This eighth grade mathematics lesson focuses on two-dimensional geometry - in particular, parallel lines and angles. It is the third lesson in a sequence of seven lessons on this topic. The lesson is 50 minutes in duration. There are 31 students in the class.

00:00:02[ Bell ]
00:00:04How about today?
00:00:08Good morning.
00:00:09Stand up.
00:00:16Sit down.
00:00:22Okay. It's supposed to be as usual so let's go on without being concerned about it.
00:00:32Anyway your face might not be taped.
00:00:34[ Laughter ]
00:00:35Actually there is just going to be a black line across your face.
00:00:37[ Laughter ]
00:00:39Don't worry.
00:00:53Okay, yesterday was about the relationship between parallel lines.
00:00:56We ended quickly yesterday by doing some problems.
00:01:00Do you remember what they were?
00:01:27The first written problem on the handout we hurried through it yesterday,
00:01:31so we were not able to summarize it well at the end...
00:01:35So today in between the parallel lines there is a line that is bent into an angle.
00:01:43So first we will obtain that angle.
00:01:46When I had you do the problems yesterday
00:01:48we had three problem solving methods that were presented...
00:01:51It's okay to use the method you think is the easiest.
00:01:57Please try to do the problem.
00:02:32After you have finished writing, please add an explanation.
00:02:37My- my method is this and this way.
00:02:39If you can also give an explanation that would be amazing.
00:03:36This, right.
00:03:38You're fast.
00:03:54If you forgot how to solve it, it is okay to look at the previous handout.
00:04:06(inaudible) let me borrow this for me.
00:04:09Let me borrow this because he needs a pencil.
00:04:13It's a little different... line... (inaudible).
00:04:22Umm. Okay.
00:04:26Are you done? Can I ask? How is it?
00:04:32People who solved it?
00:04:35Okay then... Arai.
00:04:42Okay, what did you do?
00:04:43Fifty. The 50 degree.
00:04:46The line around it. The line there extend it.
00:04:50Okay. Did you extend it like this?
00:04:53Bring that over there.
00:04:56Uh- 50 degrees. Alternate interior angles of it.
00:05:00There. That is 50 degrees so.
00:05:04This? Okay.
00:05:06The right angle is 30 degrees.
00:05:09If you add it all up it becomes 180 degrees so.
00:05:12A triangle. Yes.
00:05:14Yes. The top part is 100 degrees.
00:05:18The one before. A right angle is.
00:05:18A right angle is.
00:05:21Um, it becomes 180 degrees so.
00:05:25Okay. Not a straight line?
00:05:26Yes. A straight line becomes 180 degrees
00:05:29So X is 80.
00:05:32Okay. Is this okay? Here we go. Okay.
00:05:35We can see he was a little tense.
00:05:38Um, that was a clear method. People who did it using this method?
00:05:45Oh... okay.
00:05:48About five people? Okay, other methods.
00:05:52Okay, Bunya.
00:05:56Okay, please go ahead.
00:05:58In the middle.
00:06:00Draw a... parallel line.
00:06:07Oops? Okay.
00:06:08Then. Um, the top 50 degrees, the alternate interior angles of that.
00:06:15It is made there so.
00:06:17Yes, well- Well, okay.
00:06:21So it's 50 degrees.
00:06:23The bottom is 30 degrees.
00:06:25It can also be done in the same way.
00:06:28Add those together it becomes 80 degrees.
00:06:31Okay. That's good.
00:06:32The method was to use the alternate interior angles of the parallel lines.
00:06:38People who did it with this same method?
00:06:42Right. About 10 people. Okay.
00:06:47Are there any other methods?
00:06:50Oh. One person... Chika.
00:06:56Umm, from point B.
00:06:58From point B. Okay.
00:06:59From point B.
00:07:01Is it... L?
00:07:04From the top line it would be an L, huh.
00:07:05To the top straight line draw a perpendicular line. Then-
00:07:09Draw a perpendicular line from point B.
00:07:11Then. The B area is 60 degrees. Umm- well not really B,
00:07:16but between that perpendicular line and 30 degrees is a right angle.
00:07:22So subtract 30 degrees from 90 degrees.
00:07:25It becomes 60 degrees.
00:07:27The area A is fi- fifty degrees so.
00:07:32Umm, the straight line is 180 degrees.
00:07:37Angle A becomes 130 degrees.
00:07:42Since if you add up all of the angles of a quadrilateral it becomes 360 degrees.
00:07:47Umm, if you add-
00:07:51what is this? It becomes 280 degrees. Eight.
00:07:55Subtract 280 degrees from 360 degrees.
00:07:57X became 80 degrees.
00:08:00Okay. Yes. Here we go. Okay.
00:08:04People with the same method as this?
00:08:08Oh. Just Doi. Okay.
00:08:11Good. Are there any more?
00:08:14Think a little more.
00:08:17I'm thinking.
00:08:18Okay? I am sure there are people who also extended this one.
00:08:23Here are methods like that.
00:08:25Basically these three are the ones you have thought of.
00:08:29Now I will summarize these into main points.
00:08:35With just this diagram...
00:08:38with just this diagram we can't solve the problem.
00:08:41So Sano and others extended the line to make a triangle.
00:08:45Bunya and others drew in a parallel line in the middle.
00:08:49Chika and others drew in a perpendicular line.
00:08:51That means we can't solve it without auxiliary lines.
00:09:12The auxiliary lines become important.
00:09:15Then by the way we draw the auxiliary lines in all
00:09:20we got three types. Sano's, Bunya's, Chika's.
00:09:25They were different based on the way the auxiliary lines were drawn.
00:09:28That means, one way is to draw a parallel line and
00:09:31use the alternate interior angles.
00:09:37Another is to extend the line and make... a triangle,
00:09:42and use the fact that the sum of the angles of a triangle is 180 degrees.
00:09:49And Chika used the method that if we add the four angles of...
00:09:56a quadrilateral we get 360 degrees. These are the three we got.
00:10:03Now then, any method was okay,
00:10:06but the task was to do it using the easiest method.
00:10:10And each person seemed to have selected one out of the three we summarized yesterday.
00:10:14Today's problems are ones that can't be solved without auxiliary lines.
00:10:17We want to do problems of this kind.
00:10:21Today. The way this problem is made is that
00:10:25it is a problem of an angle bent once between parallel lines.
00:10:32The angle is bent once between parallel lines.
00:10:36Today by changing this condition, by changing this condition-
00:10:41I will have you make your own problems.
00:10:45And today we won't change the parallel lines.
00:10:50We won't change the parallel lines, but rather the middle.
00:10:52Change the middle means-
00:10:55Oh. It's okay with the outside. The outside can be good.
00:10:59Now it has one bend.
00:11:02It is even okay to bend it twice.
00:11:03If I say this I know there is going to be someone who is going to bend it like 10 times.
00:11:05[ Laughter ]
00:11:08Eda (inaudible).
00:11:09Eda difficult.
00:11:10There are always at least two or three of you thinking about it?
00:11:12[ Laughter ]
00:11:15I know there will be some.
00:11:18But twice or three times. That's the limit.
00:11:21After that nobody will understand it.
00:11:23You may be creative in between and change the number of bends, or,
00:11:26it is also okay to use the outside with one bend,
00:11:29or it is also okay to be outside with two bends.
00:11:32Please think by yourselves.
00:11:35You can't just go and bend it 10 times. Don't draw random squiggly lines. That is not good.
00:11:38Okay, umm, I have prepared two on your handout, but one is okay.
00:11:45If it's easy please do two.
00:11:48It is okay to bend twice or be outside.
00:11:51Then put in angles by yourselves and
00:11:55write in which angle is X, the one we need to obtain.
00:12:17Oh, if 30 students do it,
00:12:22even if they made only one problem each we will have 30 problems.
00:12:27It's actually pretty hard to put in the degrees.
00:12:30You can't just put in any degrees.
00:12:33Do we also write the (inaudible).
00:12:43When you turn it in you must be able to solve it.
00:12:47If somebody says he or she doesn't understand it
00:12:49you must be able to say what and why the problem is.
00:12:51Please think about how the problem can be understood.
00:13:10A moment.
00:13:14Wait a minute.
00:13:29Um? This one, too?
00:13:39Which one?
00:13:56Um- make one and try it yourself. Make sure you can solve it, okay?.
00:14:28These are the same, since there is only one bend between the parallel lines,
00:14:31it's the same as this.
00:14:35Try to change it.
00:14:40Oh, it looks difficult.
00:14:44It looks pretty tough.
00:14:52This also looks difficult.
00:15:03You can put the bends in, but does the number fit in, is the problem.
00:15:16Isn't it-
00:15:21It has to be a problem you can solve.
00:15:33It can be the second one or the outside.
00:15:38Bam, bam, and it can be bent on the outside.
00:15:42Or you can bend it once, twice.
00:15:47(inaudible) Just one?
00:16:01Whichever equals.
00:16:02If you make it too hard, well it'll be... stressful.
00:16:12How many times is this bent? One, two, three, four times it has been bent.
00:16:15It's been bent too many times.
00:16:22Is this okay for the first one?
00:16:24Is this okay for the first one?
00:16:26You can't solve this.
00:16:28Without another angle I don't think it can't be solved.
00:16:34If you try to solve it and you can solve it yourself then it is okay.
00:16:46Isn't it difficult? This is difficult.
00:16:55If you have finished.
00:16:57Where do you solve for this?
00:17:00Where do you solve for this problem?
00:17:01For this, it is for this.
00:17:03Then what is this?
00:17:07There's a reason for this.
00:17:09It starts at point A and goes through point B and-
00:17:12Ah, I see.
00:17:16Ahh, it looks hard.
00:17:18This is simple.
00:17:20Ahh, it is hard.
00:17:21(inaudible) like these.
00:17:25Like this.
00:17:28This looks hard.
00:17:29This is 90 degrees.
00:17:30Oh, I made a mistake. This is going to be hard.
00:17:34This 65 degrees is last.
00:17:39Eventually, if you extend it here is 62. This is 180 degrees.
00:17:43Two hundred, 220.
00:17:52Is this 62 degrees?
00:17:56Here, lend it to me.
00:18:03One hundred twenty.
00:18:10At 15 you can do it.
00:18:33Are you done writing? Are you finished?
00:19:09It seems hard.
00:19:12Let's try hard to do one problem. Try hard to make one problem.
00:19:56Um, if you put it on the inside maybe it'll be easier to solve.
00:20:01Wow. This is how you take off from A and bend and reach B.
00:20:08What does that mean for A?
00:20:13This is fine.
00:20:18Is it solvable?
00:20:19You can solve this.
00:20:20It is solvable.
00:20:22It won't be solvable without these angles here.
00:20:24What? Add this, like this?
00:20:26Like this. If you extend it... it's 60 degrees.
00:20:38This, isn't this 60 degrees, this... like this isn't 60 degrees?
00:20:42That doesn't work if this and this are not parallel.
00:20:47If you draw a parallel line.
00:20:49If you draw a parallel line.
00:20:50If this and this are not parallel then this angle is not 60 degrees.
00:20:55This angle, too... can't be used under that assumption?
00:20:59No. If they're parallel lines then just put in the symbol for parallel lines here. (inaudible)
00:21:16Thank you very much.
00:21:19I don't know if they'll understand this.
00:21:21It looks a little tough. This is too difficult.
00:21:25And right... I made it a little too difficult. I made it too difficult.
00:21:32Here it's hard.
00:22:06If there isn't an angle here it can't be understood. Just put something in.
00:22:21You guys seemed to have learned that any number just doesn't fit in.
00:22:25You guys seemed to have learned that any number just doesn't fit in.
00:22:46How many bends are there? One, two, three, four, five bends.
00:22:50If you don't make it at most three then it will be impossible.
00:22:53I understand.
00:23:11Let's see. What time?
00:23:12Well it seems it was a little hard. I made a mistake.
00:23:16There are many of you that are in trouble.
00:23:20Well there are not that many people with one, two done,
00:23:22but there seems to be a lot of people with one done,so,
00:23:24get in your groups and... from the problems you have made...
00:23:33pick a problem you and others think is challenging,
00:23:38and group leaders please bring them up here.
00:24:01Please check if the problem can be really solved and then bring it up.
00:24:21One problem is enough.
00:24:23There are problems that can't be solved.
00:24:43How about this one? This?
00:24:46Th- Th- This (inaudible).
00:24:57(inaudible) It seems hard.
00:25:03Can't it be solved?
00:25:06You don't have to make a new problem at this point.
00:25:09Just from the currents ones you have.
00:25:19This is a good problem.
00:25:20Is it good?
00:25:23Yes, because it's two bends, and the bends are different.
00:25:27And this is also a good problem, whether you can do it or not,
00:25:30change that part. Not too much.
00:25:36Up to number two?
00:25:56(inaudible) a quadrilateral.
00:25:59Ha ha. Indeed.
00:26:01A quadrilateral. A quadrilateral? (inaudible)
00:26:13Who made this?
00:26:20You can't solve it.
00:26:23Mmm. Can they do it or not?
00:26:54Hey, let's bring up one.
00:27:04Ha, it looks like a hard one.
00:27:05Whose is it?
00:27:07From which group?
00:27:08From group number five.
00:27:09This one?
00:27:11Group five.
00:27:36It's a hard one.
00:27:42Oh, the diagram was too small.
00:27:45You're completely wrong. Do you know what you are doing?
00:27:50This part is easy.
00:27:56Here teacher.
00:27:57This- which group?
00:27:58This one.
00:27:59This can't be used.
00:28:00It can't?
00:28:01Uh huh. Because this is bent many times.
00:28:03Teacher. This one.
00:28:04Just a little more- this is too hard.
00:28:06He said it was too hard.
00:28:08Oh, it looks like a good problem. What group?
00:28:11Group one.
00:28:12Group one?
00:28:19Group one?
00:28:20Group one.
00:28:27The diagram is a little...
00:28:56Which number?
00:28:57Number four.
00:28:58Number four- can you really solve it?
00:29:01That can be solved easily.
00:29:02We can.
00:29:03It seems.
00:29:06Is this all?
00:29:34Okay, that's good. Yes.
00:29:38Next, who's next?
00:29:41It's easy. That one.
00:29:44Don't say anything yet. Don't say anything.
00:29:47Groups that have not yet brought up one, please hurry up.
00:29:54Teacher, I solved this.
00:29:57I wonder.
00:29:58It can- it can be solved teacher.
00:30:01Can this be solved?
00:30:02I believe it can be solved.
00:30:06I think it is impossible.
00:30:09This one, this one.
00:30:10Which one?
00:30:11This one, this one. (inaudible)
00:30:13Mm. It's probably right?
00:30:19Which one is that, group three?
00:30:20Group three.
00:30:26The second time is hard but easy.
00:30:37Number one is easy, too.
00:30:50Now number four is left.
00:31:01Which number?
00:31:02Number four.
00:31:03Number four?
00:31:04Number four. It might be complex. Right, it's different.
00:31:06It might be complex, okay?
00:31:08I'll do number four just a little. Sorry, okay?
00:31:32Can it be solved?
00:31:33It can be solved.
00:31:34You're able to solve it? Hum
00:31:36Hum oh.
00:31:42What number?
00:31:45Hum. Okay.
00:31:48They are problems that we don't know if we can solve or not until we try.
00:31:54I'm sorry for taking so much time,
00:31:57but there are six problems on the right side.
00:32:06(inaudible) if we can solve it or not.
00:32:11I wonder.
00:32:20It seems impossible to do all of them in this time so.
00:32:23So we'll think about them a little next time, too.
00:32:36Please hurry up and copy the six problems.
00:32:44Imai is X here?
00:32:50Hum. (inaudible)
00:32:54Oh, I got it.
00:32:57Can you solve it?
00:32:58I got it. I got it.
00:33:09They really seem challenging, huh?
00:33:12There seems to be one that is too challenging.
00:33:19They are really challenging, like eating a dried squid.
00:33:24[ Laughter ]
00:33:27Umm, this one is not that bad.
00:33:37Umm... I'll say it's like a piece of gum.
00:33:47Oh, group two's, group two's, the one on the top.
00:33:50Umm. Group four.
00:34:01This is a rock.
00:34:03[ Laughter ]
00:34:05Then this is a rock.
00:34:31Umm. Um um um.
00:34:52Rice cracker.
00:34:53Number five. Number five.
00:34:55Isn't it easy?
00:34:59Rock, it's a rock.
00:35:05Then I'm sorry there isn't much time, but there are six problems.
00:35:10There may be ones from your own group. There may be ones that are yours.
00:35:12Please copy them.
00:35:15It can be from number one. It can also be from the hardest one.
00:35:21(inaudible) This is gum.
00:35:46That means since today's problems can't be solved without auxiliary lines...
00:35:50what kind of auxiliary lines. It can't be solved without auxiliary lines.
00:35:57If you need the auxiliary lines,
00:35:58the problem is what kind of auxiliary lines should you use.
00:36:03There were three, but which one is the easiest way to do it.
00:36:29For now please finish copying the problems by 35 after the hour.
00:36:35I think... the answers will be given next period.
00:36:41What shall I do?
00:36:44Can you solve this?
00:36:45I don't know.
00:36:49This is well... this is really hard.
00:37:10Genji's symbol is good. It's there nicely working.
00:38:40Are there any groups that were able to solve them?
00:38:43Plus 15.
00:38:48(inaudible) since it's 15 degrees, so 90. Ooh.
00:38:52Not so (inaudible).
00:38:54Oh- the rice cracker hard problem seems to be solved.
00:38:57(inaudible) fifteen. It's 30 degrees.
00:38:59Teacher. Fifteen.
00:39:01Fifteen, 15, 30, 90, 90.
00:39:02It's not 90, 90. Ooh.
00:39:04One hundred.
00:39:11It's correct. We got this. We solved this.
00:39:19The rock?
00:39:20Later. We'll check it later.
00:39:23We'll check it later.
00:39:31No, at the time- at the time I was solving it I was thinking that I couldn't solve it.
00:40:14I didn't write the group names.
00:40:35Since all of them are challenging we can't solve it just like that.
00:40:39Let's do it step by step.
00:40:43People who are done solving the problem from group one?
00:40:47Okay good. Group two's?
00:40:50A little more. Straight. Raise your hands straight. Ahh, about the same.
00:40:54Group three's?
00:40:57Oh. Group four's?
00:41:00Okay. Group five's?
00:41:03Okay. Group six's?
00:41:05Really. Group six's. Okay good.
00:41:08Number six is easy so-
00:41:12There is an even harder one.
00:41:14This one seems more likely to work out.
00:41:22I haven't gotten this one yet, but-
00:41:26It's hard?
00:41:27I just finally got it.
00:42:04Is it group one or group two?
00:42:05One hundred eighty degrees?
00:42:07Is this 40?
00:42:09Forty degrees.
00:42:15One hundred twenty degrees.
00:42:16One hundred twenty degrees.
00:42:18Why is that? (inaudible) Isn't this 47? Right here.
00:42:36This is easier. Umm. This one symbol. If you miss that one symbol...
00:42:43you can't solve the problem. This is parallel.
00:42:46This and this are parallel. Without that you can't solve it.
00:42:54That's right. That's right. That's right. If you do that. It's about done.
00:43:23Group one's is hard. I think group one's is really hard.
00:43:48How you draw the auxiliary lines makes the difference. Something...
00:43:57One hundred ninety. This is 190.
00:44:07Number eight. Number two. Number eight. Number eight. (inaudible)
00:44:19Okay, do you want to try it again?
00:44:28I'll give you a hint. Try extending this further.
00:44:32Try extending the auxiliary line further.
00:44:53Try this one. This one is easier.
00:44:59There is a symbol included here. This parallel symbol.
00:45:05Extend the auxiliary line all the way to here. Try extending it.
00:45:15Yes, yes.
00:45:18This and this are parallel. If you use (inaudible) angles this angle becomes the (inaudible).
00:45:25This is a straight line so you know it. So you somehow get this angle.
00:45:56You know this angle and this angle (inaudible).
00:46:00If you write in as much as you'll work.
00:46:18Oh that's good. They're the right auxiliary lines.
00:46:24Umm. Ahh, this is (inaudible). This is hard.
00:46:27You can't solve it without auxiliary lines.
00:46:30That's good.
00:46:33Umm, 120 degrees and then the straight line.
00:46:40Like this line and like this, use auxiliary lines and try drawing it.
00:46:44It's good to draw auxiliary lines.
00:46:48Man. This one is amazing.
00:46:58Umm, I see.
00:47:24Do it one more time.
00:47:25I did it once.
00:47:26A mistake.
00:47:27Uh I thought it was wrong. (inaudible)
00:47:29Umm um. That it. That's it. That's it.
00:48:07(inaudible)? (inaudible)
00:48:12Umm, that one.
00:48:18Umm, um, um. Divide it around here. It's the auxiliary lines.
00:48:23Auxilary lines. It's a lot different with another auxiliary line.
00:48:29But even if you didn't get the last one, you got this one, right?
00:48:33This and this are parallel,
00:48:37and here and here are corresponding angles, and you get here.
00:48:40You got there and with that...
00:49:10This is going to be a little bothersome but, well, I want to know the present situation.
00:49:14People who are done with group one's?
00:49:18You solved it?
00:49:22Umm. Group two's?
00:49:27Oh, okay. Group three's?
00:49:32Um, okay. Group four's?
00:49:35Up to where did you get to? Um. Group five's?
00:49:40Hum. Two people. This was a hard one.
00:49:42Umm. So that's it. Group six's?
00:49:45Group six's was simple.
00:49:46It was very troublesome.
00:49:49Did you do it?
00:49:50(inaudible) Okay then there are a lot of people who are using triangles.
00:50:00[ Bell ]
00:50:01That's okay, but- There were three types of auxiliary lines.
00:50:06There are other easier methods of solving... using other types of auxiliary lines.
00:50:11We will check these in the next period.
00:50:16Return your desks.
00:50:33You can never do this one.
00:50:38Who got group four's? Who said he or she got group four's?
00:50:41Fujita Today. Okay.
00:50:46Please end the class.
00:50:48There's no math class tomorrow so let's check them the day after tomorrow. Okay.
00:50:50Do we have math the day after tomorrow?
00:50:53We have math class the day after tomorrow right?
00:50:58The day after tomorrow is Thursday.
00:51:02Did it change?
00:51:04The day after tomorrow is Friday.
00:51:07Today is what day of the week?
00:51:10Today is Tuesday.
00:51:13Day after tomorrow?
00:51:15That's right. I got it. Okay.
00:51:17Go ahead.
00:51:19Stand up. Bow.