrilateral, right? If it becomes wha-
00:24:51what sided shape can you do the problem?
00:24:56If it becomes- what sided shape can you do the problem?
00:25:00If it's just as it is it's a quadrilateral, right?
00:25:02So we're to make a quadrilateral into a triangle?
00:25:04Right? If everybody had a triangle and parallel lines
00:25:06then they can do it, huh?
00:25:10That's right. You're sharp.
00:25:16That's good.
00:25:18Hm. That's right, that's right, that's right.
00:25:20You noticed a good point. That's right. That's right. That's right. That's right.
00:25:23Teacher, the one we call the triangle isn't just one?
00:25:27In addition?
00:25:28It may be that you can get many types, huh?
00:25:29Is it okay with two?
00:25:30It's okay even if it's two types or even it's three types.
00:25:34Bando. You're sharp.
00:25:36You're quite sharp.
00:25:37What do we need to do to change this into that? How can we do it?
00:25:42Triangle and yes. That's right, huh?
00:25:45Since you can't do it just as it is so.
00:25:48What are you saying.
00:25:52Oh. That's sharp, huh? You noticed a good point.
00:25:55A triangle has been made, right? Then, okay?
00:25:58When you look at it like this
00:26:00then there is a triangle
00:26:02and how can we change it, may I ask?
00:26:08For example, do you want to think about it with the triangle over he- over here?
00:26:10What did we have to do to make a triangle that has
00:26:13the same area as this triangle, may I ask?
00:26:15Can't you use what you saw on the TV screen.
00:26:19That's right, huh? That's right, huh? Yes. (inaudible)
00:26:22nicely at the bottom.
00:26:27Are there anymore other than this one?
00:26:28The fact that you can get one on this side means that...
00:26:31then if we extend it a little like this then... huh?
00:26:34The fact that you got one like this means that...
00:26:37Can't you make one on this side?
00:26:43Teacher. Teacher.
00:26:44Are you done? You're sharp.
00:26:47Then people who are done
00:26:49please go to Instructor Ebina again.
00:26:50And, uhm, then people who want hints
00:26:52I will leave hint cards here
00:26:53so please look at them and try doing it.
00:26:55It's also fine to do it with your friends.
00:26:58Okay begin.
00:27:02Uhm. It's also fine for people who don't understand
00:27:04to ask me.
00:27:08Oh. But I don't think it's limited to one (inaudible).
00:27:10There is one more.
00:27:18This is okay?
00:27:20How many? How many do I make?
00:27:21I wonder how many can you make?
00:27:23Is it okay with one?
00:27:26Yes. It's okay if first of all you get one, but-
00:27:29With one zero.
00:27:30Is one all you made?
00:27:32In all? There's three.
00:27:37How many can we make? This?
00:27:38Hm. I wonder how many all of you can mak- make.
00:27:40Oh, I see. That's what you mean.
00:27:41Yes. The answer is not just one.
00:27:53Oh. That's right, huh? Oh. That's right, huh? Oh.
00:27:59I got it.
00:28:02It's fine with this, right?
00:28:14Huh? It's wrong.
00:28:17Now. The ones I haven't got yet teacher (inaudible).
00:28:34Teacher. That's unfair.
00:28:36There is no possibility.
00:28:45Excuse me. It's okay with this, right?
00:28:46Yes. If you got one on this side then-
00:28:49(inaudible) I can get it over here and over here from this-
00:28:51You can do that too. And can't you make a triangle
00:28:53over here that is the same as this?
00:28:56Well that's right, huh?
00:28:57Right? The same as before.
00:28:58You can get it on the opposite side, right?
00:28:59Would I know all of them doing it like that?
00:29:01Hm. You'll notice them- If you notice them then that will be great.
00:29:02Am I supposed to draw all of them?
00:29:04I wonder how many ways it can be made?
00:29:06You're sharp.
00:29:25I think it's the same area as this quadrangle
00:29:28of the hint card and the notebook.
00:29:33You're advanced to a good place in the problem.
00:29:37I got it. I got it. I got it.
00:29:38Are you going to make it into an upright state?
00:29:41Are you going to fold it like this?
00:29:47If you have folded this then, okay? What kind of shape will it become?
00:29:49It won't at least overlap, right?
00:29:51I got it.
00:29:53Because (inaudible).
00:29:54Yes. You've gotten up to a good place in the problem.
00:29:55But after th- that I'm having problems.
00:29:56Yes. Then okay?
00:29:58If this corner- for example it's good if the corner here was gone, right?
00:30:03I'll draw a line longer here, okay? Ones that are same as the area of this.
00:30:08Things like these dr- are also the same, right?
00:30:10Things like these are also the same, right? There's a lot, right?
00:30:12For example, if the corner here becomes straight with this here and here then this is...
00:30:17Parallel. (inaudible)
00:30:18Does it have to be teacher Ebina who looks over my work?
00:30:19Wai- It's okay to come to me to have your work looked at.
00:30:21The fact is we're making a triangle- a quadrangle into a triangle you do what to one angle?
00:30:23Nagai. It's because there's a lot of people waiting in line for teacher Ebina.
00:30:27All you have to do is decrease it, right? That means that to reduce angles you take one-
00:30:33Yes. We have to make it straight, right? Then you get a triangle like this. A triangle like this. And also?
00:30:37(inaudible) It's okay with this line segment.
00:30:40If you get one over here then this time also on this side...
00:30:42Yes. We get the same thing. Okay? Are there any others? You can also do this side, right?
00:30:44The same thing.
00:30:48Please think about it in many ways.
00:30:49It's not just one, right? Is it okay with this?
00:30:51If you notice this one then, okay?
00:30:52You'll also notice the one over here, huh?
00:30:53You'll notice it, right?
00:30:54[ Laughter ]
00:30:55Why can't you d- it when you know it?
00:30:58Hm. Ohba, how are you doing?
00:31:00Hm. You've gotten to a good place in the problem, huh?
00:31:03The fact that we're making a quadrilateral into a triangle means what happens to the number of angles?
00:31:10I don't understand.
00:31:11Hm. The number of angles goes from four to...
00:31:14It becomes three, right? It's reduced by one, right? That means that here,
00:31:20it's good if the angle in this quadrilateral is gone. For example, if here were to become straight.
00:31:24Draw a parallel line and.
00:31:26You get a triangle like this? Do you understand? Here and here are the same, right?
00:31:31This okay? How about this side? Can't we make somewhere? Straight like this?
00:31:43How will it be I wonder?
00:31:47Are you done?
00:31:48I have gotten only one yet.
00:31:49Then would you draw four.
00:31:50Ahh. Can I borrow four. Ah. We don't have time. Ti-
00:31:51Where do I draw it?
00:31:57Teacher, I don't understand it very well, but-
00:31:59Hm, okay?
00:32:00There's eight.
00:32:01Do you have your notebook?
00:32:02I do.
00:32:04Then look at it. Teacher.
00:32:07What is it? Snobbishly. (inaudible)
00:32:13If it has become this, then can we do it?
00:32:16You can't do it as it is a quadrilateral?
00:32:19It's this.
00:32:21It's okay?
00:32:23And also?
00:32:26Don't you have to draw a parallel line to the triangle?
00:32:29It's here.
00:32:30Triangle and- hm.
00:32:34That's funny you know?
00:32:36Then- wait a moment. I'm sorry, okay?
00:32:39Then where is the tri- quadrilateral, oh,
00:32:41triangle that's equal to the area here? Hm.
00:32:44No, it's not. Since it's parallel like this so
00:32:48in the parallel lines where is the qua-... that's equal to here?
00:32:51This is a little off. Teacher.
00:32:52Hm. That's right. It becomes more off.
00:32:54Hm. (inaudible) here and here and (inaudible) to here.
00:32:57Hm. Hm. You drew this and this?
00:33:00Then okay? If you have drawn it on this side, okay?
00:33:02Isn't there one more parallel line- a triangle?
00:33:05You cut it doing like this- doing it like this-
00:33:06into two- doing it like this, right? The triangle is over here, right?
00:33:08Then also over here.
00:33:10If you cut it like this, then here with this
00:33:12one diagonal line, would you get a triangle just over here?
00:33:14I can do this.
00:33:15And it comes over here and-
00:33:17Over here is also (inaudible).
00:33:18Is it just over here? Won't it extend to this side?
00:33:24You noticed things like this on this side, right?
00:33:25Uh huh.
00:33:26Then. This area and over here are the same, right?
00:33:28Isn't there anything like this over here, okay?
00:33:30Oh. Oh, oh, oh.
00:33:32Hm? Oh. It's over here, huh? Over here. Over here.
00:33:34Things like this. (inaudible) On this side.
00:33:36Oh. That's right, huh?
00:33:37Right? If there is a right side then there is a left side also, right?
00:33:39(inaudible) How about this?
00:33:40Yes. Then triangles that are the same with this area are,
00:33:43hm� there are a lot of these, right?
00:33:46Ones like this.
00:33:47Many different kinds.
00:33:49And the fact that we're making a quadrilateral into a triangle means that we do what to one angle?
00:33:51Get rid of it.
00:33:52All you have to do is get rid of it, right?
00:33:53Please get rid of one somewhere. When it moves to where would it disappear?
00:33:54It's an interesting thought, huh? That's good, huh? I like ones like that. Well please think with this. In total there's a lot more. Good.
00:33:57Yes. Here if you make this straight like this then this angle disappears, right?
00:34:02This is one answer, right? The one before this one. Triangles like this. Also?
00:34:07If there is one on this side then how about this side?
00:34:11Hm. Where is it?
00:34:12Here if there is one then-
00:34:16It's somewhere here, huh?
00:34:17That's right. That's right. That's right. That's right.
00:34:20That's right (inaudible) keeps on bending and the angle disappears.
00:34:23Hm. Since we want to lose this angle so this... hm, isn't it a little bent?
00:34:29Did you get it?
00:34:31Yes. That's right. That's right. That's right. You're at a good point,
00:34:34and then we want to get rid of this angle, right? From four to three. If it becomes like this then this angle?
00:34:37It disappears.
00:34:38It disappears, right?
00:34:40And if you get one on this side, then this time can't you also get it on this side?
00:34:44Am I to draw all of these?
00:34:46This. This.
00:34:47Phuse. For this there are t- two more, more.
00:34:49Is it okay to change some?
00:34:51One that is a new (inaudible).
00:34:53You're kidding.
00:34:54You found a new one? That's good.
00:34:55Then would you draw it then like this? If you have another one.
00:34:57Well, it's okay with it here.
00:34:58Hm. Please draw it. It's okay, it's correct.
00:35:02Can you shrink it?
00:35:03Uh, well...
00:35:04Well you don't understand it well?
00:35:06But you were able to do it? You made this
00:35:08triangle and this the same, right?
00:35:11This. Then can't you get one on this side?
00:35:13Still. Still how do I get it?
00:35:15You don't understand it well? Is it okay?
00:35:18There is something like this here, right? A triangle, huh?
00:35:21Then how many triangles are there that are the same as this triangle?
00:35:25The ones we call triangle's ar- triangles that are the same as this triangle's area.
00:35:30There are ones like this and there are ones like this
00:35:32and the fact is there are a lot of them.
00:35:34Also a lot of them over here. Oh. This is. This.
00:35:37If you look at it like this then it's a triangle and parallel lines, right?
00:35:39And to fix triangles into quadrilaterals- Oh, I'm sorry.
00:35:41The fact that we're making a quadrilateral into a triangle means...
00:35:44all we have to do is reduce the angles by one, right?
00:35:47That means that if this becomes straight,
00:35:49then the angle here disappears, so you get
00:35:52a triangle like this.
00:35:54Then on the other side.
00:35:55Then is this the only triangle that we can get?
00:36:00Teacher. Will methods like these also work- Oh, I'm sorry.
00:36:04Yes. That's right, huh? You can also do that, right?
00:36:06Can I also do that?
00:36:11But, this is triang-
00:36:12It hasn't become a diagonal line.
00:36:14Quadrilateral. Quadrilateral.
00:36:15It's no good if it's a quadrilateral. It hasn't changed at all.
00:36:16Teacher. I'll also do one for you.
00:36:17First of all this diagonal line like this.
00:36:18Triangle like these are (inaudible) okay?
00:36:19Please do a different one.
00:36:20It's like this.
00:36:22The diagonal line isn't done, okay?
00:36:23In the end this is-
00:36:24Just now I made a little mistake.
00:36:25You made a mistake?
00:36:26Which diagonal line?
00:36:28Hey, you. This isn't good if it's the middle point.
00:36:30It's not good if it's the middle point. It's not a triangle. This.
00:36:31[ Laughter ]
00:36:32Like this, like this.
00:36:34I don't understand this.
00:36:35You drew it like this?
00:36:36It's not a triangle.
00:36:37This becomes straight and-
00:36:39That's wrong. That's wrong. That's wrong.
00:36:40Oh. Over here? Huh?
00:36:41This is where it's supposed to be straight, okay?
00:36:42That's right. Straight.
00:36:43It doesn't look straight, okay?
00:36:44This. Straight.
00:36:45Oh. That's right. That's right. That's right. Hey, it's crooked.
00:36:47Like this.
00:36:48If it's that, it's okay. Okay, okay, okay. Please have it drawn neatly.
00:36:53How many methods have you found?
00:36:55I don't know yet up to where I did.
00:36:57You don't know anymore? The reasons? Yes. Quoi.
00:36:59You've gotten to a good point in the problem. Then okay?
00:37:02Since we're making a triangle into a quadri- oh,
00:37:03I'm sorry a quadrilateral into a triangle-
00:37:04So all you have to do is get rid of one angle okay?
00:37:06For example, if you do it like this how is it?
00:37:08Is better over here?
00:37:10For example, when this becomes straight how about it?
00:37:14If it becomes like this?
00:37:17Also what do I do?
00:37:19How about a triangle like this?
00:37:20The area of the triangle over here is the same, right? This triangle and
00:37:25this triangle are the same, right? This has become straight, right?
00:37:28Then this becomes a triangle, right?
00:37:30Do you understand? The problem?
00:37:31Are you okay? Is there another one?
00:37:34Yes. Then if you make this straight over here then
00:37:36what kind of triangle will it be?
00:37:37[ Laughter ]
00:37:39Yes. Oh, you're making two triangles, huh?
00:37:43Hm? It's good if you make it into- a triangle.
00:37:45Just like before draw a line somewhere and try making a triangle.
00:37:48If you draw a line wh- where then would you get a triangle?
00:37:54It's like this, right? It's like this or like this, huh?
00:37:56Area. Okay? What's up?
00:37:59How many methods did you get?
00:38:00Still only two.
00:38:01First of all this is correct, huh?
00:38:02No. Something. I thought something might be wrong.
00:38:04Hm. Hm. Hm? You said it might be wrong.
00:38:05Why did you think it might be wrong?
00:38:06Something. The base (inaudible).
00:38:08The question is where to put the base, right?
00:38:11That's right. That's right. That's right. That's right. You're sharp.
00:38:13Huh? Huh?
00:38:14(inaudible) just us.
00:38:15You're so noisy. Why don't you shut up?
00:38:18She is so scary.
00:38:20I'm not scary.
00:38:26Chiba. How many methods did you get?
00:38:29You got two, right?
00:38:30Yes. You're done, right?
00:38:33Yes. That's right. That's right. That's right.
00:38:35It's good, okay? Then is it over with these with two?
00:38:38Then how about this side? This side?
00:38:40Isn't there one that's the same as this triangle?
00:38:48How many did you get Ryu?
00:38:50Are you worried thinking about it?
00:38:52You were able to get one? This okay?
00:38:55You got it, right? Something like this.
00:38:57You got something like this, right?
00:38:59One that is the same as this, is this, right?
00:39:01Is the area that is the same as this triangle just this?
00:39:02Sada. It's better to slide it to the other side.
00:39:05Is it this?
00:39:06Hmm. That's good, huh?
00:39:08Hm? Can't you make one on th- this side?
00:39:09This side (inaudible) a lot.
00:39:10Yes. I can.
00:39:11You can, right? With that for now it will be two methods, right?
00:39:13It doesn't look like it. It doesn't look like it. It doesn't look like it.
00:39:15Did you understand the meaning?
00:39:17Then is it just this side where we can draw parallel lines?
00:39:20Over here?
00:39:21You can draw it, right?
00:39:22It's easy to see, huh?
00:39:24How about it? Are you done?
00:39:27How many methods did you get?
00:39:28Only two yet.
00:39:32You found ones like these? Hm? This. Hm? Which is it?
00:39:36One like this and-
00:39:37And this. This.
00:39:41Oh. Over here, huh?
00:39:43Then, okay? There are parallel lines like this, right?
00:39:46If the one that is the same as this triangle is this, then also at this side,
00:39:51you got it huh?
00:39:52(inaudible) [ Laughter ]
00:39:54You want to do it?
00:39:56Then would you do one of a different type? After you look for it.
00:39:58I found eight.
00:40:00Eight. Eight.
00:40:01Eight? You noticed the fact that there are eight? Splendid.
00:40:08Oh. That's good, huh? Uh, you're doing well.
00:40:12Are there others, Taniguchi?
00:40:16Then okay? Which ones did you get? This and... one like this and one like this, huh?
00:40:22Now then, is it just this side where we can draw parallel lines?
00:40:25This side.
00:40:26You can also draw them over here, right? Then can't you transform this triangle?
00:40:30Over here.
00:40:31Yes. If you draw it over here, then the area that is the same as this triangle is...
00:40:35this original figure has become somewhat harder to understand but-
00:40:38This (inaudible).
00:40:39Yes. Then it's okay to draw one more- one more or.
00:40:48That's good. This, okay?
00:40:52You did it so it says this triangle and the triangle over here are the same, right?
00:40:54Is it just this side where you can get it?
00:40:56Over here also.
00:40:59Over here.
00:41:00On this side?
00:41:01We can do it.
00:41:02We can do it, right? Then we can do it over here, right?
00:41:04Then how about this? Is it just this side where we can do it?
00:41:05This side.
00:41:06We can do this side also, right? How about this?
00:41:08Oh. You were able to do this. Over here. Hm?
00:41:15This is- you drew it like this and oh,
00:41:17you drew it like this? Yes. Yes. Yes. This is good, huh? Oh.
00:41:20This- it's the same as this, huh?
00:41:21Draw over here and you can do this, right? If you do that.
00:41:25You've gotten to a good point in the problem.
00:41:37You found eight?
00:41:39That's amazing.
00:41:41How many did you get?
00:41:43How many did you get?
00:41:45How many? One. Did you get one? It's okay if you get one.
00:41:46I could only get one.
00:41:49Okay? You were close in getting the other, huh?
00:41:51How did you get one?
00:41:52Can I borrow this of yours, Usui?
00:41:53No way.
00:41:54Hm. Oh. It's okay, right? You got one, right?
00:41:57Then these are parallel, right?
00:41:59This. This is parallel, right? This and this.
00:42:00Huh? This?
00:42:02Then the triangle and this orange-colored object is the same, right?
00:42:04Then do you get it only on this side?
00:42:06Which side?
00:42:07This side also.
00:42:08Hm? We can get it over here also, right?
00:42:10Already you found two.
00:42:11Yeah. Oh. Is it that?
00:42:14Oh. With here (inaudible), huh?
00:42:18(inaudible) Oh. That's terrible of you. (inaudible)
00:42:22Is it okay? This?
00:42:23I can see this.
00:42:24That's terrible.
00:42:25[ Laughter ] (inaudible)
00:42:26That's terrible.
00:42:27[ Laughter ]
00:42:29That's terrible.
00:42:32The fact about selecting our class is a mistake.
00:42:33[ Laughter ]
00:42:34It's an excellent class right?
00:42:37One more. Lastly... would you do it for us, okay?
00:42:39I'll do it.
00:42:41Well, please draw one that is not already there.
00:42:43After you look for it yourself. Okay?
00:42:45You're still going to it?
00:42:48Try looking for it.
00:42:50I always do it, right?
00:42:51Hm. Look for it. One that's not there.
00:42:52Teacher. Please tell me.
00:42:53May I wash my hands?
00:42:56Teacher. How many are there left?
00:42:57I don't get it.
00:42:59Then this also? This also?
00:43:01There are eight, right? If I remember.
00:43:03There are one, two, three...
00:43:04I don't understand it.
00:43:05It's very hard to write on it.
00:43:06Five, six, seven, eight, nine, ten types.
00:43:08I don't understand it from here.
00:43:09Hm. You don't understand?
00:43:11You got one here, right?
00:43:12It's good with this as one. (inaudible) With this you got one.
00:43:13What is it?
00:43:15These are parallel, right?
00:43:16This triangle and the triangle over here are the same, right?
00:43:18And so you got a triangle here. See?
00:43:19Ten types?
00:43:20Oh. Oh. Is that true?
00:43:22Right? Then if you get one on this side then?
00:43:25Over here.
00:43:26Yes. We can get it there, right? You've already got it.
00:43:38Now then. Wait a minute.
00:43:40I don't know anymore which ones are which.
00:43:43This is (inaudible).
00:43:44Oh. You noticed a lot, huh?
00:43:47How many types did you f- notice?
00:43:49Hm. First of all, this is one, right? Two, right? Over here is three. Four.
00:43:54Okay. Now, then this time is drawing diagonal lines just on this side?
00:43:59Hm? Other than this side can't you get a triangle?
00:44:02A way to partition it?
00:44:04That's right. That's right. That's right.
00:44:05Even on this side it's good if you think about it in the same way, okay?
00:44:07This. I'm not sure, but did both of them come off?
00:44:09The other one was still stuck here but.
00:44:10I was able to make a hexagon into a triangle.
00:44:13What? You want to make a hexagon into a triangle?
00:44:15All right. Then.
00:44:16Actually I don't want to.
00:44:17Since you, Phuse, are asking for it let's do it, okay?
00:44:18You're going to make a hexagon into triangle?
00:44:31Okay. Were all of them found?
00:44:34You found them? These then are quadrilaterals, right?
00:44:35The ones we're doing right now.
00:44:37Do you want to do them all at once?
00:44:38Do you want to do them all at once?
00:44:39[ Laughter ]
00:44:40Thank you.
00:44:41I don't mind. [ Laughter ]
00:44:42Penta- pentagon. Pentagon. Pentagon.
00:44:43Six. Six. Six.
00:44:44Take six. Si-.
00:44:45If it's a pentagon isn't it (inaudible)?
00:44:47So what. (inaudible)
00:44:49Who was that who said he or she found eight?
00:44:51It's these three.
00:44:52Did you (inaudible)?
00:44:53[ Laughter ]
00:44:54Next, okay? Challenge to change a pentagon, okay? Make a pentagon into a triangle.
00:44:55I don't know. At once change to a triangle from a circle.
00:44:57Cir- circle.
00:44:58Make a circle into a triangle.
00:45:00It becomes more and more difficult.
00:45:04You found it... did you? Okay, then... one two...
00:45:06one two... three... four, okay. One, huh?
00:45:18One. Of this side. One.
00:45:23Also isn't there one like this?
00:45:26Didn't we do this?
00:45:28One two. There isn't one on this side, right?
00:45:30Is there one like this? Oh. Is it this?
00:45:32Is it this?
00:45:33That's right huh?
00:45:35Lately I've worked really hard for my life.
00:45:36Also what else?
00:45:37The line have- the lines have become messy.
00:45:38Is it over with this?
00:45:39There should be eigh- eight.
00:45:40One, two.
00:45:42What is this? This?
00:45:43We haven't done that yet.
00:45:44One two.
00:45:46There are eight of them see? There are even ten quadrangles.
00:45:47Are all of them there? One, two, three, four, five, six.
00:45:49There are only seven, see?
00:45:51What is that over there?
00:45:52I don't know anymore which is which.
00:45:54One more.
00:45:55The top of this one is this, right?
00:45:57Is it this? Then it's this. Since it's the top of this.
00:45:59I don't know anymore which is which.
00:46:01I don't know.
00:46:02It's this, right? This one's
00:46:03No. No. There is the bottom one see? Here.
00:46:04I think we don't have the top one see?
00:46:05It's two more.
00:46:06Oh. The opposite side.
00:46:08Okay. Thank you.
00:46:10Okay then everyone please return to your seats at once.
00:46:12Yasuda. Hasn't drawn it, okay?
00:46:14Okay. I'm sorry.
00:46:15Everyone, let's get to our seats.
00:46:18Okay, then... everybody, okay? Did it for us, okay?
00:46:21Did you find eight? Good.
00:46:23Uhm. I don't understand.
00:46:26Teacher, there is no time left.
00:46:29There isn't time? That's true, huh? Now then... well, okay?
00:46:36For the convenience of explanation we will put in symbols.
00:46:40There were some who were doing it already labeled them, but okay?
00:46:43We will make them A, B, C, D, okay?
00:46:45And right now in the beginning we draw
00:46:48a diagonal line through A to C and
00:46:50draw a diagonal line from A to C and we will make a triangl-
00:46:53we were able to make two, right?
00:46:55And, uhm, if you ask which ones are the triangles
00:46:58we found that go through this D are... this and
00:47:02this, right?
00:47:04This one is made like this on the bottom side like this.
00:47:08This one is on the top side.
00:47:10Since this triangle and the original triangle are the same,
00:47:13so it becomes a fact that we changed this, um,
00:47:16quadrilateral into a shape like this, right?
00:47:19Uhm, next we drew in the same way through A C but
00:47:22this time we drew the parallel line on the B side.
00:47:26Okay? Those are this and... this, okay?
00:47:30It's this and this, okay. Is everything okay?
00:47:32Then I will ask.
00:47:34Uhm. About how many people are there who says he or she found this
00:47:36i- were able to find it?
00:47:40Okay. People who say they were able to find this one.
00:47:43There are about the same number of people, huh?
00:47:44Um how about this one?
00:47:46The D one. The sharp tri-. How about this one?
00:47:49Okay. That's good. Thank you. Okay.
00:47:50I don't understand all of them.
00:47:51What is it? You don't understand?
00:47:52On this side this time we will draw it on the B D side.
00:47:54Then in the same way, um, which one is it?
00:48:00Ones that we drew through B D and drew a parallel line through vertex A are
00:48:03this and... this, okay?
00:48:06Those are the triangle that we make on this side
00:48:08and the triangle we make on the top, right?
00:48:11And also the ones where we drew a line through B D
00:48:14and we drew a parallel line on the C side are
00:48:16this and...
00:48:18which one is it?
00:48:21Which one is it?
00:48:23Is there none?
00:48:24Is this it?
00:48:27It's wrong? Is this it?
00:48:29This? Huh? Is it?
00:48:31Okay then... for the time being, okay?
00:48:33We don't have much time
00:48:36but with the computer...
00:48:38I will explain it a little, okay?
00:48:40It's something. We have done it, but-
00:48:42Well, the screen is not showing up.
00:48:43It's showing.
00:48:44It came out.
00:48:45It's showing. Uhm, right now,
00:48:47well, the symbols are different from over there, but on B D...
00:48:50for now... we'll draw a line.
00:48:53And we will connect A.
00:48:55The parallel line is created. It's two parallel lines.
00:48:58Then there are a lot of them
00:49:01like this which are the same area as this triangle, okay?
00:49:04There are many of them,
00:49:06but� within these...
00:49:08since, the fact is, we're making the quadrilateral into a triangle-
00:49:11Meaning all we have to do is lose one of the angles. The angle here, okay?
00:49:13When it becomes a straight line then
00:49:15that will become a triangle.
00:49:18if it's this side
00:49:20when this becomes straight...
00:49:22we get a triangle.
00:49:24Then we have done quadrangles.
00:49:26Next what do you want to do? (inaudible)
00:49:28Five pentagon.
00:49:29Six hexagon.
00:49:30[ Laughter ]
00:49:32Phuse. You [ Bell ] want to do six by all means.
00:49:33Then five. Transform pentagons into triangles.
00:49:39A pentagon's- try drawing a pentagon of your liking. Then penta-
00:49:41I don't have a clue.
00:49:43That's impossible.
00:49:44It's impossible?
00:49:45It's impossible.
00:49:46You don't know if it is impossible until you try it.
00:49:47Okay? Then let's try making the pentagon-
00:49:48[ Laughter ]
00:49:49into a triangle.
00:49:52Right now?
00:49:53No. I'll make that homework.
00:49:55Okay? People who are interested
00:49:57you can do ten-angled or twenty-angled or one hundred-angled or anything.
00:49:59We- well, they were talking about doing a circle.
00:50:00A circle?
00:50:01A circle.
00:50:02You'll make a circle into a triangle? It's not impossible.
00:50:06Label it A B C D E and quadr-
00:50:09Cut the pentagon into a quadrilateral, quadrilateral and-
00:50:12(inaudible) the quadrilateral.
00:50:13Oh. Yuki has just now told us the method of doing it.
00:50:14The fact is... won't it be okay if,
00:50:16first of all, you make the pentagon into a quadrilateral,
00:50:17and then make the quadrilateral into a triangle-
00:50:19is all. Okay then you worked very hard...
00:50:22amazingly hard. Okay?
00:50:25Then today let's get something to drink at the (inaudible).
00:50:26It's wonderful, huh?
00:50:27What? [ Laughter ] Okay, then we will end class.
00:50:32Stop doing (inaudible).
00:50:33Reward. Reward.
00:50:40Okay. Then let's say the farewell properly.
00:50:45Thank you very much.