# SW2 INTRODUCING ALGEBRA

This eighth grade mathematics lesson focuses on terms and variables. It is the first lesson on this topic. The lesson is taught in Swiss German and is 45 minutes in duration. There are 25 students in the class.

TimeCaption
00:00:38Eh, Mr., can we watch the video sometime?
00:00:42Sure.
00:00:44At the end of the lesson?
00:00:46No, hardly.
00:00:55All right, close your books, please. We are going to begin a new chapter in mathematics today.
00:01:03It is our first algebra chapter in third grade secondary school. I have written two, three things at the board,
00:01:12which are important in this algebra chapter. It is about terms and variables.
00:01:19Terms, that's a- that word derives from Latin, terminare means determine. So you could say that we are going to determine
00:01:28something with terms. Another word that will be important, is the word variable. A variable is, another word for it is...
00:01:40Not yet heard of it? A substitute symbol, a placeholder. We are then going to operate with these placeholders or, as we learned before, variables.
00:01:48And to operate in mathematics means nothing other than to calculate. We are going to calculate with variables.
00:01:56Okay, that's what we will have to expect in the next couple of math lessons.
00:02:00Now I have drawn a sketch at the blackboard. What is it?
00:02:08I heard lines. More specific. Anja?
00:02:12Two lengths.
00:02:13Correct, that's two lengths.
00:02:16And, if I write the number five next to this length and the number three next to the other length,
00:02:26then it is already somewhat clearer. I now have two lengths and I am now able to operate with them. For example...
00:02:48I can illustrate this equation with these two lengths. The equation five minus three equals two. I want to show you,
00:02:57how it works, and I will give another equation afterwards, which you will solve by yourselves using this one as an example.
00:03:08So now I am going to present this equation five minus three equals two algebraically.
00:03:17In order to do so, I will draw two straight lines, which have a distance of two unit squares. I will determine a starting point on the upper straight line.
00:03:38With the compass... I will... mark... this length five... like that.
00:04:10All right... five... from the starting point... until here.
00:04:20Minus three.
00:04:23I take the compass... and adapt the length three to it.
00:04:37Minus three, that means that I will go in the opposite direction, to the left again.
00:04:57I get the rest of the length... from the starting point until here.
00:05:18The length two.
00:05:22I have now illustrated the equation five minus three equals two.
00:05:27By first drawing the length five to the right. Two square units down the length three to the left,
00:05:36because it is minus, and marked the result two in color. Can you understand that?
00:05:46In that case I will give you another equation... That was our first problem, the one I explained to you at the blackboard.
00:05:57Our problem: three... times five, minus... two times three equals nine.
00:06:13That's problem one, page 40. Take textbook and notebook and solve this problem in your notebook now.
00:06:51One glance at page 40 in the book will show you at the top in problem one, what I have been explaining to you.
00:07:00The two lengths are also drawn in the book. Now you don't have to take the protractor to measure a length of five.
00:07:09You must take the compass and do the same thing in the book as I did at the blackboard.
00:07:18Namely, adapt the length to the compass and then mark it.
00:07:50I can see that a lot of you are first drawing two lines with a pencil in a distance of two square units. That is absolutely correct. Do it that way.
00:08:03How long these two lines are is not significant.
00:08:09What is significant is what you will draw on these straight lines. And that's what the equation will tell us.
00:08:16Three times five minus two times three equals nine.
00:09:46Who can say in words, how he is going to mark three times five on the upper straight line?
00:10:03Anja?
00:10:04I draw this length twice- eh- three times next to each other.
00:10:09With the compass. Yes, exactly. You will adapt the length five into the compass and mark it next to each other.
00:11:27Don't forget the arrowheads, they indicate the direction.
00:11:38With the number five we mark that and show that we're dealing with the length, with the number five.
00:11:51Now explain to me what I should do on the other straight line... Bal- Rafael?
00:11:58I take the compass of three centimeters and mark it from the right to the left.
00:12:04What exactly are you doing with these three... with the length three that you have been adapting to the compass?
00:12:13Eh- I subtract it from- from the lower straight line from the right to the left.
00:12:18How many times?
00:12:20Twice.
00:12:21Yes, correct.
00:12:24Once, twice.
00:13:17Left is a piece that we will mark with yellow color. We also want to give it a so-called name. Which one?
00:13:28Anja?
00:13:29Nine.
00:14:26I would like to finish this first problem. Are there any questions concerning it? Nothing.
00:14:54All right, now let's move to another problem.
00:15:01On the left side of the blackboard I have, instead of lengths, put up these... colored paper sheets. The blue ones stand for a length of five,
00:15:17and yellow ones for a length of three. Now try to illustrate the following number with these sheets. The number 17.
00:15:34Right here at the blackboard. What is there to do? How can you present the number 17 with blue sheets of five,
00:15:41and with yellow sheets of three. Give it a try, Fabian. Comment on what you're doing.
00:15:50Well, you have to, eh- five, plus five... plus five... plus five... minus three.
00:16:19Well, can I present it in one step? Without minus.
00:16:23Eh, I would like to have that judged first, what he did. Is it okay like that? Esti?
00:16:31Yes, I think so. Well, maybe you should- ehm- move over- ehm- the yellow one. So that it's (inaudible) with the line... Yes, exactly.
00:16:43Okay... Now where can the number 17 be spotted? Can you show it, Tamara?
00:16:52Ehm- down there.
00:16:53Yes.
00:16:58Here.
00:17:00From the left margin of the blackboard to the margin of the sheet on the right.
00:17:04Yes.
00:17:05Good. Thanks. Do you have another idea?
00:17:08M-hm, yes.
00:17:09Namely?
00:17:10Eh- five, and four times the three... times the five.
00:17:15Also equals 17. Good. Well done.
00:17:23New number.
00:17:38Sheets of five, sheets of three. Present the number of 26.
00:18:10Sandra, please also comment on what you're doing.
00:18:14Five... times three...
00:18:34Plus three times three...
00:18:39Christoph.
00:18:51Plus three.
00:19:03Jenny?
00:19:04Equals 27.
00:19:07Yes, on the one hand it equals 27. And further, Andi?
00:19:12Ehm- you ehm- have to, if you calculate three times five, you have to take the three five times and not- one sheet of five and one of three.
00:19:20Otherwise it would mean that this is the length of 15.
00:19:27But that isn't the length of 15. What is the length in this case here? Tamara?
00:19:32A length of eight.
00:19:33That would be a length of eight, because the length five plus the length three, would add up to the length eight.
00:19:42You considered another operation. Multiplication. You said that this was nine.
00:19:49Three times three. In reality that would be another length, Patrick?
00:19:55Six.
00:19:56And what would be the total length now? Not 27 or 26, but a total length would be here?
00:20:07(inaudible) teen.
00:20:10Seventeen, right. Eight, six, three. So we have to choose another way of doing it. Who is going to try?
00:20:31Who will give it a try? Twenty-six... Adrian.
00:20:43At the blackboard.
00:21:00So five times five... equals- equals 25.
00:21:06Stop! Now I expect a reaction from the rest of the class. Stephanie?
00:21:11Equals ten.
00:21:13Well, can't I multi- eh, multiply that?
00:21:17Why not? All right, Adrian tried to do it the same way as Sandra did before. With a multiplication, right.
00:21:26What do we disagree with there? Desiree?
00:21:30He should put up all sheets of five, because that's the only way they can be 25.
00:21:36Because we also have to count five and five and five.
00:21:41Well... That way- yes, I will let Anja explain, too.
00:21:45Ehm, five is just like a length of five centimeters and five, five is 10 centimeters. But you want 25.
00:21:55One moment, do you understand that?
00:21:57Yes, ah the (inaudible).
00:21:58The length five, and once again the length five adds up to the length 10. You have marked it twice and you want 25.
00:22:07Anja gave you the hint. Now you could think about if it makes sense to mark the length five, five times.
00:22:18What do you think? We want 26 at the end. Michelle?
00:22:23I would only mark it four times.
00:22:25So there are two opinions opposing each other: to mark it five times or four times?
00:22:31Four times.
00:22:32Four times. Adrian... Now here's the suggestion from Michelle and the confirmation from Fabian, you should only mark it four times.
00:23:07Yes.
00:23:08[ Laughter ]
00:23:13I would have put the two threes under...
00:23:17Under. What do you mean by under? Do it.
00:23:22Adrian can go back to his seat.
00:23:26No, it's wrong.
00:23:28Then let's drop it. All right. Five plus five plus five plus five equals 20, plus three equals 23,
00:23:38plus three equals 26. Now you can solve such a problem by yourselves, in your notebook, and again with the values
00:23:48that you have in problem one, with the lengths from the textbook, adapt the lengths to your compass.
00:23:55Choose example D. The number four, so to speak.
00:24:05That is problem two, page 40, letter D. Present the number, with the lengths five and three.
00:24:48Was that clear, before?
00:24:50Yeah, yeah.
00:24:52Good. That's correct. Just write down: three. Because that is the length three that you have marked twice here,
00:25:01three, write three.
00:25:22Are you aware of why you're making arrowheads?
00:25:25No. Not really.
00:25:26What does the arrow indicate?
00:25:29Oh yes, I know. Where it goes through, well- it goes this way and afterwards at the lower line it continues that way.
00:25:35M-hm, yes, well the-
00:25:36Yes.
00:25:37So now what indicates the direction to the right? What calculating operation?
00:25:43Plus.
00:25:44Yes exactly. And the counter operation of plus is, as you know, minus. That's why you go in the opposite direction with minus.
00:25:48Minus.
00:26:01It becomes even clearer, Silvio, if you trace the result with color. Here it would be nine, hm.
00:26:09M-hm, yes.
00:26:10Again with the arrow.
00:26:25Anja is done. Could you also do the problem at the blackboard?
00:26:38With the sheets?
00:26:39No. With the compass.
00:26:49That would be five again, and that would be three. You can adapt these two lengths to the compass. Down here.
00:27:04Push it down a little bit so that- that we have something different here.
00:27:27For those who have finished: Think about the other problems in number two, without solving them in your notebook,
00:27:35but in a way that you would be able to explain how to proceed afterwards.
00:28:07So now I'm going to ask you how to present problem two A, the number 15. Fabian?
00:28:14Eh, mark the five three times ehm- on one line.
00:28:18Good. B, the number 12. Esti?
00:28:24Also three times the five, and then one times the three on the second line.
00:28:30In the...
00:28:31In the counter direction.
00:28:32Good. C, 13? Carmen?
00:28:37Two times the five, and in the counter direction one times the three... That is minus. Plus... eh- (inaudible).
00:28:52Good. Everything in the same direction. Twice the five you said, and once the three. Now we can do the check at the blackboard.
00:29:00Anja is almost done with this presentation of the number four... Please evaluate it.
00:29:26Eh, not so exact, but it is correct.
00:29:28Well she did the right step. If you say not so exact, what is it that is not so exact?
00:29:33Eh, the lines.
00:29:34M-hm, I see.
00:29:35Not so exact.
00:29:36Good. All right, so those who are not used to working with a compass and chalk at the blackboard... then it's maybe no- not possible to present
00:29:46it the way that you would like to have it, Rafael, if it's that what you meant. But basically it's correct.
00:29:51Eh...
00:29:52Two times the five, and then in the counter direction two times the three. Check your solution in the notebook.
00:30:01And then we will go one step further.
00:30:26I would like to do an example form number three. Patrick, please read it out loud.
00:30:35(inaudible) are drawn.
00:30:44All right. And what lengths do we have at our disposal now.
00:30:52Which lengths do you have at your disposal now?
00:31:05Well what are they called. Five? Three?
00:31:07X, Y, and Z.
00:31:09Aha.
00:31:13So now you haven't got numbers anymore, but so-called variables, placeholders instead of a number, maybe.
00:31:22And we want to do a presentation again, as we did before. Draw a length which will correspond to the term X plus Y minus Z.
00:31:34You don't have to do anything in your notebook now. So how do we present the term X plus Y minus Z... Fabian.
00:31:49First the- just draw a line somewhere a (inaudible).
00:31:52One moment, one moment! Explain to me- okay I will just do a sketch for now. Exactly the way that you instruct me to.
00:32:00Then put a starting point somewhere.
00:32:02Yes.
00:32:03And then mark the length X.
00:32:14From this point mark the length Y.
00:32:27And then a second length...
00:32:30Line.
00:32:31Eh, line, and then from this point at the Y (inaudible) ehm, in the counter direction Z.
00:32:50Thank you Fabian. There you have a yellow piece of chalk. Show the term X plus Y minus Z on your drawing... Fabian.
00:33:21Menhir.
00:33:23It is correct.
00:33:26What would you change? Or did you think it was correct?
00:33:30Yes.
00:33:31It is correct. Maybe just the arrow, hm. Well done. X plus Y minus Z, that's the term. Good. Jenny? You don't agree.
00:33:45Yes, explain.
00:33:47That, eh- that's the, eh- the last length, which Fabian drew just now.
00:33:52Yes.
00:33:53He- he actually should have drawn the length X, the length Y and the length Z.
00:34:11Hrm, no, it's correct. Because...
00:34:19Because X plus Y minus Z is equal to a length, and that length you have to in-
00:34:26And with white, what Fabian did with white chalk, is the way to get there. The idea, X plus Y minus Z and then it's exactly like the
00:34:39result, the way you would like to eh- name it. Now this is exactly what is called a term, X plus Y minus Z. Good.
00:34:48You're now going to get different problems, which you are going to sketch up at the blackboard afterwards, in pairs... Problem three.
00:35:05B, C, D... E, B, C, D, E, B, C, D, E, B.
00:35:29Discuss with each other, you don't have to write anything in your notebooks. Decide together how you want to present the solution at the blackboard.
00:36:06Are you finished? E. E, yes. Think about D like Daniel... Finished? You too. Think about E.
00:36:18E?
00:36:19E, yes.
00:36:35Very well. Finish your discussions. Problem B. Five times Y.
00:36:50You've got that. No? Who has got B? Here, good... Show us.
00:36:59Only sketches, without compass. Christoph and Rafael, your task as well as for all of the others: evaluate them.
00:37:14It will come out even better, Fabian and Menhir, if you comment on what you are doing.
00:37:18Eh, well, here we have drawn a straight line and eh- a starting point, and now we will mark Y here, the first time.
00:37:30From this point once more... once more...
00:37:45And...
00:37:55The yellow here, that's the length five times Y.
00:37:58Could you also name the partial lengths?
00:38:14Rafael?
00:38:15That's correct.
00:38:18For all of you? All right... Problem C was solved by Miriam and Sandra.
00:38:39You have? Which one?
00:38:42C.
00:38:43Also C.
00:38:46We mark the length Z three times.
00:39:36Could you also mark in yellow what three Z is? From where to where does three Z go?
00:39:50Mh, from here to here.
00:39:51Show it with yellow color.
00:40:02Thanks. Three times Z.
00:40:09Christoph?
00:40:10It's correct.
00:40:11That's right. Christina. Are you ready for D- Y plus four times Z? Please go ahead then.
00:40:30I'm making a length again.
00:40:33Length?
00:40:35Eh, a straight line. And mark the starting point. Then I'm marking Y.
00:40:53And four times Z.
00:41:14Now that's the- the length.
00:41:18What length?
00:41:19Y plus four times Z.
00:41:21Good... Do you agree? You're nodding, good. Then we will proceed to the last one, problem E. Who would like to show it?
00:41:36Desiree and Anja, please, yes. Two times X plus Z.
00:41:46All right, I draw a straight line... Put a point somewhere. Then I mark Y twice.
00:42:14And then Z once again.
00:42:45And with yellow you showed, from here to here, that's the term two times X plus Z. Yes. Good. Comments? Jenny?
00:42:57It's correct.
00:42:58Is correct. Good. Now you have noticed so far in this lesson that with these variables you are also able to present something figuratively.
00:43:06But to operate doesn't only mean to present figuratively, it also means to calculate.
00:43:12The following problem four is going to show you, for the first time, how you can calculate with variables. It is a swimming pool problem.
00:43:19I will read from the book and you are supposed to be at the same place, page 40 number four, at the swimming pool, meaning-
00:43:25The letter E does not stand for a length anymore, but in our case, Debbie?
00:43:35Exactly. Which means that- the letter E stands for one adult. It doesn't say how much it costs.
00:43:45Three Francs, four Francs. Just the admission fee for one adult. In exactly the same way the letter K stands for, Pascal?
00:43:53Eh, the admission fee for a child.
00:43:55The letter A means in our case, Tamara?
00:44:01Yes, if somebody wishes to rent a locker, he has to pay an additional fee. And finally the letter B, according to our problem, Menhir?
00:44:10If somebody uses a cabin, a further additional fee has to be paid. Now if a family, here in the text book the name is Fuchs,
00:44:17pays for the following: Two times E plus three times K... plus three times A... plus B... What can we make out of that?
00:44:39The family Fuchs goes to the swimming pool and pays the following amount at the cash register: two times E plus three times K plus three times A plus B.