This eighth grade mathematics lesson focuses on geometrical solids. It is the first lesson where students use the Pythagorean Theorem to explore solids. The lesson is taught in Swiss Italian and is 50 minutes in duration. There are 17 students in the class.

00:01:07All right, good morning everybody.
00:01:11Well, let's imagine today to- eh, we are getting closer to Christmas time.
00:01:16We want to send a present to our friends. A present that has a long and narrow shape.
00:01:24It could be a thin straw, it could be I don't know-
00:01:29Has anybody an idea? I heard that- Joel you were about to say something?
00:01:32No, no, no, nothing
00:01:34No idea? A long and narrow object. So we ask at the post office, has anybody noticed,
00:01:40at the post office there are boxes with different shapes. And these are the six shapes you can see over there.
00:01:46There is box number zero, box number one, number two, three, four,
00:01:53and five, and those boxes have different shapes determined by the post office.
00:01:59Now ever- every group- you sit together in groups of three and one group of two-
00:02:04will receive a box. Start observing this box a little bit, there are
00:02:10some instructions in common that they give together with the box. Look to see what
00:02:15your box number is, the measurements, the dimensions of that box.
00:02:19And then take one of these straws, try to find out the position in which the straw
00:02:27can be placed to obtain the longest straw. That means you insert the
00:02:32whole straw, it's very possible that it doesn't fit even in the biggest box,
00:02:36and try to obtain the longest straw by cutting the straw from the start
00:02:43of the point in which you think you received the longest straw.
00:02:46That means to insert it that way, insert it horizontally. Try to find out which is
00:02:51the longest straw that you can thread into this box. Once you have found out
00:02:56the position in which the straw is the longest, cut it-
00:03:00cut it, you can keep the rest of the straw if it helps you for further observations you can
00:03:05also cut it. Measure it, you have a tape measure, you have a ruler, measure the maximum length that you can obtain.
00:03:16At this point you have a measure. Check with the calculations if the measure you found is correct.
00:03:25To do this, you have to observe the box a little bit and see how you can find
00:03:30the length of this straw. The calculations, do them, eh, one copy per group.
00:03:37Write a draft of your calculations trying out the reasoning that you need.
00:03:42Then everybody will receive a worksheet and will complete the space where the number of your
00:03:49box is, by writing the solution that you have found. After each group is done, I'll pass by and
00:03:56you will complete a transparency with the results you have found. Therefore the materials that you
00:04:01need are: Calculator, mmh, a draft sheet per group. And you work in
00:04:06groups of three or two. Only one person can move the chair, okay?
00:04:34You're welcome.
00:04:41To do the calculations you have to find (inaudible) calculations so that you can find this measure of the straw?
00:05:12Madam, but- and when is not (inaudible).
00:05:51Also observe your instructions a little bit, eh?
00:06:08Madam, can we already cut it?
00:06:09Of course you can already cut it, yes.
00:06:16Madam, Madam, can we (inaudible)?
00:06:27Cut the straw already. Have you found the best position?
00:06:30Eh, (inaudible).
00:06:36Well, cut it.
00:06:38And if it's wrong?
00:06:39In case it's wrong you take another piece and then you use the rest of the straw.
00:06:50Eh, it's forty point five.
00:06:55Eh, yes.
00:07:11You do it that way?
00:07:12Put it a little (inaudible).
00:07:13You measure forty?
00:07:14Forty point five.
00:07:15Well, well, this is the length (inaudible).
00:07:20Look, you have instructions, eh, that are already given.
00:07:23Ah, already.
00:07:28Try to cut it and then if it doesn't work you'll take another piece of straw and- try.
00:07:33We only cut one piece.
00:07:35Find the longest straw that you can place in your box, m-hm, yes.
00:07:38Madam, like this it's longer (inaudible).
00:07:42(inaudible) calculate it like this.
00:07:44Try to think about- it isn't that complicated.
00:08:02So did you already cut the straw?
00:08:06Go on, cut it and also if you fail, you can do it again. Look, you have already got the- the instructions.
00:08:13Yes, yes, but we have already found-
00:08:14We have already found it.
00:08:16You have it already- then- okay. Complete the sheet.
00:08:19But don't we have to do the calculations?
00:08:23We- we've got it
00:08:24You have inserted, cut and measured it.
00:08:28Okay. Then now you can try with the calculations to check if this length corresponds to what you have measured.
00:08:38(inaudible) the diagonal like this so (inaudible).
00:08:46So complete: Here with your situation, with your straw, you have the box number two and you put it- do it again here.
00:08:58Ah, we have to do-
00:08:59And then start doing this, call me to have a look at it when you've finished.
00:09:02Like this, Madam (inaudible).
00:09:07Now have you found the straw and have you measured it?
00:09:10Now observe the box then put it aside.
00:09:17Insert the straw: What- if I- so in your opinion, what is the longest straw laying in the box?
00:09:28No, no shorter.
00:09:35One that's shorter.
00:09:37Like this.
00:09:39So in reality I could take a very shallow box, if I place the straw
00:09:44that way- a very shallow box, it's of no use that the box is that high.
00:09:48Could I have it low like this? (inaudible) a box this big but it's of no use.
00:09:56Try to think about if you somehow can use the fact that it
00:09:59is high, that you have a box high like this.
00:10:05If not, I'll give you another straw, if you need one.
00:10:09We have finished.
00:10:10We have finished.
00:10:11So, then. Try to complete this sheet. Everybody has his own.
00:10:18That means together?
00:10:19Try if you reach- yes, yes together. Here you write the calculations you've done,
00:10:23drawing the straw and then try to do the calculations from the other boxes too,
00:10:29those that aren't yours. So in a few words, with the straw inserted like this,
00:10:34in reality it isn't of any use that the box is high, it could also be a box that's very flat.
00:10:43Yes, good.
00:10:44It's already very flat, yours is really.
00:10:47But in reality you won't use the fact that- that it's high. Try to think about if you utilize the fact that this one is high.
00:10:53You can do it like this.
00:10:54Place it like this. Raise it, raise it.
00:10:58Like this, yes.
00:10:59No. It's not like that, perhaps here.
00:11:02Like this.
00:11:03So from you we won't make up for a lot, but in fact...
00:11:07This distance here.
00:11:09How do you do it? With the height?
00:11:12Eh, calculate now think- le- leave this like it is. Try to
00:11:17do the other. Now you may have to replace it. Try now to cut
00:11:21the right one here again. Like you said (inaudible).
00:11:24(inaudible) like this.
00:11:25Try to cut it.
00:11:26Be careful.
00:11:36It's a little bit too much. You have to cut a bit more.
00:11:43Yes, without forcing it.
00:11:45Eh, (inaudible) like this (inaudible).
00:11:49Now how can I calculate the length of this straw?
00:11:55No, you do it, here.
00:11:56So, do you have the instructions of the box. Where are the instructions?
00:12:00Then you know how long it is, how wide it is, how high it is.
00:12:02(inaudible) we find.
00:12:04Now reason within the group, explain it to them.
00:12:10Madam, in what sense are these calculations-
00:12:12Have you found the longest straw?
00:12:13Yes, yes.
00:12:14So then, how is it?
00:12:15Like this.
00:12:16Without bending it, eh, you can't bend the straw.
00:12:20We did it that way.
00:12:21No, no, no, without bending it, eh, without bending it...
00:12:27You haven't found it yet, try to thin- try to place the straw into it without bending it, eh?
00:12:37No. First (inaudible).
00:12:38Once (inaudible) and once here (inaudible).
00:12:42Because it's written here.
00:12:45So I- have you found the straw? What is the position?
00:12:48Like this.
00:12:49This one. Now are you calculating its length?
00:12:52How do you do it?
00:12:54We measure.
00:12:56Okay, measure, that was the first operation. Now you have to calculate the length of the straw.
00:13:05With the Pythagorean theorem you can calculate (inaudible).
00:13:08And because we placed it oblique-
00:13:10But try to cut the straw corresponding to the diagonal and insert
00:13:14the straw corresponding to the diagonal.
00:13:20Yes, but-
00:13:21It's from there to there to here.
00:13:24No, wait, wait, wait.
00:13:36Like this we see (inaudible). Let it go. Good, cut it down here. No (inaudible).
00:13:49Okay. This is the diagonal, more or less. Also if no- now however you, with Pythagoras-
00:13:54I can calculate this, okay, with this and this, you have also the measures.
00:13:58But now I'm interested in this. But it- I know-
00:14:02Ah, but now if I have this and this- if I have this and this I'll find it.
00:14:03I- I, I know that you need to do the- the- the- the- the-
00:14:09But no, but because this one here is practically the diagonal, if you do her above (inaudible), that's it.
00:14:14No, it's right...
00:14:15Listen to Boris and then try to observe this sheet and complete it.
00:14:21Okay, listen (inaudible).
00:14:26Eh, the diagonal here (inaudible).
00:14:29M-hm, yes.
00:14:49We try to be as exact as possible and not lose the precision in our calculations.
00:15:01Or as we said: Instead of writing this, I can write-
00:15:06Directly. M-hm, yes.
00:15:11You can also write here afterwards.
00:15:24But now try- you are calculating other boxes.
00:15:27Yes, but-
00:15:28You are already at the second one. After some time it seems to be the same thing.
00:15:29Only that (inaudible).
00:15:31Look a bit at the measurements. Look here, they have the same length...
00:15:38The width, instead look at number zero, pay attention to number one.
00:15:44It's quite a lot wider, eh.
00:15:46But indeed it's right?
00:15:48And then there's the- and then there is the-
00:15:49Ah, already, because it's longer.
00:15:51Afterwards there is the height, which is different, but the- afterwards we will see that it's right.
00:15:52But it's right indeed.
00:15:55Now try a bit- you already did two calculations- try to see if there is a faster way to calculate. It's correct, eh, what you
00:16:02are calculating, but I'd like to see if you can shorten the calculation- also write the unit, eh.
00:16:09Madam, we have found it.
00:16:12Well we used- we have the Pythagorean theorem (inaudible).
00:16:15M-hm, yes.
00:16:16(inaudible) that we know (inaudible) and now we got (inaudible).
00:16:25Reason. If you cut that right one, go on- this one you can leave down like this.
00:16:32If you cut this one it's too short. You should have found- well, I won't tell you.
00:16:39I don't know if everybody failed, it doesn't matter, we will correct it later. So complete
00:16:44the sheets, this part here and- and it tells you what to do with your box. Then here,
00:16:51there are the dimensions of the other boxes, from the other groups. Try afterwards to
00:16:56calculate those too: the diagonals that you got from the other groups.
00:17:00Because here (inaudible).
00:17:02The? How do you complete the title? What is it that you apply?
00:17:07By doing these calculations, what did you apply?
00:17:09The Pythagorean theorem.
00:17:10Pythagorean theorem, eh. Pythagorean theorem in solid geometry.
00:17:24So now complete your solution in your exercise book: The calculations in blue, the solutions in red in the middle.
00:17:32Will you write it, or shall I write it?
00:17:34In your exercise book, eh.
00:17:37Do we have to fill in all of them?
00:17:40Start filling in your part. When you have finished yours, continue filling in from the other groups too, yes.
00:17:46(inaudible) in scribble or do we have (inaudible)?
00:17:49No, no, there is space here.
00:17:54Now. You are at a good point, you only have to check if it's correct. Complete this
00:18:02and draw and do your calculations which Andrea has now written here. Everybody write it on their own sheet.
00:18:08And then there you have the dimensions of the other boxes. There is the box zero, the bo- and start doing the calculations like this
00:18:13for the others, seeing what the others found with their boxes.
00:18:20We are inexact.
00:18:21You are inexact, what do you mean?
00:18:22Well, we have practically found that the longest straw is this-
00:18:24The diagonal from here to here.
00:18:29We found it with the diagonal.
00:18:33From here to here.
00:18:34From here to here.
00:18:35Mine worked, but the straw is this other one.
00:18:36It's logical because if you put it from here to here it will get longer.
00:18:41That means that you haven't finished calculating yet. It means that what you calculated isn't the length of the straw.
00:18:46So we, what...
00:18:47But it's only the diagonal from the box- from- from the circumference of the base from the box, from the rectangle, let's say the base.
00:18:55We have to find from here to here, many (inaudible).
00:18:57M-hm, yes... Use this- the measure you have already found.
00:19:03Thirty-nine point three.
00:19:09You can write it with a pen because I think you did it right.
00:19:16Do it like this, maybe: Cut the straw that you measured, the one placed on the base. Cut it. That way you can observe it.
00:19:23So now, is this one here the right one?
00:19:26Also cut the one placed on the base.
00:19:28Ah, that way?
00:19:29That way?
00:19:30Yes, with the- but don't- the straw after, no- yes, m-hm, yes.
00:19:34Ah, that.
00:19:44Place it on the base. Eh, cut it again a bit more, that...
00:19:59Ah, still too long.
00:20:06Okay. It's good, it's good. Now place the other. It gets placed in the same...
00:20:13Like this? In the same trajectory.
00:20:14In the same posit- eh.
00:20:19Now here you have already calculated?
00:20:22This? We calculated this.
00:20:25You calculated this. Now observe how to calculate the other.
00:20:29Using the Pythagorean theorem.
00:20:30Using the Pythagorean theorem.
00:20:34So it's thirty-nine point six squared.
00:20:40You have to find the hypotenuse.
00:20:41Plus twelve (inaudible).
00:20:43(inaudible) this one (inaudible) here.
00:20:45And then you have to do this plus this (inaudible).
00:20:47Ah, it's true (inaudible).
00:20:50Your calculations, write them here so that you have more space. Afterwards you can also solve them underneath it.
00:20:56Ah, but (inaudible)?
00:20:58But here, eh, here.
00:20:59But it's right (inaudible).
00:21:00No, it's okay. Afterwards write it also here, but start- afterwards write it again.
00:21:01Madam, but here (inaudible) do we have to write the same things like here?
00:21:26Have you completed the transparency?
00:21:27Yes. We didn't get as far as finding what (inaudible) fast (inaudible) the fast calculations.
00:21:33It isn't because you're slow, but you can do it with a single calculation, eh.
00:21:36Even faster?
00:21:38Madam, it's true that here (inaudible) the Pythagorean theorem (inaudible)?
00:21:42Yes. Think about it for a moment. You square it plus square this.
00:21:48And find a result. Then take this, and form the other. Didn't I tell you to do it like this yesterday? Isn't it like I told you yesterday?
00:21:54Ah, right.
00:21:55To already use the (inaudible).
00:21:56Eh- I'll take that. I and then- in reality instead of writing this.
00:22:06Eh, but why we (inaudible).
00:22:07We have to take the square root of this.
00:22:09Try to think about what I have to write.
00:22:10Madam, do we now have to check the boxes with the others?
00:22:13Wait, eh. Tell me.
00:22:15We have found it.
00:22:16You have found it. Then put- take these papers. Now write, well, everybody calculate your calculations.
00:22:26In this area here. Write your calculations, draw the straw, read what's written and start doing this-
00:22:34Tell me.
00:22:35I've been thinking.
00:22:36Have you finished, have you completed- here you write me- here you haven't completed yet, eh?
00:22:41Ah- do we also have to do this too?
00:22:42M-hm, yes.
00:22:52Afterwards yes. Number three you can- you only have to fill in the same thing.
00:22:57Madam (inaudible).
00:22:59All the calculations?
00:23:00M-hm, yes.
00:23:14Madam, by doing the question we got a result that doesn't (inaudible). Why?
00:23:20Well, I think there will be something wrong.
00:23:28We're done.
00:23:29You're done. Then now complete in your exercise book where there is- what do you have? Box number? Two?
00:23:37Complete case number two with your solution.
00:23:43Shall I leave it here?
00:23:50But here you can directly-
00:24:32Will you complete the title? What did you apply to find these measurements?
00:24:38The Pythagorean theorem.
00:24:39M-hm, yes, then the Pythagorean theorem in solid geometry.
00:24:45Sit down correctly, please.
00:24:56And here write down the calculations of your box too, eh.
00:24:59I've written it here.
00:25:00Never mind, you can transfer it- the idea was writing it above and transferring it down here...
00:25:11Here, label it I. Maybe their- in the drawing- draw it. The I- Your I is only this one, isn't it?
00:25:17The hypotenuse, which (inaudible).
00:25:18It's the hypotenuse. Draw it in and write the I.
00:25:24No, but then you won't understand well if I write.
00:25:26No, do it in colors.
00:25:28Eh, yes.
00:25:29No, here afterwards, give it another name, also if there- this is also a hypotenuse. Call it straw or call it...
00:25:33Is diagonal of the base, okay?
00:25:34Or the diagonal of the base, that's the same, yes.
00:25:43Ready, Lucia?
00:25:45Yes, yes, yes.
00:25:46Do you still need the box?
00:25:48So box, tape measure, instructions and scissors. Bring everything to the table, then try to complete the rest of the table
00:25:55with the measurements from your partners.
00:26:05Have you finished yours, eh?
00:26:08So in the transparency you have the box number zero.
00:26:12Madam, do we have to color them? All of the (inaudible).
00:26:16Yes, yes.
00:26:17Madam, but (inaudible).
00:26:19What (inaudible)?
00:26:20If we've made mistakes with the calculations?
00:26:22I don't know if it's wrong. [ Laughter ] Complete it and then we'll correct them. Complete it here with your calculations.
00:26:29There: In your exercise book, in zero. Then if you don't need it any more, bring you box, scissors, tape measure to the table.
00:26:39Look how easy it is: Only changing all the measurements.
00:26:43Also the straws are to be brought here, eh, not to be played with.
00:26:56Madam. Here I have this (inaudible) has this.
00:27:01M-hm, yes. It's okay.
00:27:17Try to think a little bit while calculating. That means every time you square a measurement, add the square of the other, then the
00:27:25square root. Then you take this number here, it- square it again and add the third dimension in the square.
00:27:35Think a bit now if this result that I have here, I square it, if I can't instead of taking, he...
00:27:43That (inaudible).
00:27:44Take the first calculation.
00:27:46Eh, but... (inaudible).
00:27:50Now yes, but if you have finished bring me the material over there, straws a- and stuff like this, and do the measurements of the others.
00:27:58If you have already filled in the transparency- no, I'll bring it to you now, eh, to fill in.
00:28:02You need to (inaudible).
00:28:04Now you do the same calculations with the other boxes.
00:28:07We take (inaudible).
00:28:08No, don't take- look, as many measurements are written here, the reasoning is always the same.
00:28:12Bring the material over there, now I'll give you the transparency to complete.
00:28:32Do we have to do it in red?
00:28:33No, in blue.
00:28:37No, not this. It's this.
00:28:41Yes, (already).
00:28:42And now how should I do?
00:28:44Doesn't matter, doesn't matter.
00:28:51And then, Isabella, put the box in its place, eh. Wait, I'm coming. Eh- you, complete in your exercise book
00:29:01(inaudible) the calculations (inaudible) similar...
00:29:03Tell me.
00:29:04Is it right like this?
00:29:05Try to do it with your own and look: If you find the same result, it's right. If you don't find the same result, that means
00:29:11it's wrong.
00:29:16Have you done yours?
00:29:23But it gives you a straw of 880 centimeters?
00:29:26Eh, yes.
00:29:27But was it that long? Was- which one was it? This here?
00:29:33No, this.
00:29:34Here inside.
00:29:38This one here? But take it out.
00:29:42But also if you take it out a bit, this one here is 880 centimeters?
00:29:46There's something that doesn't work.
00:29:49We can try to do the calculations again. Try to do this piece here. It equals 48.
00:29:53But take care that- Let's look at your- what did you do before, explain it to me in words. Marisa, what did you do?
00:30:00Okay, thirty-eight point five squared plus 25 squared (inaudible) to find this diagonal here.
00:30:06M-hm, yes.
00:30:09Now it's more.
00:30:12But what did Pythagoras say? Having a rectangular triangle?
00:30:20Right angle side, right angle side (inaudible).
00:30:26Eh- in that triangle which is the right angle side? Which is the hypotenuse?
00:30:29The hypotenuse is this one.
00:30:30Eh, yes.
00:30:31And these are the other two sides.
00:30:33But this triangle here.
00:30:34No. This is a side and this is the other.
00:30:37This is a side, that is a side, and this is the hypotenuse. So if you have found this, you already have calculated it.
00:30:45It's 55.
00:30:46Squared plus (inaudible) 16 squared and then extracting the root of everything.
00:30:52Do you agree?
00:30:53M-hm, yes.
00:30:54We have found it.
00:30:55Is it right like this?
00:30:59I get an expression.
00:31:01And where can I simplify again?
00:31:03No. [ Laughter ]
00:31:04[ Laughter ]
00:31:06Is it right or can you simplify something else? Are you done with the transparency?
00:31:13This goes ahead, two remains. Have you already completed the transparency?
00:31:19(inaudible), ah, no, the transparency not yet.
00:31:20The transparency. You have to complete with- are you doing number three or number four?
00:31:24Three. Complete it with your solution. If you have finished with the box, put it back over there, box, scissors. What?
00:31:33No, put it over there- put it over there.
00:32:27Eh, yes, I think so.
00:32:29And some of you bring back the box, please. In the meanwhile, Andrea-
00:32:33Madam, may I ask you (inaudible).
00:32:35In the meanwhile put them away. Later I'll give them to you. Go on, put it inside, Claudio (inaudible).
00:32:54Assio, it's your pen.
00:32:55No, mine.
00:32:57No, why (inaudible).
00:33:07Finished? Now try to think a while, to see if you can find a faster way to do- to find
00:33:17the same result. Instead of doing two calculations every time, do one calculation only.
00:33:26What do I do? Sqaure this plus fourteen point seven squared, root.
00:33:33Ah- I can use the properties of powers.
00:33:36Be careful, I squar-
00:33:37Plus, eh, you can't use the properties of powers.
00:33:41Barbara is right. Here is the square root, now the sum, then extract the root, I square.
00:33:50But first (inaudible).
00:33:52First we extract the root, then we square it.
00:33:56Ah, so you can take away the root, it's this.
00:33:59Which root can I take away?
00:34:02No. This, because squar- taking away the square root and then squaring it isn't of any use.
00:34:08But the (inaudible).
00:34:10No, pay attention, eh. Here I extract the root of it all, and then I square it again.
00:34:17Dividing by two doesn't make sense.
00:34:20Try to write the simplified version. Have you done?
00:34:24Here we have to do it in the simplified way or-
00:34:26Now what have you found? The measurement corresponds to? Have you measured it?
00:34:30Okay. So now write here in your- you are the box number four? Write the solution here on the transparency.
00:34:35The result you found.
00:34:37The same thing that we wrote down here.
00:34:38Yes, exactly.
00:34:39Now here you extend the same scheme.
00:34:40Yes, and afterwards try to do what the others have done with the other- other boxes.
00:35:10And pay attention, extract the root but you have to extract what else?
00:35:12Eh- (inaudible)
00:35:14I told you.
00:35:31Why do you take the box?
00:35:32To do the faster thing.
00:35:34I told you that- however it's enough to look at the calculations, eh, not- the calculations you are doing. Try to reason with some of the numbers you take
00:35:41if you couldn't do it in an easier way.
00:36:10If you would have to write a title. What did you apply in solid geometry?
00:36:15Boh! I don't know.
00:36:16The Pythagorean theorem.
00:36:17The Pythagorean theorem. So you can write it... Finished?
00:36:31(inaudible) square root direct.
00:36:33And that without- eh- without.
00:36:35The direct square root, what- what is that?
00:36:37What is the direct square root?
00:36:38Is the (inaudible) and then you have to square it and immediately extract the root.
00:36:39This (inaudible) the last time.
00:36:44Do you remember?
00:36:46Eh, yes and no. Yes, it hits a bit. But instead of doing two calculations try to see if you could do it with a single one.
00:36:54It's enough to do (inaudible).
00:36:55Put it a bit (inaudible) below.
00:36:56Try to write it.
00:37:01Try to write it.
00:37:02Let's do it here, go on.
00:37:03What (inaudible)?
00:37:04What you just told me now.
00:37:06Now here, I'm going to write the same (inaudible)?
00:37:09Yes, now you are doing more then the rest. In the meantime the others, you have to write the same thing here.
00:37:13If not, then- try to do- from now on let's continue all together.
00:37:21Okay, stop, interrupt- shh- [ silence ] Andrea. Interrupt the things you are doing. All of you have found
00:37:31the length of the straw to be longer then it was in the real box. Now there is- who has already finished calculating?
00:37:38All the others who are still calculating, please go back to your seats now, not in groups anymore, and then look forward.
00:38:44All right, first of all let's complete the title. Not everybody has done that. What have we applied in solid geometry
00:38:53that we haven't ever done, Neluca?
00:38:54The Pythagorean theorem.
00:38:56The Pythagorean theorem. So whoever hasn't written it yet can add it.
00:39:13The Pythagorean theorem in solid geometry. Okay, now we will observe the transparency that you have completed.
00:39:22It seems to me that all the groups have understood what the method was, let's say, to apply. Let's start with box number zero,
00:39:31which was the most shallow box. To find the longest straw that was the straw... let's call it diagonal.
00:39:46And now not anymore the diagonal of an area, but the diagonal of the box. I did it like this, first of all I calculated the diagonal
00:39:55of the base's surface and then by using the height I found the diagonal of the box.
00:40:04So all of you- look- looking at the transparency, what do you have? First you calculated the diagonal of a surface and then
00:40:14by taking this diagonal, you have, with the height of the box, you have found the length of the diagonal of the box.
00:40:21All the groups did it this way. For those who have completed the sheets, you can see the various measurements.
00:40:35Everybody used the same method: Always the diagonal of the base and then the diagonal of the box.
00:40:46Now one of you was finished a bit earlier. He tried to find a single calculation to find the length
00:40:56of the diagonal. So that instead of doing one calculation and then doing the second, he has found a single possible calculation to find
00:41:07this length. Now let's take the example- so Barbara and Andrea were able to find it. You had box number?
00:41:13Eh, one.
00:41:15One or two?
00:41:17Box number one. So their calculation here, first diagonal of the base and then diagonal of the box.
00:41:28Well Barbara, would you like to try to come and explain it at the blackboard? Do it at the blackboard.
00:41:39Yes. So, first- eh- in practice we put together the two calculations. Do I have to write down the two calculations here?
00:41:47Write it again here? Yes... So, observe the difference between the written calculation on the overhead and the calculation that she is writing now on the blackboard.
00:42:19In practice we put together, eh, the first calculation that- that we have- eh- we have put together the two calculations-
00:42:28eh- but we simplify because in practice the calculation was like this at the beginning. Only extracting the root and then squaring it,
00:42:39isn't of any use. So we simplified, we- we took away the root and we squared it.
00:42:51The result?
00:42:52The result is the same.
00:43:01Good, thanks. Now I don't know if it's clear for everybody, so that instead of writing here- look the second passage of the transparency:
00:43:08On the second passage, what did we write? Twenty-seven point two squared plus nine point five squared.
00:43:16Now instead of writing twenty-seven point two here, they wrote the calculation that allowed me to find twenty-seven
00:43:26point two. Instead of writing the number directly, they wrote the calculation that allowed me to find
00:43:33twenty-seven point two. Then Barbara correctly said: I do the root and then I square it, I can annul it because
00:43:42one is the inverse operation of the other. Therefore the shorter way to find this diagonal is length of the box squared,
00:43:51plus width of the box squared, plus the height of the box squared, and at the end extract the root.
00:44:02They are right, eh, also all of the ways that you all did it. This one here makes it easy to understand the reasoning I used.
00:44:08That I first have to find the diagonal of the base, then the diagonal of the box. You can find this method after doing some
00:44:15calculations. At the beginning it isn't that evident to find this- this calculation.
00:44:20So now, whoever, eh, still has to finish the sheet, complete this sheet with the measurements of the other boxes
00:44:29that your partners had. On the other hand, whoever's finished, I'll give everybody a sheet of exercises. Try to work on them with your partner.
00:44:36I'll leave the solutions here, who- you can come to correct the solutions as usual.
00:44:43But first complete this, without looking, then, eh, check if it's right.
00:45:15Yes, eh... Yes, I'm raising it now.
00:45:19First do it without looking. That means first complete the sheet, then...
00:45:25The overhead projector doesn't work well.
00:45:27But can we?
00:45:29Have you finished?
00:45:31Checked? Are they correct?
00:45:32Yes, yes.
00:46:12Have you understood the shorter version? I saw you looking with the face a bit.
00:46:19M-hm, yes, yes. Now I say if you have understood how you did this, do the others. You can always use this method, can't you?
00:46:25You don't have to do the other, don't (inaudible)- start doing this, afterwards, in case, we'll see next time to- to work on it again.
00:46:31Well I didn't under- well I didn't understa- I didn't understand (inaudible) how to arrange (inaudible).
00:46:35Practically, what did you do here? This number here you transferred here.
00:46:38M-hm, yes.
00:46:42Do you agree? This number, you transferred here. On the other hand if you have to transfer this number here, I can transfer the
00:46:48calculation that allowed me to find this number.
00:46:52And I used that to find this number here.
00:46:55Which calculation did I do? I did this, which in reality was this.
00:46:58M-hm, yes.
00:46:59Instead of writing this number, I could write this calculation.
00:47:07M-hm, yes.
00:47:08It's the same thing because this calculation gives me the same result. So I can, instead of writing two thousand one hundred fourteen point five-
00:47:12[ Bell ]
00:47:16I can write this squared plus this squared.
00:47:23And then?
00:47:24(inaudible) plus 166, which would be 13 squared.
00:47:31(inaudible) okay, I'll do it that way (inaudible).
00:47:33M-hm, yes, yes.
00:47:34Madam, (inaudible).
00:47:49But it's the same as you told me last time (inaudible) is 2,000. Do you remember that you told me already last time that there-
00:47:56Was this problem with your calculator?
00:48:04What am I doing wrong? I should extract the root.
00:48:09Ah, it's true (inaudible).
00:48:11Now you have to extract the root, yes. (inaudible) the result down here, if you extract the root.
00:48:15And the root (inaudible).
00:48:34Did you understand well here? Do it on a sheet by side, on a squared sheet of paper.
00:48:43Vera, got it all right?
00:48:57Boris, got it?
00:49:03Is it right now?
00:49:04Yes. But down here? Yes.
00:49:08Two plus eight, then it depends on how you approximate.
00:49:22When do I have to finish the lesson?
00:49:49Now stop what you are doing, put away everything and we'll continue next time.