Welcome to the Math Kangaroo webinar for level five, six. Hope everyone enjoyed a nice Thanksgiving break last Sunday. Thank you for coming back this Sunday. I appreciate that. Glad that you're returning to solve more problems and learn more about math problem solving. Last time was two Sundays ago, we worked on solving some problems with two-dimensional geometry. This time we're gonna do some three-dimensional, 3D geometry and some spatial and visual thinking. So not just solving, you know, volume problems in geometry, but also really analyzing what shapes look like from different directions and how we can change them and turn them in different things in three dimensions. Okay? We're still helping students join. So be a little patient as Jacob, your TA, and I click around and try to make that happen. How many bricks were taken out from the wall in the picture? So it's a brick wall and some bricks are removed. I do have quite a few polls today, and this is a question where I have a poll. So I'll go ahead and launch it right away. As you're working on this problem, you can put your answer into the poll. You can also put your answers into the chat if you have questions. You can ask our assistant Jacob or you can ask me in the chat. If you've just joined in the last few seconds, welcome to our Math Kangaroo Level 5-6. We have a warm-up problem. You can read that on your screen. All right so this problem I don't think I don't think we're trying to trick you with this problem it's pretty much can you figure out how many are missing by counting them up or by adding so you can kind of fill this back in and you can see that there's one missing from this row there are three missing in this row this row we have one two three it's a little harder to tell four five six seven and then we knew that there are three here eight nine ten so there are ten missing in this big row one then this is going to be three plus two is five this is back to one two three four in the bottom row and then three in the row above that so by some very careful counting you can see that we have one plus three plus five plus ten plus three plus four and that should equal 26 blocks you can correct me if I misadded that but I think I've double checked it right so if we end our poll we can share the results and most of you did a really good job two-thirds of you have said it's 26 the others might have done a little bit of counting off a little bit sometimes it's hard to figure out how many are gone from these little spaces here but you can clearly see one and one two three and then if you take two more that's five so this is five plus another one two three four five makes ten by kind of comparing to the row above and below you can be able to see how many are missing in each So we're moving right along in our lessons. This is our eighth webinar. Next week, we have one with a little hands-on experimentation. And then in our final webinar, we start combining techniques. So it's more complicated questions that use several different pieces of these topics together. So that's kind of a fun lesson where we see if we can master everything and select the correct tools and use them at the right time. So what are 3D shapes? In geometry, a three-dimensional shape can be defined as a solid figure or an object that has length, width, and height. Unlike the flat two-dimensional shapes, these are gonna have a thickness or a depth, okay? So you could think of it kind of as the difference between a piece of paper and a book. Although obviously a piece of paper must have some depth. It's just very, very small. Otherwise, when we stack them, we wouldn't get the stack of papers on your desk, right? 3D shapes have specific properties such as the number of edges, faces, and vertices. Hopefully you've heard the word vertex and vertices is the plural. That means the corners. 3D shapes are solid and 2D shapes are flat. And sometimes you can create 3D shapes by turning, rotating, or sliding around a 2D shape. So you might be able to see flat 2D shapes in the 3D shapes. That can be fun. So here are several examples of our three-dimensional shapes. You should be very familiar with a cube, a rectangular prism. That could be like your shoe box at home is a rectangular prism, or your refrigerator is a rectangular prism. A triangular prism is a triangle on top and bottom and rectangles on the sides. A pyramid is any shape with a point on one end, a vertex on one end, and the base of the pyramid determines what we call it. So if it has a triangle on the bottom, we'll call it a triangle pyramid. If it has an octagon, like a stop sign on the bottom, we'll call it an octagonal pyramid. A tetrahedron, a cylinder is kind of like a prism, but the top and the bottom are circles. The cone is kind of like a pyramid, but the bottom is a circle. And then the sphere is what happens if you had a circle and you spun it all around and turned it into a solid. There are formulas for finding the surface area. That's the sum of all of the surfaces and the volume. I'm not gonna have you memorize these, but these are things that by fifth and sixth grade, you should be doing some of these things in school. So some of this should look kind of familiar to you. You can always come back to the recording if you wanna write some of these down, or you can look these up in your math books. I'm sure you'll find the answers there. So in our math kangaroo problems, our number one step is always to read the problem statement sometimes more than once to understand everything it's asking. So make sure when you see these vocabulary words, you know what they mean. A face is one of the flat surfaces on a three-dimensional shape. The vertex is, I said, the corner. The edges is the line segment where faces join up with each other, kind of like the hinge on a laptop or the spine of a book. The net is if you were to open up a three-dimensional shape and turn it into a flat, and then you could fold the net back up in order to make it into your three-dimensional shape. Surface area, volume, you probably know. And base, like I said, is either the base of one of those prisms or a base of a triangle. It's gonna be kind of what you would call the bottom side. How many shortest distances along the edges of the cube are there that connect vertex A with the opposite vertex B? So we have to walk along the edges. You can almost pretend that you're a tiny little bug trying to creep around the square and get from A to B, and you can only walk on the edges, not on the middle of the faces. This question also asks about the shortest distance, so you do not want to go the long way all around this cube, you want to go as short a distance as possible. Here's the poll that goes with this question, and remember when you answer a poll, it is anonymous. So there's no risk of any embarrassment if you get it wrong. Don't worry about it, just always try your best. Would anybody else like to put the response in the poll? Remember, I know I repeat this every class, but the math kangaroo does not penalize for an incorrect guess. But leaving a blank is a definite 0. So the odds, if you guess, are better than the odds if you leave it blank. I will share the results. We have some students saying 3, and half of you saying 6. So let's take a look and see if it's 3 or if it's 6. So I think we can try to determine how many sides you need to travel to get from A to B, how many edges in the shortest distance possible. So you can go up, across, and that's the easiest one for me to see. Again, you're an individual. So I can see that. So it looks like I need to travel along three edges. I can never get there if I travel along only two edges. And if I try any other path, it's going to not be a shortest path. So now, if from point A I go up, how many ways can I use three edges? I can go up and to the right like I drew in green, or I can go up and along these like I drew in blue. That's also a three-edge path. So if I go up, there are exactly two paths. If I go to the right from A, I can go either along this path, or I can go up. And then I have to continue closing up the gap to B. So there are also two paths if I choose to go to the right. If I choose to go along this back diagonal, the bottom corner, then I have the choice of going up or going across when I get to the back corner. But then I have to finish coming into B. So there are also two choices if I travel in that direction along the floor or whatever you would call that direction. So it is going to be 2 plus 2 plus 2 is there are six shortest paths. And I'm going to use the symmetry of the cube as well. All the sides are the same length. So because it's a cube, all the edges are the same. And it doesn't matter if I had gone up to the right or back for my first step. All right, Michael paints the following solids made out of identical cubes. Their bases are made of eight cubes. Which solid needs the most paint? So remember, we're only painting the outsides. We're not painting inside. We're not separating these cubes. We're painting outsides. There is a pole. But I'm going to wait for the pole because it might be easier for some of you to see it as this larger screen than in the pole. There's an interesting question. A student is asking if they are painted on the bottom surfaces as well. And they might be. It doesn't say that they are not. But they all have bases of eight cubes. So that's not going to make any difference between the different options. All right, some students want the poll. These figures do appear in the poll, so hopefully you can see them. I apologize, it looks like I did not update the answers. Just assume it's A through E on the answers. So just the first one would be A, then B, C, D, and E. All right, thank you. We have quite a, most of you answered the poll, so I'll stop there and share the results. Again, I apologize. Normally I fix these. The answer is A, and most of you have that. So let's take a look. I'm glad that you are doing it correctly. So one way to do it is to count how many outside faces there are for each. So on the bottom, they all have eight. So from the bottom, they're all eight. Don't really need to think about it since it's all the same, but. If I was looking from this right side, they also all have eight. Every single one has eight from the right. So I can look at this and there's one, two, three, four, five, six, seven, eight, if I count them that way, right? And also from the other side, there's always eight. So I don't have to think about that as well. That's always eight. If I look from the top of all of them, I'm gonna see one, two, three, four, five, six, seven, eight. Even on this one, one, two, three, four, five, six, seven, eight, so there's always eight from the top. Top, bottom, left, or right. Where there is a difference is if I'm looking from the front or I'm looking from the back. Okay, so I need to look from the front and from the back. Gonna change colors a little bit. From the front on this one, I can see one, two, three, four, five. On this one, I see one, two, three, one, two, three, four, one, two, three, four, one, two, three, four. If I look from the back, from the back over here, I'm gonna have four, but I have another one over here. This is the fifth. From the back here, I have three. From the back here, I have one, two, three, and then this is the fourth. Same idea for this one, and of course, there's gonna be four from the back here. So where we see a difference is we see a difference, see which color, I see a difference that I have five and five. This is the least because I have three from the front and three from the back, and then these are all the same. But they're asking which has the most, and since this one has five, the correct answer is A is the most. So most of you did it correctly in that poll. All right. One corner of a cube was cut off. Which of the figures below represents the net of the cube after unfolding? Sorry, for some of these questions, I could not put them in a poll because I can't really put those answer choices into the poll. It becomes like that last one, I can't put more than one figure. If I put the figure of the cube, I wouldn't be able to put these answer choices in there. Just put your answer into the chat, put your questions into the chat, Jacob and I will be happy to help you. Remember, the net is if this is a hollow cube and you opened up all these sides, it's hollow inside so you could flatten it like if you folded it out of paper. I'm getting some great responses in the chat. So thank you and thanks for understanding. I can't pull every question. One of the ways I was eliminating possibilities very quickly here is I looked at this and I said, three faces are missing corners. Right, because there's this face, this face, and this face that are missing corners. So when I look at this one, only two faces are missing corners, so this is a bad choice. I have only two of the faces missing corners, Now I'm down to choosing between D and E, and I love it when I can eliminate, can make it faster. When you fold the neck together, the three faces that are missing corners have to join each other. If I'm doing this, I can see these faces would be nice together, but this one is gonna be on the opposite side of the cube. So these three missing pieces don't join each other. So this one's out. If I look at this one, these two are gonna join, and then this one would come and make the third join, so the correct answer is E. If you have difficulty visualizing these problems, the best thing to do is to practice with some nets and to practice folding them. Okay, so you could cut these out and you could try to fold them up. It would be a really great option. Okay? Each of the figures, A to E shown below, is made of exactly five blocks. Which of them can you not make if you start with the figure on the right and are allowed to move only one block? So remember, reading is always really important. You can move only one block, and it says, which can you not make? There is no poll put your answers in the chat to Jacob. I'm gonna step away from my computer for just a minute I'll be right back Yeah, I'm seeing some really great answers. Hopefully, Jacob was able to help encourage you. So one of the things you'll notice is there's this kind of core that looks like this. And most of these figures have that same core of three. You can still see that core right here of three. Here's that core of three, right? It's actually a core of one, two, three, four. So pardon me. It's a core of four. Everyone. Sorry, got a little discombobulated everybody. It happens. So you'll see that core of four. You'll see that core of four over there. You can see that one, two, three, four core here. You can see that four core here. So once you have the four core, then if you move this block, you're still moving just one block. So in this one, we moved it from the side to the top. Here we moved it along over here. We moved it into this space here. But C does not have this core of one, two, three, four. We've actually had to move two blocks to get C. Did you want to lead this one, Jacob? Yeah, sure. So Morton wants to put the figure shown on the right into a regular box. What will the dimension of the smallest box you can use? I'll give you guys some time to think about this. And this, imagine the shape is rigid, it's not moving. It has to stay the way it is. All right, Jacob, it looks like they're ready for the poll. I have students asking for it. Okay. Thank you for joining us today. It looks like most students have answered the poll, Jacob. Do you want to share the results and help them through the problem? Yeah. Okay. So it seems that most of us have answered three by four by five, which is good. So to figure out the dimension of the smallest box, we kind of want to look. So remember, like a box, it looks like this. So remember it has a length and a width and a height. So we want to find the length, the width and height for the box. So to find the length, you want to find like the amount of squares from here to here, we can see that we have this square here. So this square, we also have. So this is like one. We also have this square here. There's another one. Now we can go up here, but that still counts as the second, the square below it. So it doesn't really matter for the length of the box. Then we have this square here. So that's another length. We have this square here. That's a fourth square for the length. And then we have this square here. So that's a fifth square for length. So in total, we have five. Does it mention the length of the small box should be five. So now we want to find what the width would be. So the width would be here. Notice that we have this square here. We have this square here, and then we have these two squares here. So the width should be one plus one plus two, which is four. And then, so we know that the length is five and we know the width is four. Now we want to find the height, but the height should be. So we have one block here. So right here, we have another block that's added to this, to the bottom layer. So that's right here. And then we have this set of three blocks at the top. So in total, we have the height should be three. So the height here should be three. So the dimension of the box should be three by four by five. So the answer should be C. Thank you. I think the tricky part here is to remember you can't rearrange the shape in any way. It has to stay stuck this way. Some students try to rearrange the blocks to make a smaller box, but you cannot do that. Okay. Did you want to do this number 14 or was. Oh yeah, sure. I didn't realize there were two number 14. Yeah, I can do that one as well. So yeah. So by what amount will the surface area of the rectangular box shown in the picture decrease if the rectangular section is removed as shown? I'll give you guys some time to think about the zone as well. Yeah, I think the trick here is to remember you can't rearrange the shape in any way. It has to stay stuck this way. Alright, I think it's a good time to launch the poll, it should have all the prompts and everything, so yeah. Okay, so hopefully everyone has put an answer for the poll. Yeah, so I think most people have put in 54. Another answer that most people put in was 126, which I think I know how people got that, so I'll explain that at the end. So, we want to figure out by the amount that the surface area of the rectangle block decreased in surface area. So, notice that this block here has the exact same surface area as the block at the top. And same thing for here. This block here has the same surface area as this rectangle here. So, other than that, the surface area doesn't really change, so we just need to figure out the area of this here and this rectangle here. So, notice that the side length is 9 and 3. So, it's a 3 by 9 rectangle. So, we just want to find the area of a 3 by 9 rectangle and then multiply that by 2 since we have 2 rectangles on each side. So, to do that, we have 2 times 3 times 9, which is 54. So, that should be our answer. Some people got 126, but that's actually the result when you add all the faces that are dotted out or changed. But remember that you still will obtain this rectangle down here, so you have to actually add that back. So, yeah, you end up getting 54. A really useful technique is to kind of be able to transpose or slide a surface from its beginning position to its ending position. And I think that's what Jacob is trying to get you to see, is that this flat section, I'm calling it steps, the flat section on the lower part of the step, that would have the same area as the original. It just kind of like dropped down instead of being up at the top. So, you can think of it as just having moved up and down or back and forth. And that's a really useful technique so that you can see which things are the same when you are comparing a shape that's had a change. Okay. The 3x3x3 cube in the picture is made of 27 small cubes. And you know that because you know how to find the volumes, right? You have to just multiply 3 times 3 times 3. How many small cubes do you have to take away to see the picture on the right as a result when looking from right, from above, and from the front? So, you might call this the front. You might call this the right. It's up to you. And you might call this above. But any sides that are kind of like that. This problem doesn't specifically say it, but it says, how many do you have to take away? So assume it's the minimum number you have to take away. I think most of you answered the poll now. I'm going to end it here and share the results with you. And what you'll see is that we've had responses for every answer. I'm not going to tell you which one is correct, but you can see that there's a really good mix here. So this problem requires us to explore it a bit more. And I really enjoy problems like that because what it means is there's a lot we can learn, right? So we're not all getting it right away. And the whole point of coming to a webinar like this is to learn something new. So we're going to get that experience right now. OK, so remember I had said that this direction we're going to call looking at it from the front. If I look at it from the front and I want to see nothing on the top, I'm going to have to remove these three blocks from the top in order to see just this shape, right? That will look like it because those three will be missing. So far from the front, I've had to remove three. If I'm looking at it from the right, I also want to see that shape, which means I'm going to have to remove this block in the front. The block in the center is already missing, but I have to remove that one. So there'll be two blocks that I have to remove. Now, if I take a look at this from the top, I'm still going to see, even though these blocks have been removed, I don't know that I can actually draw that myself. So even though these blocks have been removed, we can see the blocks that are in the second layer. So if you look at it from the top, you will see the second layer. So it would still look as if it still has everything from the top. We need to remove everything down one of these columns. It doesn't really matter which side we do this on. So we just do it on a side we can see. You can see that if I remove this one and this one, then from the top, as I look down, I would finally be able to remove that block and see the ground. So from the top, I still have to remove two more blocks. It doesn't really matter because of the symmetry because I've removed all of those top pieces. It doesn't matter which way I go down, but it's still two. So 3 plus 2 plus 2 equals 7. I would have to remove seven blocks in order to be able to see the missing piece shape from all three directions. What does the object in the picture look like when viewed from above? I'll give you a hint. Pay attention to whether it's black or gray. There is no poll because I couldn't put the pictures as answer choices so please put your answer into the chat. All right, I'm going to start to go through this. There are a couple of ways to do this. We're going to do some eliminating options. We're going to do some looking at things that are consistent or inconsistent. We are, remember, looking from above. And we can see that in the center hexagon, we should have a black, a gray, a black, a gray, a black, a gray. And all of the answer choices meet that requirement. So these pieces are not really going to help us make the determination as to whether the answer is A, B, C, D, or E. We can look at the bottom hexagon. And if we look at it from the top, this is going to be our clockwise direction. I'm just going to call that clockwise so that you can tell which one I'm doing. So remember, you're looking from above. So clockwise, I should go black. I should go black, black, gray, black, gray, gray. So I can look and I can see black, black, gray, black, So B, black, black, gray, black, gray, gray. The only one that doesn't meet that clockwise is black, black, gray, gray. So E, I can cross off the list right there. So E is gone. All right, let's look at these diagonal pieces. From one of these, I'm going to have two blacks. So I have to look and see from, it's going to be, let's see if I can do, let's see how I can explain it in words. Visual questions get a little difficult. So we have to have black, black, gray, gray, black, gray, going around this way. And we have to start in the correct place. So how about if I start where I have the two blacks, and so I'd be saying black, gray, gray, black, gray. So let's see that. So here's the two blacks. And I have black, gray, black, gray, gray, black, gray, gray. And then I should have black, black, black, gray, gray, black. This should be a gray. Black, gray, gray, black. This should be gray. So this one is out because this one should be gray, and it's not. If I look at number two, if I start here, I do have the black. Before I have the black, black. So I have black, and then I should have gray. I should have gray. I should have black. I should have gray. And I should have black. So this one matches the inner hexagon, the inner hexagon, the outer hexagon, and the diagonals. So the correct answer is B. And you can try the others to see if they match, but you'll find a mistake in each one of those. So I got lucky, and I tested. By the second one, I had one that matched. We are officially at time is up. I think I will just read this one so you can see this problem. Alice forms four identical numbered cubes using the net shown. She then glues them together to form a 2 by 2 by 1 block as shown. Only faces with identical numbers are glued together. Alice then finds the total of all the numbers on the outside surfaces of the block. What is the largest total that Alice can get? So for those of you who might want to jot this down, you can do that. I'm going to move ahead and do the wrap-up. If you would like to stay late, we can discuss this problem. I don't know if Jacob can stay late. I know he had wanted to go over this problem with you. But I don't want to force you to stay late, so let's do our wrap-up. The three-dimensional shapes are usually made up of some two-dimensional shapes put together, unless you get something like the sphere and then you're rotating it, right? Our two-dimensional shapes are flat, but our three-dimensional shapes, we are adding depth. Make sure you know about faces, edges, and vertices. We had that cube where we had to walk along the edges. We might ask you about the numbers of vertices. You should get very comfortable working with nets, with surface areas, with volumes. So make sure you study some of those surface area and volume formulas for at least the very basic shapes. So you notice we had some areas of rectangles today, the problem Jacob did with you where we cut a piece away. You needed to know some of the surface area problems. And then also conceptually, we had the one that I was doing where we were looking at it from the different sides doing the painting. So you didn't have to actually calculate, but you were being able to count and look at conceptually, how did the surface areas vary? Don't forget the best way to practice is to actually go ahead to your Math Kangaroo account, find some practice contests. If you have signed up to take the contest, you have discount codes in there. You have discount codes also in your registration for this webinar series. All of the webinars that we have had so far are available in your registration. If you will log into your account, you go to where you said, I registered for my webinar, what is my courses? There is a box in that course that says content. In the content box, you can find the link for the videos for webinars one through seven. And in a day or two, you will find the link for this one, number eight as well. If you do need to leave now, thank you for joining us. And I'll see you again next week. If you want to stay and do this last problem, you're welcome to do so. I'm gonna put it back up. Jacob, do you have time to stay and help them with this one? Yeah, so Alice forms four identical numbered cubes using the net shown. She then glues them together to form a two by two by one block as shown. Only faces with identical numbers are glued together and Alice finds then the total of all the numbers on the outside of the block. What is the largest total that Alice can get? I guess I'll give you guys some time to figure out this. Okay, so I think it's kind of explained. So, notice that each, the numbers are going to be shown on the outside. There's going to be, for each cube, there's four squares that's going to be shown on the outside. Then you're going to have like two squares on the inside, but these two squares are going to be adjacent squares on the cube. So, they're going to be like sharing an edge. So, we want to find the largest sum that Allison get. So, in other words, that means that we want to find like the smallest sum that Allison get in the, between like two faces of the cube that are like adjacent. So, first we want, we want to find the smallest sum of two adjacent faces. So, first we want to see if, let's say one and two work, but notice that one and two, they're here and here, but one and two, they both share, they're not going to share a face, right? Because the cube is going to look like something like this. So, you're going to have like a five here and then you're going to have like a one here, but then underneath the five, you're going to have two. So, one and two are actually going to be on opposite sides. So, you can't ever have them be adjacent, but for one and three, you can, right? Because three will share an edge with everything else except two. So, one and three are adjacent and that's actually going to be our smallest possible sum that we can get between two adjacent faces. So, at this point, we want to find the largest total that Allison gets. So, the sum of the four faces on the outside of the box, so like here, here, and then you have like one here, and then underneath, those squares will contain the numbers two, four, five, and six, right? And then these sum to 17. So, 17 times four, because we have four cubes, is 68. So, the answer should be B. This is a problem that could almost be in our last week of lessons because it combines the spatial reasoning, it combines using a net for making a 3D shape, but it also combines some of the logical thinking, mathematical logic to hide the small numbers, but it has to be adjacent numbers. So, it combines a couple of different tools. Thank you for explaining to everybody, Jacob. I hope you've enjoyed the lesson and I'll see everyone next week. Bye.

Math Kangaroo

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three-dimensional geometry

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3D shapes

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