SW3 PYTHAGOREAN THEOREM
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.
| Time | Caption |
|---|---|
| 00:01:07 | All right, good morning everybody. |
| 00:01:11 | Well, let's imagine today to- eh, we are getting closer to Christmas time. |
| 00:01:16 | We want to send a present to our friends. A present that has a long and narrow shape. |
| 00:01:24 | It could be a thin straw, it could be I don't know- |
| 00:01:29 | Has anybody an idea? I heard that- Joel you were about to say something? |
| 00:01:32 | No, no, no, nothing |
| 00:01:34 | No idea? A long and narrow object. So we ask at the post office, has anybody noticed, |
| 00:01:40 | at the post office there are boxes with different shapes. And these are the six shapes you can see over there. |
| 00:01:46 | There is box number zero, box number one, number two, three, four, |
| 00:01:53 | and five, and those boxes have different shapes determined by the post office. |
| 00:01:59 | Now ever- every group- you sit together in groups of three and one group of two- |
| 00:02:04 | will receive a box. Start observing this box a little bit, there are |
| 00:02:10 | some instructions in common that they give together with the box. Look to see what |
| 00:02:15 | your box number is, the measurements, the dimensions of that box. |
| 00:02:19 | And then take one of these straws, try to find out the position in which the straw |
| 00:02:27 | can be placed to obtain the longest straw. That means you insert the |
| 00:02:32 | whole straw, it's very possible that it doesn't fit even in the biggest box, |
| 00:02:36 | and try to obtain the longest straw by cutting the straw from the start |
| 00:02:43 | of the point in which you think you received the longest straw. |
| 00:02:46 | That means to insert it that way, insert it horizontally. Try to find out which is |
| 00:02:51 | the longest straw that you can thread into this box. Once you have found out |
| 00:02:56 | the position in which the straw is the longest, cut it- |
| 00:03:00 | cut it, you can keep the rest of the straw if it helps you for further observations you can |
| 00:03:05 | also cut it. Measure it, you have a tape measure, you have a ruler, measure the maximum length that you can obtain. |
| 00:03:16 | At this point you have a measure. Check with the calculations if the measure you found is correct. |
| 00:03:25 | To do this, you have to observe the box a little bit and see how you can find |
| 00:03:30 | the length of this straw. The calculations, do them, eh, one copy per group. |
| 00:03:37 | Write a draft of your calculations trying out the reasoning that you need. |
| 00:03:42 | Then everybody will receive a worksheet and will complete the space where the number of your |
| 00:03:49 | box is, by writing the solution that you have found. After each group is done, I'll pass by and |
| 00:03:56 | you will complete a transparency with the results you have found. Therefore the materials that you |
| 00:04:01 | need are: Calculator, mmh, a draft sheet per group. And you work in |
| 00:04:06 | groups of three or two. Only one person can move the chair, okay? |
| 00:04:33 | Thanks. |
| 00:04:34 | You're welcome. |
| 00:04:41 | To do the calculations you have to find (inaudible) calculations so that you can find this measure of the straw? |
| 00:04:45 | Yes. |
| 00:05:12 | Madam, but- and when is not (inaudible). |
| 00:05:15 | (inaudible) |
| 00:05:51 | Also observe your instructions a little bit, eh? |
| 00:06:08 | Madam, can we already cut it? |
| 00:06:09 | Of course you can already cut it, yes. |
| 00:06:16 | Madam, Madam, can we (inaudible)? |
| 00:06:19 | No. |
| 00:06:27 | Cut the straw already. Have you found the best position? |
| 00:06:30 | Eh, (inaudible). |
| 00:06:36 | Well, cut it. |
| 00:06:38 | And if it's wrong? |
| 00:06:39 | In case it's wrong you take another piece and then you use the rest of the straw. |
| 00:06:50 | Eh, it's forty point five. |
| 00:06:55 | Eh, yes. |
| 00:07:11 | You do it that way? |
| 00:07:12 | Put it a little (inaudible). |
| 00:07:13 | You measure forty? |
| 00:07:14 | Forty point five. |
| 00:07:15 | Well, well, this is the length (inaudible). |
| 00:07:20 | Look, you have instructions, eh, that are already given. |
| 00:07:23 | Ah, already. |
| 00:07:28 | Try to cut it and then if it doesn't work you'll take another piece of straw and- try. |
| 00:07:33 | We only cut one piece. |
| 00:07:35 | Find the longest straw that you can place in your box, m-hm, yes. |
| 00:07:38 | Madam, like this it's longer (inaudible). |
| 00:07:42 | (inaudible) calculate it like this. |
| 00:07:44 | Try to think about- it isn't that complicated. |
| 00:08:02 | So did you already cut the straw? |
| 00:08:04 | No. |
| 00:08:06 | Go on, cut it and also if you fail, you can do it again. Look, you have already got the- the instructions. |
| 00:08:13 | Yes, yes, but we have already found- |
| 00:08:14 | We have already found it. |
| 00:08:16 | You have it already- then- okay. Complete the sheet. |
| 00:08:19 | But don't we have to do the calculations? |
| 00:08:23 | We- we've got it |
| 00:08:24 | You have inserted, cut and measured it. |
| 00:08:27 | Yes. |
| 00:08:28 | Okay. 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). |
| 00:08:46 | So complete: Here with your situation, with your straw, you have the box number two and you put it- do it again here. |
| 00:08:58 | Ah, we have to do- |
| 00:08:59 | And then start doing this, call me to have a look at it when you've finished. |
| 00:09:02 | Like this, Madam (inaudible). |
| 00:09:07 | Now have you found the straw and have you measured it? |
| 00:09:09 | Yes. |
| 00:09:10 | Now observe the box then put it aside. |
| 00:09:17 | Insert the straw: What- if I- so in your opinion, what is the longest straw laying in the box? |
| 00:09:28 | No, no shorter. |
| 00:09:29 | Shorter. |
| 00:09:35 | One that's shorter. |
| 00:09:37 | Like this. |
| 00:09:39 | So in reality I could take a very shallow box, if I place the straw |
| 00:09:44 | that way- a very shallow box, it's of no use that the box is that high. |
| 00:09:48 | Could I have it low like this? (inaudible) a box this big but it's of no use. |
| 00:09:56 | Try to think about if you somehow can use the fact that it |
| 00:09:59 | is high, that you have a box high like this. |
| 00:10:05 | If not, I'll give you another straw, if you need one. |
| 00:10:09 | We have finished. |
| 00:10:10 | We have finished. |
| 00:10:11 | So, then. Try to complete this sheet. Everybody has his own. |
| 00:10:18 | That means together? |
| 00:10:19 | Try if you reach- yes, yes together. Here you write the calculations you've done, |
| 00:10:23 | drawing the straw and then try to do the calculations from the other boxes too, |
| 00:10:29 | those that aren't yours. So in a few words, with the straw inserted like this, |
| 00:10:34 | in reality it isn't of any use that the box is high, it could also be a box that's very flat. |
| 00:10:43 | Yes, good. |
| 00:10:44 | It's already very flat, yours is really. |
| 00:10:45 | Yes. |
| 00:10:47 | But 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. |
| 00:10:53 | You can do it like this. |
| 00:10:54 | Place it like this. Raise it, raise it. |
| 00:10:58 | Like this, yes. |
| 00:10:59 | No. It's not like that, perhaps here. |
| 00:11:00 | (inaudible) |
| 00:11:02 | Like this. |
| 00:11:03 | So from you we won't make up for a lot, but in fact... |
| 00:11:07 | This distance here. |
| 00:11:08 | (inaudible) |
| 00:11:09 | How do you do it? With the height? |
| 00:11:12 | Eh, calculate now think- le- leave this like it is. Try to |
| 00:11:17 | do the other. Now you may have to replace it. Try now to cut |
| 00:11:21 | the right one here again. Like you said (inaudible). |
| 00:11:24 | (inaudible) like this. |
| 00:11:25 | Try to cut it. |
| 00:11:26 | Be careful. |
| 00:11:36 | It's a little bit too much. You have to cut a bit more. |
| 00:11:43 | Yes, without forcing it. |
| 00:11:45 | Eh, (inaudible) like this (inaudible). |
| 00:11:49 | Now how can I calculate the length of this straw? |
| 00:11:55 | No, you do it, here. |
| 00:11:56 | So, do you have the instructions of the box. Where are the instructions? |
| 00:11:59 | Here. |
| 00:12:00 | Then you know how long it is, how wide it is, how high it is. |
| 00:12:02 | (inaudible) we find. |
| 00:12:04 | Now reason within the group, explain it to them. |
| 00:12:10 | Madam, in what sense are these calculations- |
| 00:12:12 | Have you found the longest straw? |
| 00:12:13 | Yes, yes. |
| 00:12:14 | So then, how is it? |
| 00:12:15 | Like this. |
| 00:12:16 | Without bending it, eh, you can't bend the straw. |
| 00:12:20 | We did it that way. |
| 00:12:21 | No, no, no, without bending it, eh, without bending it... |
| 00:12:24 | Aha. |
| 00:12:27 | You haven't found it yet, try to thin- try to place the straw into it without bending it, eh? |
| 00:12:37 | No. First (inaudible). |
| 00:12:38 | Once (inaudible) and once here (inaudible). |
| 00:12:41 | Why? |
| 00:12:42 | Because it's written here. |
| 00:12:45 | So I- have you found the straw? What is the position? |
| 00:12:48 | Like this. |
| 00:12:49 | This one. Now are you calculating its length? |
| 00:12:51 | Yes. |
| 00:12:52 | How do you do it? |
| 00:12:54 | We measure. |
| 00:12:56 | Okay, measure, that was the first operation. Now you have to calculate the length of the straw. |
| 00:13:05 | With the Pythagorean theorem you can calculate (inaudible). |
| 00:13:08 | And because we placed it oblique- |
| 00:13:10 | But try to cut the straw corresponding to the diagonal and insert |
| 00:13:14 | the straw corresponding to the diagonal. |
| 00:13:20 | Yes, but- |
| 00:13:21 | It's from there to there to here. |
| 00:13:24 | No, wait, wait, wait. |
| 00:13:36 | Like this we see (inaudible). Let it go. Good, cut it down here. No (inaudible). |
| 00:13:49 | Okay. This is the diagonal, more or less. Also if no- now however you, with Pythagoras- |
| 00:13:54 | I can calculate this, okay, with this and this, you have also the measures. |
| 00:13:58 | But now I'm interested in this. But it- I know- |
| 00:14:02 | Ah, but now if I have this and this- if I have this and this I'll find it. |
| 00:14:03 | I- I, I know that you need to do the- the- the- the- the- |
| 00:14:09 | But no, but because this one here is practically the diagonal, if you do her above (inaudible), that's it. |
| 00:14:14 | No, it's right... |
| 00:14:15 | Listen to Boris and then try to observe this sheet and complete it. |
| 00:14:21 | Okay, listen (inaudible). |
| 00:14:26 | Eh, the diagonal here (inaudible). |
| 00:14:29 | M-hm, yes. |
| 00:14:30 | Right? |
| 00:14:31 | Yes. |
| 00:14:49 | We try to be as exact as possible and not lose the precision in our calculations. |
| 00:15:01 | Or as we said: Instead of writing this, I can write- |
| 00:15:05 | This. |
| 00:15:06 | Directly. M-hm, yes. |
| 00:15:11 | You can also write here afterwards. |
| 00:15:24 | But now try- you are calculating other boxes. |
| 00:15:27 | Yes, but- |
| 00:15:28 | You are already at the second one. After some time it seems to be the same thing. |
| 00:15:29 | Only that (inaudible). |
| 00:15:31 | Look a bit at the measurements. Look here, they have the same length... |
| 00:15:38 | The width, instead look at number zero, pay attention to number one. |
| 00:15:44 | It's quite a lot wider, eh. |
| 00:15:46 | But indeed it's right? |
| 00:15:48 | And then there's the- and then there is the- |
| 00:15:49 | Ah, already, because it's longer. |
| 00:15:51 | Afterwards there is the height, which is different, but the- afterwards we will see that it's right. |
| 00:15:52 | But it's right indeed. |
| 00:15:55 | Now 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 |
| 00:16:02 | are calculating, but I'd like to see if you can shorten the calculation- also write the unit, eh. |
| 00:16:09 | Madam, we have found it. |
| 00:16:11 | Found? |
| 00:16:12 | Well we used- we have the Pythagorean theorem (inaudible). |
| 00:16:15 | M-hm, yes. |
| 00:16:16 | (inaudible) that we know (inaudible) and now we got (inaudible). |
| 00:16:25 | Reason. If you cut that right one, go on- this one you can leave down like this. |
| 00:16:32 | If you cut this one it's too short. You should have found- well, I won't tell you. |
| 00:16:39 | I don't know if everybody failed, it doesn't matter, we will correct it later. So complete |
| 00:16:44 | the sheets, this part here and- and it tells you what to do with your box. Then here, |
| 00:16:51 | there are the dimensions of the other boxes, from the other groups. Try afterwards to |
| 00:16:56 | calculate those too: the diagonals that you got from the other groups. |
| 00:17:00 | Because here (inaudible). |
| 00:17:02 | The? How do you complete the title? What is it that you apply? |
| 00:17:04 | (inaudible) |
| 00:17:07 | By doing these calculations, what did you apply? |
| 00:17:09 | The Pythagorean theorem. |
| 00:17:10 | Pythagorean theorem, eh. Pythagorean theorem in solid geometry. |
| 00:17:24 | So now complete your solution in your exercise book: The calculations in blue, the solutions in red in the middle. |
| 00:17:32 | (inaudible) |
| 00:17:32 | Yes. |
| 00:17:32 | Will you write it, or shall I write it? |
| 00:17:33 | Madam. |
| 00:17:34 | In your exercise book, eh. |
| 00:17:37 | Do we have to fill in all of them? |
| 00:17:40 | Start 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)? |
| 00:17:49 | No, no, there is space here. |
| 00:17:54 | Now. You are at a good point, you only have to check if it's correct. Complete this |
| 00:18:02 | and draw and do your calculations which Andrea has now written here. Everybody write it on their own sheet. |
| 00:18:08 | And then there you have the dimensions of the other boxes. There is the box zero, the bo- and start doing the calculations like this |
| 00:18:13 | for the others, seeing what the others found with their boxes. |
| 00:18:20 | We are inexact. |
| 00:18:21 | You are inexact, what do you mean? |
| 00:18:22 | Well, we have practically found that the longest straw is this- |
| 00:18:24 | The diagonal from here to here. |
| 00:18:28 | Yes. |
| 00:18:29 | We found it with the diagonal. |
| 00:18:33 | From here to here. |
| 00:18:34 | From here to here. |
| 00:18:35 | Mine worked, but the straw is this other one. |
| 00:18:36 | It's logical because if you put it from here to here it will get longer. |
| 00:18:41 | That means that you haven't finished calculating yet. It means that what you calculated isn't the length of the straw. |
| 00:18:46 | So we, what... |
| 00:18:47 | But 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. |
| 00:18:55 | We have to find from here to here, many (inaudible). |
| 00:18:57 | M-hm, yes... Use this- the measure you have already found. |
| 00:19:03 | Thirty-nine point three. |
| 00:19:09 | You can write it with a pen because I think you did it right. |
| 00:19:16 | Do it like this, maybe: Cut the straw that you measured, the one placed on the base. Cut it. That way you can observe it. |
| 00:19:23 | So now, is this one here the right one? |
| 00:19:25 | Yes. |
| 00:19:26 | Also cut the one placed on the base. |
| 00:19:28 | Ah, that way? |
| 00:19:29 | That way? |
| 00:19:30 | Yes, with the- but don't- the straw after, no- yes, m-hm, yes. |
| 00:19:31 | Cut. |
| 00:19:33 | This. |
| 00:19:34 | Ah, that. |
| 00:19:44 | Place it on the base. Eh, cut it again a bit more, that... |
| 00:19:59 | Ah, still too long. |
| 00:20:06 | Okay. It's good, it's good. Now place the other. It gets placed in the same... |
| 00:20:13 | Like this? In the same trajectory. |
| 00:20:14 | In the same posit- eh. |
| 00:20:19 | Now here you have already calculated? |
| 00:20:22 | This? We calculated this. |
| 00:20:25 | You calculated this. Now observe how to calculate the other. |
| 00:20:29 | Using the Pythagorean theorem. |
| 00:20:30 | Using the Pythagorean theorem. |
| 00:20:34 | So it's thirty-nine point six squared. |
| 00:20:36 | Minus. |
| 00:20:37 | Plus. |
| 00:20:38 | Minus. |
| 00:20:39 | Plus. |
| 00:20:40 | You have to find the hypotenuse. |
| 00:20:41 | Plus twelve (inaudible). |
| 00:20:43 | (inaudible) this one (inaudible) here. |
| 00:20:45 | And then you have to do this plus this (inaudible). |
| 00:20:46 | (inaudible) |
| 00:20:47 | Ah, it's true (inaudible). |
| 00:20:50 | Your calculations, write them here so that you have more space. Afterwards you can also solve them underneath it. |
| 00:20:56 | Ah, but (inaudible)? |
| 00:20:57 | No. |
| 00:20:58 | But here, eh, here. |
| 00:20:59 | But it's right (inaudible). |
| 00:21:00 | No, it's okay. Afterwards write it also here, but start- afterwards write it again. |
| 00:21:01 | Madam, but here (inaudible) do we have to write the same things like here? |
| 00:21:04 | Yes. |
| 00:21:08 | Done? |
| 00:21:09 | Yes. |
| 00:21:18 | Madam. |
| 00:21:26 | Have you completed the transparency? |
| 00:21:27 | Yes. We didn't get as far as finding what (inaudible) fast (inaudible) the fast calculations. |
| 00:21:33 | It isn't because you're slow, but you can do it with a single calculation, eh. |
| 00:21:36 | Even faster? |
| 00:21:38 | Madam, it's true that here (inaudible) the Pythagorean theorem (inaudible)? |
| 00:21:42 | Yes. Think about it for a moment. You square it plus square this. |
| 00:21:47 | Yes. |
| 00:21:48 | And 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? |
| 00:21:54 | Ah, right. |
| 00:21:55 | To already use the (inaudible). |
| 00:21:56 | Eh- I'll take that. I and then- in reality instead of writing this. |
| 00:22:06 | Eh, but why we (inaudible). |
| 00:22:07 | We have to take the square root of this. |
| 00:22:09 | Try to think about what I have to write. |
| 00:22:10 | Madam, do we now have to check the boxes with the others? |
| 00:22:13 | Wait, eh. Tell me. |
| 00:22:15 | We have found it. |
| 00:22:16 | You have found it. Then put- take these papers. Now write, well, everybody calculate your calculations. |
| 00:22:26 | In this area here. Write your calculations, draw the straw, read what's written and start doing this- |
| 00:22:34 | Tell me. |
| 00:22:35 | I've been thinking. |
| 00:22:36 | Have you finished, have you completed- here you write me- here you haven't completed yet, eh? |
| 00:22:41 | Ah- do we also have to do this too? |
| 00:22:42 | M-hm, yes. |
| 00:22:50 | Here. |
| 00:22:52 | Afterwards yes. Number three you can- you only have to fill in the same thing. |
| 00:22:57 | Madam (inaudible). |
| 00:22:58 | Yes. |
| 00:22:59 | All the calculations? |
| 00:23:00 | M-hm, yes. |
| 00:23:14 | Madam, by doing the question we got a result that doesn't (inaudible). Why? |
| 00:23:20 | Well, I think there will be something wrong. |
| 00:23:28 | We're done. |
| 00:23:29 | You're done. Then now complete in your exercise book where there is- what do you have? Box number? Two? |
| 00:23:37 | Complete case number two with your solution. |
| 00:23:41 | Yes. |
| 00:23:43 | Shall I leave it here? |
| 00:23:45 | Yes. |
| 00:23:50 | But here you can directly- |
| 00:24:32 | Will you complete the title? What did you apply to find these measurements? |
| 00:24:38 | The Pythagorean theorem. |
| 00:24:39 | M-hm, yes, then the Pythagorean theorem in solid geometry. |
| 00:24:45 | Sit down correctly, please. |
| 00:24:56 | And here write down the calculations of your box too, eh. |
| 00:24:59 | I've written it here. |
| 00:25:00 | Never mind, you can transfer it- the idea was writing it above and transferring it down here... |
| 00:25:11 | Here, label it I. Maybe their- in the drawing- draw it. The I- Your I is only this one, isn't it? |
| 00:25:17 | The hypotenuse, which (inaudible). |
| 00:25:18 | It's the hypotenuse. Draw it in and write the I. |
| 00:25:24 | No, but then you won't understand well if I write. |
| 00:25:26 | No, do it in colors. |
| 00:25:28 | Eh, yes. |
| 00:25:29 | No, here afterwards, give it another name, also if there- this is also a hypotenuse. Call it straw or call it... |
| 00:25:33 | Is diagonal of the base, okay? |
| 00:25:34 | Or the diagonal of the base, that's the same, yes. |
| 00:25:43 | Ready, Lucia? |
| 00:25:45 | Yes, yes, yes. |
| 00:25:46 | Do you still need the box? |
| 00:25:47 | No. |
| 00:25:48 | So box, tape measure, instructions and scissors. Bring everything to the table, then try to complete the rest of the table |
| 00:25:55 | with the measurements from your partners. |
| 00:26:05 | Have you finished yours, eh? |
| 00:26:07 | Yes. |
| 00:26:08 | So in the transparency you have the box number zero. |
| 00:26:12 | Madam, do we have to color them? All of the (inaudible). |
| 00:26:16 | Yes, yes. |
| 00:26:17 | Madam, but (inaudible). |
| 00:26:19 | What (inaudible)? |
| 00:26:20 | If we've made mistakes with the calculations? |
| 00:26:22 | I don't know if it's wrong. [ Laughter ] Complete it and then we'll correct them. Complete it here with your calculations. |
| 00:26:28 | Here? |
| 00:26:29 | There: In your exercise book, in zero. Then if you don't need it any more, bring you box, scissors, tape measure to the table. |
| 00:26:39 | Look how easy it is: Only changing all the measurements. |
| 00:26:43 | Also the straws are to be brought here, eh, not to be played with. |
| 00:26:56 | Madam. Here I have this (inaudible) has this. |
| 00:27:01 | M-hm, yes. It's okay. |
| 00:27:17 | Try to think a little bit while calculating. That means every time you square a measurement, add the square of the other, then the |
| 00:27:25 | square root. Then you take this number here, it- square it again and add the third dimension in the square. |
| 00:27:35 | Think a bit now if this result that I have here, I square it, if I can't instead of taking, he... |
| 00:27:43 | That (inaudible). |
| 00:27:44 | Take the first calculation. |
| 00:27:46 | Eh, but... (inaudible). |
| 00:27:50 | Now 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. |
| 00:27:58 | If you have already filled in the transparency- no, I'll bring it to you now, eh, to fill in. |
| 00:28:02 | You need to (inaudible). |
| 00:28:04 | Now you do the same calculations with the other boxes. |
| 00:28:07 | We take (inaudible). |
| 00:28:08 | No, don't take- look, as many measurements are written here, the reasoning is always the same. |
| 00:28:12 | Bring the material over there, now I'll give you the transparency to complete. |
| 00:28:32 | Do we have to do it in red? |
| 00:28:33 | No, in blue. |
| 00:28:37 | No, not this. It's this. |
| 00:28:41 | Yes, (already). |
| 00:28:42 | And now how should I do? |
| 00:28:44 | Doesn't matter, doesn't matter. |
| 00:28:48 | Thanks. |
| 00:28:51 | And 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... |
| 00:29:03 | Tell me. |
| 00:29:04 | Is it right like this? |
| 00:29:05 | Try 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 |
| 00:29:11 | it's wrong. |
| 00:29:16 | Have you done yours? |
| 00:29:17 | Yes. |
| 00:29:23 | But it gives you a straw of 880 centimeters? |
| 00:29:26 | Eh, yes. |
| 00:29:27 | But was it that long? Was- which one was it? This here? |
| 00:29:31 | Yes. |
| 00:29:32 | No. |
| 00:29:33 | No, this. |
| 00:29:34 | Here inside. |
| 00:29:38 | This one here? But take it out. |
| 00:29:42 | But also if you take it out a bit, this one here is 880 centimeters? |
| 00:29:46 | There's something that doesn't work. |
| 00:29:49 | We can try to do the calculations again. Try to do this piece here. It equals 48. |
| 00:29:53 | But take care that- Let's look at your- what did you do before, explain it to me in words. Marisa, what did you do? |
| 00:30:00 | Okay, thirty-eight point five squared plus 25 squared (inaudible) to find this diagonal here. |
| 00:30:06 | M-hm, yes. |
| 00:30:09 | Now it's more. |
| 00:30:12 | But what did Pythagoras say? Having a rectangular triangle? |
| 00:30:20 | Right angle side, right angle side (inaudible). |
| 00:30:26 | Eh- in that triangle which is the right angle side? Which is the hypotenuse? |
| 00:30:29 | The hypotenuse is this one. |
| 00:30:30 | Eh, yes. |
| 00:30:31 | And these are the other two sides. |
| 00:30:33 | But this triangle here. |
| 00:30:34 | No. This is a side and this is the other. |
| 00:30:37 | This is a side, that is a side, and this is the hypotenuse. So if you have found this, you already have calculated it. |
| 00:30:44 | Yes. |
| 00:30:45 | It's 55. |
| 00:30:46 | Squared plus (inaudible) 16 squared and then extracting the root of everything. |
| 00:30:52 | Do you agree? |
| 00:30:53 | M-hm, yes. |
| 00:30:54 | We have found it. |
| 00:30:55 | Is it right like this? |
| 00:30:59 | I get an expression. |
| 00:31:01 | And where can I simplify again? |
| 00:31:03 | No. [ Laughter ] |
| 00:31:04 | [ Laughter ] |
| 00:31:06 | Is it right or can you simplify something else? Are you done with the transparency? |
| 00:31:11 | Yes. |
| 00:31:13 | This goes ahead, two remains. Have you already completed the transparency? |
| 00:31:19 | (inaudible), ah, no, the transparency not yet. |
| 00:31:20 | The transparency. You have to complete with- are you doing number three or number four? |
| 00:31:23 | Three. |
| 00:31:24 | Three. Complete it with your solution. If you have finished with the box, put it back over there, box, scissors. What? |
| 00:31:32 | (inaudible) |
| 00:31:33 | No, put it over there- put it over there. |
| 00:32:26 | Found? |
| 00:32:27 | Eh, yes, I think so. |
| 00:32:29 | And some of you bring back the box, please. In the meanwhile, Andrea- |
| 00:32:33 | Madam, may I ask you (inaudible). |
| 00:32:35 | In the meanwhile put them away. Later I'll give them to you. Go on, put it inside, Claudio (inaudible). |
| 00:32:49 | Wait. |
| 00:32:54 | Assio, it's your pen. |
| 00:32:55 | No, mine. |
| 00:32:57 | No, why (inaudible). |
| 00:33:07 | Finished? Now try to think a while, to see if you can find a faster way to do- to find |
| 00:33:17 | the same result. Instead of doing two calculations every time, do one calculation only. |
| 00:33:25 | (inaudible) |
| 00:33:26 | What do I do? Sqaure this plus fourteen point seven squared, root. |
| 00:33:33 | Ah- I can use the properties of powers. |
| 00:33:36 | Be careful, I squar- |
| 00:33:37 | Plus, eh, you can't use the properties of powers. |
| 00:33:41 | Barbara is right. Here is the square root, now the sum, then extract the root, I square. |
| 00:33:50 | But first (inaudible). |
| 00:33:52 | First we extract the root, then we square it. |
| 00:33:56 | Ah, so you can take away the root, it's this. |
| 00:33:59 | Which root can I take away? |
| 00:34:01 | This. |
| 00:34:02 | No. This, because squar- taking away the square root and then squaring it isn't of any use. |
| 00:34:08 | But the (inaudible). |
| 00:34:10 | No, pay attention, eh. Here I extract the root of it all, and then I square it again. |
| 00:34:17 | Dividing by two doesn't make sense. |
| 00:34:20 | Try to write the simplified version. Have you done? |
| 00:34:24 | Here we have to do it in the simplified way or- |
| 00:34:26 | Now what have you found? The measurement corresponds to? Have you measured it? |
| 00:34:28 | Yes. |
| 00:34:29 | Yes. |
| 00:34:30 | Okay. So now write here in your- you are the box number four? Write the solution here on the transparency. |
| 00:34:35 | The result you found. |
| 00:34:37 | The same thing that we wrote down here. |
| 00:34:38 | Yes, exactly. |
| 00:34:39 | Now here you extend the same scheme. |
| 00:34:40 | Yes, and afterwards try to do what the others have done with the other- other boxes. |
| 00:34:45 | Okay. |
| 00:35:10 | And pay attention, extract the root but you have to extract what else? |
| 00:35:12 | Eh- (inaudible) |
| 00:35:14 | I told you. |
| 00:35:15 | Yes. |
| 00:35:31 | Why do you take the box? |
| 00:35:32 | To do the faster thing. |
| 00:35:34 | I 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 |
| 00:35:41 | if you couldn't do it in an easier way. |
| 00:36:10 | If you would have to write a title. What did you apply in solid geometry? |
| 00:36:15 | Boh! I don't know. |
| 00:36:16 | The Pythagorean theorem. |
| 00:36:17 | The Pythagorean theorem. So you can write it... Finished? |
| 00:36:31 | (inaudible) square root direct. |
| 00:36:33 | And that without- eh- without. |
| 00:36:35 | The direct square root, what- what is that? |
| 00:36:37 | What is the direct square root? |
| 00:36:38 | Is the (inaudible) and then you have to square it and immediately extract the root. |
| 00:36:39 | This (inaudible) the last time. |
| 00:36:43 | Ah- |
| 00:36:44 | Do you remember? |
| 00:36:46 | Eh, 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. |
| 00:36:54 | It's enough to do (inaudible). |
| 00:36:55 | Put it a bit (inaudible) below. |
| 00:36:56 | Try to write it. |
| 00:37:01 | Try to write it. |
| 00:37:02 | Let's do it here, go on. |
| 00:37:03 | What (inaudible)? |
| 00:37:04 | What you just told me now. |
| 00:37:06 | Now here, I'm going to write the same (inaudible)? |
| 00:37:09 | Yes, now you are doing more then the rest. In the meantime the others, you have to write the same thing here. |
| 00:37:13 | If not, then- try to do- from now on let's continue all together. |
| 00:37:21 | Okay, stop, interrupt- shh- [ silence ] Andrea. Interrupt the things you are doing. All of you have found |
| 00:37:31 | the length of the straw to be longer then it was in the real box. Now there is- who has already finished calculating? |
| 00:37:38 | All the others who are still calculating, please go back to your seats now, not in groups anymore, and then look forward. |
| 00:37:49 | :00] |
| 00:38:44 | All right, first of all let's complete the title. Not everybody has done that. What have we applied in solid geometry |
| 00:38:53 | that we haven't ever done, Neluca? |
| 00:38:54 | The Pythagorean theorem. |
| 00:38:56 | The Pythagorean theorem. So whoever hasn't written it yet can add it. |
| 00:39:13 | The Pythagorean theorem in solid geometry. Okay, now we will observe the transparency that you have completed. |
| 00:39:22 | It 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, |
| 00:39:31 | which was the most shallow box. To find the longest straw that was the straw... let's call it diagonal. |
| 00:39:46 | And 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 |
| 00:39:55 | of the base's surface and then by using the height I found the diagonal of the box. |
| 00:40:04 | So all of you- look- looking at the transparency, what do you have? First you calculated the diagonal of a surface and then |
| 00:40:14 | by taking this diagonal, you have, with the height of the box, you have found the length of the diagonal of the box. |
| 00:40:21 | All the groups did it this way. For those who have completed the sheets, you can see the various measurements. |
| 00:40:35 | Everybody used the same method: Always the diagonal of the base and then the diagonal of the box. |
| 00:40:46 | Now one of you was finished a bit earlier. He tried to find a single calculation to find the length |
| 00:40:56 | of the diagonal. So that instead of doing one calculation and then doing the second, he has found a single possible calculation to find |
| 00:41:07 | this length. Now let's take the example- so Barbara and Andrea were able to find it. You had box number? |
| 00:41:13 | Eh, one. |
| 00:41:14 | One. |
| 00:41:15 | One or two? |
| 00:41:16 | One. |
| 00:41:17 | Box number one. So their calculation here, first diagonal of the base and then diagonal of the box. |
| 00:41:28 | Well Barbara, would you like to try to come and explain it at the blackboard? Do it at the blackboard. |
| 00:41:39 | Yes. So, first- eh- in practice we put together the two calculations. Do I have to write down the two calculations here? |
| 00:41:47 | Write 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. |
| 00:42:19 | In practice we put together, eh, the first calculation that- that we have- eh- we have put together the two calculations- |
| 00:42:28 | eh- but we simplify because in practice the calculation was like this at the beginning. Only extracting the root and then squaring it, |
| 00:42:39 | isn't of any use. So we simplified, we- we took away the root and we squared it. |
| 00:42:51 | The result? |
| 00:42:52 | The result is the same. |
| 00:43:01 | Good, 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: |
| 00:43:08 | On the second passage, what did we write? Twenty-seven point two squared plus nine point five squared. |
| 00:43:16 | Now instead of writing twenty-seven point two here, they wrote the calculation that allowed me to find twenty-seven |
| 00:43:26 | point two. Instead of writing the number directly, they wrote the calculation that allowed me to find |
| 00:43:33 | twenty-seven point two. Then Barbara correctly said: I do the root and then I square it, I can annul it because |
| 00:43:42 | one is the inverse operation of the other. Therefore the shorter way to find this diagonal is length of the box squared, |
| 00:43:51 | plus width of the box squared, plus the height of the box squared, and at the end extract the root. |
| 00:44:02 | They 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. |
| 00:44:08 | That I first have to find the diagonal of the base, then the diagonal of the box. You can find this method after doing some |
| 00:44:15 | calculations. At the beginning it isn't that evident to find this- this calculation. |
| 00:44:20 | So now, whoever, eh, still has to finish the sheet, complete this sheet with the measurements of the other boxes |
| 00:44:29 | that 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. |
| 00:44:36 | I'll leave the solutions here, who- you can come to correct the solutions as usual. |
| 00:44:43 | But first complete this, without looking, then, eh, check if it's right. |
| 00:45:15 | Yes, eh... Yes, I'm raising it now. |
| 00:45:18 | Thanks. |
| 00:45:19 | First do it without looking. That means first complete the sheet, then... |
| 00:45:25 | The overhead projector doesn't work well. |
| 00:45:27 | But can we? |
| 00:45:29 | Have you finished? |
| 00:45:30 | Yes. |
| 00:45:31 | Checked? Are they correct? |
| 00:45:32 | Yes, yes. |
| 00:46:12 | Have you understood the shorter version? I saw you looking with the face a bit. |
| 00:46:17 | (inaudible) |
| 00:46:19 | M-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? |
| 00:46:25 | You 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. |
| 00:46:31 | Well I didn't under- well I didn't understa- I didn't understand (inaudible) how to arrange (inaudible). |
| 00:46:35 | Practically, what did you do here? This number here you transferred here. |
| 00:46:38 | M-hm, yes. |
| 00:46:41 | Yes. |
| 00:46:42 | Do you agree? This number, you transferred here. On the other hand if you have to transfer this number here, I can transfer the |
| 00:46:48 | calculation that allowed me to find this number. |
| 00:46:52 | And I used that to find this number here. |
| 00:46:53 | Yes. |
| 00:46:55 | Which calculation did I do? I did this, which in reality was this. |
| 00:46:56 | This. |
| 00:46:58 | M-hm, yes. |
| 00:46:59 | Instead of writing this number, I could write this calculation. |
| 00:47:07 | M-hm, yes. |
| 00:47:08 | It'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 ] |
| 00:47:16 | I can write this squared plus this squared. |
| 00:47:23 | And then? |
| 00:47:24 | (inaudible) plus 166, which would be 13 squared. |
| 00:47:31 | (inaudible) okay, I'll do it that way (inaudible). |
| 00:47:33 | M-hm, yes, yes. |
| 00:47:34 | Madam, (inaudible). |
| 00:47:39 | Why? |
| 00:47:40 | (inaudible) |
| 00:47:49 | But 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- |
| 00:47:55 | Yes. |
| 00:47:56 | Was this problem with your calculator? |
| 00:47:57 | (inaudible) |
| 00:48:02 | Eh... |
| 00:48:04 | What am I doing wrong? I should extract the root. |
| 00:48:09 | Ah, it's true (inaudible). |
| 00:48:11 | Now you have to extract the root, yes. (inaudible) the result down here, if you extract the root. |
| 00:48:15 | And the root (inaudible). |
| 00:48:34 | Did you understand well here? Do it on a sheet by side, on a squared sheet of paper. |
| 00:48:43 | Vera, got it all right? |
| 00:48:57 | Boris, got it? |
| 00:48:58 | Yes. |
| 00:48:59 | Cinzia? |
| 00:49:03 | Is it right now? |
| 00:49:04 | Yes. But down here? Yes. |
| 00:49:08 | Two plus eight, then it depends on how you approximate. |
| 00:49:22 | When do I have to finish the lesson? |
| 00:49:49 | Now stop what you are doing, put away everything and we'll continue next time. |