time. Welcome to another webinar session. I am happy to feel that you are here. I cannot see that you are here, but I mean you are here, literally. Anyway, today we got to talk about three-dimensional objects. It's geometrically related. I will try to show you like as many as visuals as possible. It depends on the question. Anyway, and this is our warm-up question. Please check this out. And share your answer or your ideas in the chat box, please. Remember, you don't necessarily solve or find the answer. That's totally fine for us. Even if you give us some ideas, that works too. The question says, rectangular prism was made out of three blocks, each consisting of four cubes here. Each of the blocks below has the same shape as the white block. As you see, we have some blocks with different colors and different shaded parts. Each of those blocks, those three blocks, made of four cubes. Alright, it has been 25 seconds. We got one for A. Okay, someone says A. And then a second one got A. Okay, A got two answers so far. And it has 45 seconds. One person says B. Alright, B got one vote. What about others? It has been one minute, guys. I wonder if we get more answers. You got a third vote for A. All right, guys, let's start. I believe we have only four answers so far. Someone says A, someone says B, and so far. So here, let's just start from B. Let's say the shape should look like this one. Well, as you see, guys, here we have little curve part here. So to be able to fit that, if I fit the object like that, I had to have another box. It's kind of hard to draw it. Robot, you got the point, I hope. OK. We will get another box here, guys, another little box. As you see, it doesn't match with B. If I try to rotate that, in that case, I would get another block. It's here. So there is no way we can make it B. Since no one said C, D, or E, I was going to skip those. But I only need to give you some hint about other shapes, why C or D doesn't make sense. Because they also have those corner piece here and here. But since the question is a little tricky, or at least they expect us to think of different ways, let's just go from those. So let's say you think it is C, guys. If I rotate the shape, just with that little L shape, the last part should be here at the bottom. So I am covering this piece and the kind of base part here. If I just rotate this shape and just fit it here. But if I cover that piece, I am saying behind that little shaded piece is here, white piece. If he says answer should be like that, then look at that little kind of gray shape. We have to have only one, two, three cubes. Because of that, white blocks and other shade blocks would be covering and cutting everything in the half between. Remember, they say rectangular prism was made out of three blocks. Yes, you got to have only three of them. And each of them consists of four cubes. If there is any case that we are getting only three cubes, that's not the right answer. I remember that some of you might think that. Anyway, if you think about A, guys, many of you say A. Let's see if A is going to make any sense. Remember, I still have that little L piece. I kind of rotate this one. I would make not necessarily clockwise or counterclockwise from front to back if I just make it that 90 degree and then rotate another 180 degree. This piece has to be at the bottom. And the shape should be something like that, if I can draw. Besides those blocks, there is going to be one extra block here at the bottom, something like that. So if you rotate this shape only. And if you fit that here, you are going to cover this piece. And still, that gray box has another one piece in the middle here. That means each of those blocks has four cubes. Look, I wish I had some sort of Legos or stuff to be able to show you, but I don't have those. Extra stuff, 3D objects with me. So the question is, does it make sense, guys? Or let me ask this way. If you have any questions about this one, please say yes. I mean, not say yes. Type why in the chat box and put your question as well. Then Soham is going to check that. Soham, I believe we have no questions. I'm going to move on. Let's move on. All right. Thanks, Soham. All right, as you see, we are in lesson 6. And we're going to talk about 3D dimensional shapes today. Just a reminder about 4-step problem solving. Make sure you understand that. Plan how to solve. For this topic specifically, I recommend you to draw the shape and try to visualize whatever shapes we are talking about. Carry out your plan. Make sure it's going to make sense. And look back and check and reflect. You can just plug in the value, whatever you find, to make sure your answer is going to be right. All right. So three-dimensional geometric studies objects in a 3D shape, such as cube, rectangular prism, sphere, cone, and cylinder. Larger objects can be constructed by combining or stacking smaller ones. As you remember from warm-up question, guys, we had those interesting pieces, like different blocks. Anyway, from that one, when you stack smaller cubes, you just make those blocks. This is what they are saying. Spatial reasoning focuses on the weaves or studies of an object after it's rotated, folded, unfolded, cut, stacked, taken apart, et cetera. We're going to talk about spatial reasoning. Many problems can be solved if you understand the three-dimensional weave or representation of a 3D object. Sometimes we've got to talk about how to unfold a cube, guys. While you're telling about the cubes, a net is a three-dimensional figure that can fold up to a form 3D solid, which is the common. We are talking about cube. When a 3D solid is opened and laid flat, the result is a net. We call them a net, those little yellow pictures. A cube can be opened to result in different nets. As you see, guys, please try to imagine in your mind, when you cut that shape of piece of paper and then fold that, you are going to have a shape of cube. Or you can just try different representations. For example, this one is a really interesting one. This one and that one, these are the common ones. I'll just show you the point art. But if you cut the paper with the same measurement, I mean, same ratio, not necessarily the same size, and fold that, you are going to get, you are going to create a cube, form a cube from every single of those nets. If it doesn't make sense, I recommend you try that. Opposite faces or opposite sides of a cube do not share a common vertex. Well, think about it. We kind of try to show here, guys. Opposite faces here at the top and bottom, we just kind of color them orange. Or when you think about right and left, these are also opposite sides with blue color. Or you can think about front or back faces or back sides. These are also opposite sides. As you see, they do not have any common vertex. Vertex is those little sides, as you see. And they are separated by another face in the net exactly. You can see that between those two opposite faces, there are another face. I hope it does make sense because we got to check some questions in terms of those rules. All right. Some of the sides can be easily identified. As you see from the picture, we just label the opposite sides by the same letter. This x and x represents opposite sides when you fold the shape and make the cube. As you can imagine, this one also represents y and y opposite. Let's just say front on back, back side. And those x's, kind of right and left. And these empty ones represent the top and bottom faces of the cube. These sides can be easily identified. Others, by deduction or changing the net in the diagram, x and y pairs are easily identified because there is a face between them, as you see. So by deduction, the two unlabeled sides also must also be opposite face after folding. We also have another little trick for you to use, side pairing, guys. It's a powerful method to change the net from the same cube. It helps identify opposite sides and matching patterns and orientation. Note that you can only rotate 90 degree, but not 180 degree to pair up sides. To pair up sides means, guys, basically, think about, we literally color them for you with red, and with same color, basically. Because when you fold that shape, you are going to see that these are going to intersect. These things are going to touch, you know. Also here, as you see, another face. Green color. When you fold the shape, these little top piece and another piece next to that, next to the red one, also touch, OK? And another one here is, let's just make it blue. They are going to match as well, guys, when you fold the shape, OK? As you see, after you start with the green, you just figure out other colors. If you use the coloring method, it's going to help you to see which two faces are going to match. It's going to be important. I mean, while we are giving different type of questions about the numbers of the cube and look at the face of different object and so forth. And here, as you see, those greens are going to match, OK? And here, also, you can just think about another side or inverse. These red sides still matches. With the same logic, these faces are going to also match, guys. They also give us others. Any questions about this one? Or we are good? OK, so you're saying nothing. I believe we are good with that. I'm going to move on. All right. All right, guys. That means here is your first challenge, the question from 1998 and number 9. Let's see how many of our friends think which one is the answer or which ones are going to get some votes. Let's start the timer. In the diagrams below, house X is shown four times and house Y is shown just one. Each picture shows the house Y. As you see, four of those pictures represent only one house, the same house, from different angles, it's obvious. And only one of them doesn't match. They are saying which one is not matching. That was the question. By the way, it is 1 minute and 10 seconds. Here, I can say that, guys. You are trying to imagine that, let's say, you are in front of the house, and you just keep moving around, trying to see every different side of those walls, you know? The one person who says B. Who says B? OK. What about others? I shouldn't have only one person, guys. All right, guys, let's start. I believe there is no more answers yet. Let's see, guys. If you start from answer choice A, OK. And this is the first shape I am going to walk. I'm going to start walking around. So from the front face, as you see there, there is a door. I just go to this side from right. And I am going to rotate the shape. I am going to walk around the house or the shape, whatever. So here, if you keep walking, guys, the next shape should be, look at those two windows. We have windows, supposed to go on the left if I just keep walking. And I am going to have some sort of another face, not necessarily the door, because door is always the front face. They should give us different type of windows or door or whatever. So if I just keep moving, only D represents after I keep moving, because remember, after you just pass this part, and you got to go this corner, and you will see both windows. And that's the back window, I guess. After starting from shape, the first shape represents by A. You should see the shape represents by D. There is no logical other explanation. So after that, if you just move a little bit from another corner, as you see, we have one of the little window at the back. We have only two shapes with the little window. And other side should represent by E if you just keep moving around. And if I just keep moving around until I just go to that corner, I should see one of the little, not little actually, one of the big window. Plus, I should see the door, because we got to go back, remember, by seeing four different corners, basically. So do we have any shape represents that kind of huge window with the door? Yes, we have this one. And lastly, you should see this part of the view of the house. So then, as you see, no matter what you do, you cannot see the picture B. Also, if you think about those shapes, and you can think about your previous knowledge about mirror image, mirror image represents reflection. Remember, when we are talking about reflection, guys, those reflected pictures cannot represent the same shape. One of them is supposed to be original, and another one is supposed to be either mirror image or different shape. If you rotate, yes, you can get the same thing. But for those cases, since we are talking about four sides are different, the reflection cannot represent the same thing. OK? In that case, we have only one option, which is B. Any question about that? If there is nothing, I'm going to move. Can I move, guys? Type something. I am not giving up on you. I believe I am going to move on that. It seems that no one is in. OK. You would ask me the questions anyway if you had any questions. This is what I assume. All right, number two, we get 2,000. The question from 2,000 and number 19. Let's see what you're going to think about that one. Iron made a steel using identical wooden cubes. One diagram shows the view from above the steel, here as you see, and the other view from one of the sides. Remember, it doesn't say from back, side, right, or left. We don't know. One of them. However, it's not known from which side the side beam was taken, that one. What is the largest number of cubes that iron could have used? We're going to focus largest number, remember? And we're going to count the numbers. Let's start the timer. It has been one minute guys, so I'm going to give you a lot more time guys. We got one for C. Okay, we got one for C. All right, guys. It has been 1 minute and 40 seconds. Let me just give you a hint, guys. Look. Let's see. This is the, this picture has been taken from front and this is the top. I'm going to start from front view. Then I'm going to continue with the right side. Let me just actually write it down so you get that. Front, right, look at my awesome handwriting, back and left view. We just going to go around from that object. Let's say this picture represents front. I'm going to focus here only. So as you see, we have two blocks seen and we have one, two, three top views. If I exactly put here this one, it matches. Every single of those one, two, three blocks represents by two. Every single of only this one, by the way, from top view represents by one, two, three, four, as you see. Every single of these two block represents by only one, you see. And every single of those three blocks represents one, two, three. Well, we cannot just add two plus four plus one plus three. Remember, we have two times one, two, three, plus four times one only, plus one times one, two, and three times three equals what? So we have six plus four is 10, and 12, and 21. As you see, guys, we got 21. And one friend found even more numbers. And we still have some other answer choices. Right now, kiddos, I want you to focus on right side, back, and left, OK? So we got two votes for B and two total votes for C. All right, guys. Let's see if I rate this one. I'm going to give you more time to work with right, back, and left, please. If you already get that, you can share your ideas with us, guys. If you imagine this is from right side, I'm going to try to match those numbers with right side here. What would be the total number? Remember, when I say right side, I meant this piece, this part. OK, let me keep giving you another hint. Look, from right side, guys, as you see here, this piece and this piece match. We have one, two blocks here. That means this piece represents, and each of those represents by two. And we have one, two, three, four. Each of these represents by four. We have one. Each of those two represents by one. And last one, each of those two represents by three. So let's just multiply them. We have two times three, three of those, plus four times two, plus one times two, plus three times two. Let's see, we have six, as you see. Four times two is eight. Give us total 14, plus two, 16, plus six, 22, guys, as you see. What else we can find? And who would like to try back and give me some numbers? Remember, in that case, I am going to say that this piece is two, you know, because I'm going to start from here. Then this piece is four, and this piece is one, and that piece is three. I am going to continue multiplying and adding them. Two times three is here, plus four times two, one times one, three times two. And it gives us what? Let's see. You get six, eight, you add them, you get 14, 15, plus nine, 24, guys. We can try with left as well. I don't think anyone is saying anything. Otherwise, Soham would let me know. OK, from that two, that two starts here, guys, if you think it's the left side. And this one, four goes here. We are on the one block here, and we get three blocks here. So two times two, we have plus four times two, plus one times two. I am matching those, you know. And three times three. Three with multiplying with those three blocks gives us what? We get four, plus eight, 12. We get 14, and 23, I believe. Yeah. So as you see, guys, some parent says 23, and some parent says 24. Answer was 24, because this is what's the largest number of cubes she could have used. Any questions about this one? Or not really? So if there is nothing, I'm going to move on. No questions. All right, thanks so much, sir. Guys, if you say you got it, then you got it. I mean, I cannot read your mind. I believe you are good. OK, number three, from year 2002, and question number 20 is your next challenge. A cube was made out of the clam shown in picture. This is the net of the cube, OK? The greatest sum of the dots on three sides with a common vertex. Remember, we're going to find three sides with a common vertex. Then we add the little dots on the three faces to be able to find the greatest sum. Let's see. All right. And someone immediately answers A. Oh, OK. And another person answers D. Answer D, all right. We have some really fast friends. But guys, if I say you, well, well, well, answer's supposed to be none of them. Are you going to do anything? one person says c all right i see a c and d has given by one friends what about others okay we got one more vote for d all right so d has two so far c and a has one vote so far i believe a second vote for c all right she got two right now let's see i'm gonna find c or d which one makes more sense we got a second vote for a okay another vote for a so one more person has to respond so they're the deciding vote all right yeah the last one here probably just says that person is gonna make the call here okay yeah either 15 13 or 12 maybe that person's gonna say 14 who knows okay guys by the way you can just give us your reasoning or you can just explain a little bit on the chat chat box so you can get the last vote or a couple of more votes who knows or maybe some friends would like to change their answer that's the possibility too so we got one for that one person who answered d change their answer to a oh so right now we get a is three and b has one okay i'll just put minus one by anyway all right guys so so far a has three c has two and d has one answer and one answer let's see what's gonna happen guys they say a cube we are looking for the greatest number of dots on three side with a common vertex we are going to find those three sides with the common vertex so can i think about my for example here right if i circle those can i say that those three sides are supposed to have common only one vertex here if i add those numbers i should get eight but as you see guys there is no answer says eight what about another one let's see we just have this one i'm discussing so we get six plus five plus one is six plus six which is twelve another option they had the same little vertex here and and what if we even talk about those three they they come vertex here we get six plus five plus two we'll do 13 guys you can check other options but there is no way guys you can find number greater than 14 i wonder how those friends find 15 because let's just find the biggest sides and biggest numbers six plus five plus four you are seeing 15 but that four is here guys there is no common vertex here yes the face represents six and face represents four as common vertex here but not with another one we are going to have only one common vertex anyway that's the reason that answer would be 13 guys whoever said 13 they get right i am proud of you i am proud of every single people i am a little bit more proud of those other friends who says 13 any question for number three okay i'm gonna move on then you would type your questions for so long to see anyway but you didn't i assume you are good anyway okay number four from 2019 number 16 question number 16 katie fold the square sheet of paper exactly in half here twice as you see once and twice then cut in the middle twice here and here okay as shown in the diagram how many of the pieces that she obtains are squares look if you have a paper and scissors i would try that you know to be honest then find out the answer because we will explain that one i am going to you know draw the picture hopefully it's not too much mess you know let's start the timer guys It has been 15 seconds. We'll get some answer. OK, guys, we have 47 seconds. I am going to give you a hint, guys. For those questions, you need to work backward, OK? I'm going to give you a little hint, and you will go from there. The last shape was this one. And yes, they just cut the pieces. Let's just use the same logic with just little dot lines. Shows that we cut the pieces. But when you unfold, instead of folding, you are going to have the deflection gaze, like a mirror image, you know, OK? So this is what we get so far. it is 1 minute and 40 seconds okay I just gave you hints hopefully someone can do the rest. Work backward guys. We got two votes for B. Okay. One for E and actually we got three for B. Oh we get three for B. Okay. Look at that. We get three for B and one for E you said. Am I right? They changed the E person changed their answer to C. Oh okay. And E get one minus one then. What about others? So far, we have four participants says B should be 75% and C with 25%. What about others? Here they fold the square sheet of paper exactly in half, twice, as you see, folded like that, then folded like that, then cut in the middle twice here, middle, two times, as you see. How many of the pieces that she obtains are squares? We're looking for square shapes, guys. OK, guys, I'm going to start explaining, guys. So as you see from here, we get one little piece here and one another one is here. I cannot put the middle one because it's not the cut part. You kind of just unfold. And I'm going to do the same thing here. From right to left, just use the reflection, guys. You will get something like that. And these little dot lines are going to extend. OK, so right now, let's use different color to identify. Can you see those little tiny pieces of square, guys? We get four of them. And I assume you can see that little piece in the middle. It also represents a square. Sorry, I am not perfect in that drawing stuff. It doesn't make sense. But some of you may say, but, sir, we have two little squares here. They are actually rectangles, guys, as you see, because you just unfold the paper. You do not actually cut these pieces. From here to here, please pay attention to this part. I didn't put any line here in the middle, because these are just the same piece of paper. All right, guys, as you see, those green pieces represents a rectangle. And those kind of purplish pieces represents squares. And we have five squares and four rectangles. Some of friends says answer of the four. Remember, guys, if they ask us to find how many little rectangles we have, your answer would be right. But right now, it's a different case. Any question about number four? I assume that there's nothing. I'm going to move on, OK? Because if you had anything you would ask us, check number five, please. All right guys it has been two minutes. I hope we get some answers with explanations. Let me just give you some hints guys here. This piece and this piece. If you just fold that diagonally and if the rectangular strip is three centimeters here, that height has to be three as well guys. There is no another way to be able to fold perfectly. Okay so we know that height of little those trapezoids is three. What else? Do you think we can find some information from that 27 centimeters? okay so you're saying three people answer d this is for this one okay okay many of our friends says answer should be d but i really wonder guys how do you get d because think about d and c are really close to each other why do you think it's d but not c or so far well someone says that the each of the trapezoids bottom base lengths are nine so the longer base is nine on each trapezoid okay so let's just test how we can get this is you know nine with the same logic guys if this is three this piece is also three am i right this piece is also three this is also three and this is also three am i right so we have one two three four five of those trees basically and let's just make this piece little piece x x x x and x we can say well five times three plus four x is equal to 27 we get 15 plus 4x is 27 you gotta make take out 15 both sides 4x is 12 x is 3 like your classmate says guys if x is 3 the bottom piece of those little uh trapezoids are going to be nine because three plus three plus three is nine days okay so he says that the trapezoid contributes nine to the length of the strip this the rectangles are six and the squares are three okay so then what do we get six times two plus three times three plus nine times four is 57 okay i wish i got how they find those because i really gotta i don't need to show you guys on the shape anyway so this piece guys let's just start from here this piece is six centimeter am i right and you said that and you also find that this piece is nine am i right i just keep going from here then from here then from here as you see because even they fold i do not change the full sides if i am working with those those little pieces and sides okay so we get six here nine here another six here as you see we got x equals three oh i shouldn't explain more and we get another nine here another six is here another three here and since those great trapezoids are the same these piece the left or has to be same as well i mean the concurrent this should be six and this little piece has to be three as well okay when we add them we get six plus nine plus i'm just starting here six plus three plus another six plus nine plus six plus three plus three plus six so what do we get here so six nine gives us 15 guys six three six give us another 15 nine six give us another 15 and this one give us another 15 am i right no this is 6 9 12 my bad is 12 yeah and when you add those 15 30 45 uh 55 and 57 okay they get all of them you should get seven days does it make sense class i wish you could talk you know i mean you could say something then i will feel that i am explaining everything to so how more the screen or whatever so why will they say nothing that means we are good i'm gonna go next one next question okay all right guys all right next question says it's here all right this one is from 2000 and question number 29 think about this is the almost last question it's my little trick okay so the pictures below show the patterns of five cubes these are all represented cubes okay which of them has the property that wherever two sides touch the color along those edges is the same okay of course this doesn't mean that the whole side is one color it doesn't have to be but you need to focus here guys wherever two sides touch the color along those edges is the same we're gonna focus here for this one let me give you a hint guys which one was that okay think about that rule guys we gotta use side pairing it's a powerful method to change the net for the same cube it helps identify opposite sides and matching patterns you gotta find matching sides note that you can only rotate 90 degree here remember those sides matches i just put that with the red color these sides i have to use different color come on okay these sides matches if you just look at it carefully after those greens blues are matched as well these blues are gonna match guys when we move back to question number six what's that one okay remember when you use similar methods you are going to match the sides and those matching sides has to have same color the question is what do you think about that we got one person who says a and one person who says d all right interesting we get a and b what else we get a we get d so far a and d get one vote let's see what's gonna go with what's gonna go with b c or e we have a second vote for a okay a got two votes so far and we get nothing b is so sad c is crying over there e and e is upset a has a third vote all right look at that a is going to be winner i guess what about b c d and e a is a fourth vote yay All right, guys, do we have anyone else? One person who answered A changed to D. Oh, OK, interesting. Minus 1, and D is 1. I'm going to put another 1. All right, so far, A get 3, and D get 2 votes. And then another person switched to D as well. OK, another minus 1, another plus 1. So far, we got 3, and A is only 2. Anyone else want to change or mix or whatever? One of the B person says C. Or get rid of C and say E, anyone, or what? Another A person says D. Basically, what I'm doing is, as soon as people answer, I tell them whether their answer is correct or incorrect. Oh, you're telling them. OK, that's nice. All right, so that means we get A is 0, because we don't like A. And everyone says D. D has infinite votes. Am I right or what? OK, guys, so let's just, it's almost time. Actually, it says time, but I can just give one minute for that. All right, guys, let's just use the rule. There is a reason I show you that slide, guys. Look, the red, actually, the pink would show us better. Guys, when you do side pairing, look at here. That says, whenever two sides touch, the color along those edges is the same. Look at that pink. When those two sides touch, the color along those sides are the same. No, answer cannot be A. With the same logic, look at B here. As you see, it's not possible. So when we look at C from here, well, as you see, we have kind of white part and shaded part. Nope. When you look at E with the same logic here, or here, we can match it here as well. You know, you can just choose different parts as well. They are not matching. They have opposite colors. Well, it give us only reasonable answer was D. Look, no matter which part you try to kind of match, it's going to be showing us same colors. This is what I was trying to tell you. Any questions so far for number six? So I believe everyone changed to D. Am I right? Yeah. All right, guys. I enjoyed our class. I hope you enjoyed it too. And until next time, please take care. And I'm going to see you next time. All right, bye. Bye.

three-dimensional geometry

rectangular prism

spatial reasoning

3D visualization

nets

side pairing

visual puzzles

backward reasoning

paper folding

logical deduction